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Abstract. A survivorship curve is shaped by the differential survivability of the organisms within the population, and a change in a survivorship curve with a stressor reflects the differential response of the organisms to the stressor. Quantifying this linkage in a simple, rigorous way is valuable for characterizing the response of populations to stressors and ultimately for understanding the evolutionary selection of individuals exposed to stressors. To quantify this stressor-individual-population linkage with as few parameters as possible, I present a simple mechanistic model describing organism survival in terms of age-dependent and age-independent mortality rates. The age-independent rate is represented by a Poisson process. For the age-dependent rate, a concept of vitality is defined, and mortality occurs when an organism's vitality is exhausted. The loss of vitality over age is represented by a continuous Brownian-motion process, the Weiner process; vitality-related mortality occurs when the random pr ocess reaches the boundary of zero vitality. The age at which vitality-related mortality occurs is represented by the Weiner-process probability distribution for first-arrival time. The basic model has three rate parameters: the rate of accidental mortality, the mean rate of vitality loss, and the variability in the rate of vitality loss. These rates are related to body mass, environmental conditions, and xenobiotic stressors, resulting in a model that characterizes intrinsic and extrinsic factors that control a population's survival and the distribution of vitality of its individuals. The model assumes that these factors contribute to the rate parameters additively and linearly.

The model is evaluated with case studies across a range of species exposed to natural and xenobiotic stressors. The mean rate of vitality loss generally is the dominant factor in determining the shape of survival curves under optimal conditions. Xenobiotic stressors add to the mean rate in proportion to the strength of the stressor. The base, or intrinsic, vitality loss rate is proportional to the -- 1/3 power of adult body mass across a range of iteroparous species. The increase in vitality loss rate with a xenobiotic stressor can be a function of body mass according to the allometric relationship of the organism structures affected by the stressor. The model's applicability to dose-response studies is illustrated with case studies including natural stressors (temperature, feeding interval, and population density) and xenobiotic stressors (organic and inorganic toxicants). The model provides a way to extrapolate the impact of stressors measured in one environment to another environment; by characterizing ho w stressors alter the vitality probability distribution, it can quantify the degree to which a stressor differentiates members of a population.

Key words: allometry; dose-response studies; environmental properties; mortality rates, age dependent and age independent; organism vitality as basis of survival model; selection; stochastic equation; survival; vitality; xenobiotic stresses.


Life history, natural selection, allometry, toxicology, extinction, ecosystem management, and endangered-species recovery are all ecological topics that have as a central element the survival of populations. Survival models used in these applications can be simple or complex but for any one model to be useful it needs to be connected to the intrinsic and extrinsic factors that ultimately determine the survival of the population. A common approach is to describe empirical relationships between the relevant factors and the survival model's basic parameters. For example, the simplest survival model, the [LC.sub.50], is only a single parameter (LC = lethal concentration), the dose that produces 50% mortality in 96 h. In the more complex "hit" model, in which a toxic chemical damages the organism, the rate of mortality, the hazard function, can be described as a power-series function of the dose (Sielken 1985). Survival models with as few as two parameters, such as the Gompertz and Weibull models, are quite succe ssful at fitting survival curves. The Gompertz model parameters can be related to adult mass, giving survival dependent on this single property (Calder 1996). More complex models describe survival in populations in which individuals are differentiated by traits such as age, size, and developmental stage (Tuljapurkar and Caswell 1997). Even more complex models describe growth and survival of individuals in terms of their bioenergetics (Metz and de Roos 1992, Kooijman 1993). These examples dot the spectrum from simple to complex approaches that connect survival to the intrinsic and extrinsic factors of a population and its environment.

In this paper a relatively simple but mathematically robust three-parameter survival model (Anderson 1992) is extended to relate survival to external and intrinsic factors. The model is midway between simple models that describe only population survival and complex models that treat each organism and their individual dynamics. In models with few parameters, including the Gompertz and the "hit" models, the external factors are equated to the rate of mortality. The vitality-based model presented here is unique in that it characterizes an underlying process, vitality, and seeks to relate the intrinsic and external factors to the rate of loss of vitality. Mortality, in this approach, is not the underlying dynamic, but is the consequence of vitality going to zero. As a result the model tracks both the survival of a population and the distribution of vitality within it. In this paper a number of hypotheses are developed to describe the rate of loss of vitality in terms of the external and intrinsic factors. The re sulting dose--response model provides some unique points of view that are explored in case studies.


Vitality survivorship model

In the basic model (Anderson 1992), vitality is an abstract property that changes with the moment-to-moment experiences of an organism. An organism's resistance to disease, level of stress, behavior, success and failure in feeding, frequency of predator attacks, mating, parental care, and habitat choice all induce incremental changes in vitality. By definition the vitality-based mortality is stochastic, and mortality occurs when vitality reaches zero. Mortality can also occur independent of an organism's vitality through an accidental-based mortality. Examples may include harvest of a population, and catastrophic events that induce mortality equally within a population and independent of the different past histories of the individuals. The important feature is that the accidental mortality rate is independent of past conditions of the organisms.

The survival distribution, S, can be expressed in terms of the product of the probability of survival according to the organism's vitality, [P.sub.v], and the probability of survival in avoiding random (accidental) mortality events, [P.sub.a]. The probability of being alive at any age t is thus

S(t) = [P.sub.v](t)[P.sub.a](t). (1)

For vitality-related survival the dynamic of vitality is represented by a continuous random process, the Wiener process, which is expressed by the stochastic differential equation

[frac{dv}{dt}] = -[rho] + [sigma]W(t) (2)

where [rho] is a deterministic representation of the rate of change of vitality, W(t) is a white-noise process describing rapid fluctuations of the rate, and [sigma] is the intensity of the random fluctuations.

Since vitality varies randomly over time, an individual's vitality can only be described through the conditional probability density of v at time t, which is dependent on the initial vitality, [v.sub.0], at time 0. This is denoted

p = p(v, t\[v.sub.0], 0). (3)

The conditional probability distribution of a process defined by Eq. 2 can be expressed by a Fokker-Planck equation, which describes the change in probability of measure v according to the rate of growth of its mean and variance (for reference see Gardiner [1985]). Requiring that [rho] and [sigma] represent life-history averages of the vitality-rate processes, they are then constant and the Fokker-Planck equation for the rate of change of the conditional probability density is

[frac{[partial]p}{[partial]t}] = [rho][frac{[partial]p}{[partial]v}] + [frac{[[sigma].sup.2]}{2}] [frac{[[partial].sup.2]p}{[partial][v.sup.2]}]. (4)

Mortality is defined as entering a zero-vitality state and mathematically it is expressed by the absorbing boundary condition

p(v, t\0, 0) = 0. (5)

No maximum value is required for vitality, so the upper boundary condition is set at infinity and is expressed mathematically as

p([infty], t\[v.sub.0], 0) = 0. (6)

With these conditions the solution of the Fokker-Planck equation is (Goel and Richter-Dyn 1974)

p(v, t\[v.sub.0], 0) = [frack{1}{[sigma][sqrt{2[pi]t}]}][exp [lgroup][frac{-[(v - [v.sub.0] + [rho]t).sup.2]}{2t[[sigma].sup.2]}][rgroup]

-exp[lgroup][frac{-[(v + [v.sub.0] + [rho]t).sup.2]}{2t[[sigma].sup.2]}] + [frac{2[rho][v.sub.0]}{[[sigma].sup.2]}][rgroup]] (7)

where [v.sub.0] is the initial vitality, [rho] is the deterministic rate of change of vitality, and [sigma] is the intensity of the stochastic rate of change of vitality. This is the probability density function; it can be interpreted as the probability distribution of vitality at a given age t for an individual or as the distribution of vitality within a cohort. Assuming that organisms are initially identical, compared to the divergence of cohort members over life, the individual- and cohort-level interpretations should be functionally the same. In practice we can only extract survival information at the cohort or population level and in this model we assume individuals are identical and independent but acting randomly so that the two levels are functionally the same.

The probability that an organism is alive at a given age is determined by integration of Eq. 7 over the vitality range v [greater than] 0, so

[P.sub.v](t) = [[[integral of].sup.[infty]].sub.0] p(v, t\[v.sub.0], 0) dv. (8)

The resulting equation for the vitality-related probability of survival is

[P.sub.v](t) = [frac{1}{2}][erfc[lgroup][frac{-1 + rt}{s[sqrt{2t}]}][rgroup] - exp[lgroup][frac{2r}{[s.sup.2]}][rgroup]erfc[lgroup][frac{1 + rt}{s[sqrt{2t}]}][rgroup]] (9)

where erfc is the complimentary error function (Korn and Korn 1968) and the vitality rate parameters are normalized by the initial vitality, giving

r = [frac{[rho]}{[v.sub.0]}] s = [frac{[sigma]}{[v.sub.0]}] (10)

where r is a normalized rate with a dimension of [t.sup.-1] and s is a normalized variability with a dimension [t.sup.-1/2].

Although the model is applied at a cohort, or population, level the interpretation of the model parameters to individuals contains some caveats. In particular, the variance term s will be affected by the model assumptions. Violations in the assumption that all members have the same [v.sub.0] should be reflected in both s and r at the population level. Furthermore, since each organism has an r and s set, the population-level r and s will reflect the variance among the individuals. Generally the population estimate of s should be greater than the individual s, reflecting differences in r between individuals.

Organisms that do not die of vitality-related causes die of accidental causes, which are independent of their history. Assuming accidental deaths are randomly distributed, the rate of accidental mortality can be defined by a Poisson process. Then the probability that an orgnism is alive, excluding the vitality process, is the probability of observing zero mortality events to age t and can be expressed as

[P.sub.a](t) = [e.sup.-kt] (11)

where k is the accidental-mortality rate coefficient with a dimension of [t.sup.-1].

Now the probability of survival to age t from vitality-related and accidental causes, [P.sub.v], and [P.sub.a], are multiplied as defined by Eq. 1. The survival equation also can be expressed in terms of the more familiar cumulative normal distribution [Phi] by noting that

erfc(z) = 2[Phi](-z[sqrt{2}]). (12)

Now the survival distribution becomes

S(t) = [[Phi][lgroup][frac{1}{s[sqrt{t}]}] (1 - rt)[rgroup] - exp[lgroup][frac{2r}{[s.sup.2]}][rgroup][Phi][lgroup]-[frac{1}{s[sqrt {t}]}](1 + rt)[rgroup]][e.sup.-kt]. (13)

The survival equation can be expressed in several other forms. The survival equation in Anderson (1992) is related to Eq. 13 by defining [V.sub.0] = 1/s and R = -r/s.

The model is developed from the premise that mortality, at its most reduced level, has two processes: one dependent on the past history of the organism and the second that is independent of the past history. The first process is defined by the Weiner or Brownian-motion process and characterizes a directed random walk of the abstract variable, vitality, to its boundary at which death occurs. The minimum number of parameters that can define this process is two: one to describe the average rate of loss of vitality and a second to describe the variability in the rate. This age-dependent process is in fact 1 minus the inverse Gaussian survival distribution. The age-independent process essentially says that mortality occurs randomly at any age and is independent of the condition of the organism. This process is defined by the simplest survival model, the Poisson or exponential model, and is characterized by a single parameter, the average time between mortality events, which is 1/k. Distinguishing mortality proces ses according to their dependence on age is a fundamental division. Along this line of reasoning any mortality process that has age-cumulative and age-independent parts will require at a minimum three parameters to characterize. One- or two-parameter survival models, such as the Weibull or the Gompertz do not adequately capture these two distinct types of processes. In contrast the vitality model based on a random-walk process and a random failure can be associated with abstract but biological mechanisms.

Relative vitality distribution

Besides defining survival of the population over age, a distribution of relative vitality within the population can be defined as a function of age. Normalizing vitality defined by Eq. 7 by the initial vitality [v.sub.0], gives a relative vitality v = v/[v.sub.0] with a distribution as a function of age defined as

[p.sub.v](v, t) = [frac{1}{s[sqrt{2[pi]t}]}][exp[lgroup][frac{-[(v - 1 + rt).sup.2]}{[2ts.sup.2]}][rgroup] - exp[lgroup][frac{-[(v + 1 + rt).sup.2]}{[2ts.sup.2]}] + [frac{2r}{[s.sup.2]}][rgroup]]. (14)

If this relative vitality distribution is taken to characterize the distribution of vitality of individuals within a population, then it becomes a measure of the distribution of weak and strong individuals as a function of age.

Model properties

Density and hazard function.--To explore the statistical properties of the survival distribution, note that the distribution can be expressed in terms of the inverse Gaussian, which is the cumulative distribution for the first passage time in Brownian motion (Chhikara and Folks 1989). Equivalent expressions to Eq. 13 are

S(t) = R(t)[e.sup.-kt] = (1 - G(t))[e.sup.-kt](15)

where R(t) is a reliability function, which is the probability of having no vitality-based mortality before age t and is

R(t) = [[Phi][lgroup][frac{1}{s[sqrt{t}]}](1 - rt)[rgroup] - exp[lgroup][frac{2r}{[s.sup.2]}][rgroup][Phi][lgroup]-[frac{1}{s[sqrt {t}]}](1 + rt)[rgroup]]. (16)

G(t) is the inverse Gaussian cumulative distribution,

G(t) = [Phi][[frac{[lambda]}{t}][lgroup][frac{t}{[mu]}] - 1[rgroup]] + exp[lgroup][frac{2[lambda]}{[mu]}][rgroup][Phi][- [frac{[lambda]}{t}][lgroup][frac{t}{[mu]}] + 1[rgroup]] (17)

and [Phi] is the cumulative normal distribution; the vitality model parameters are related to the inverse Gaussian parameters by

r = [frac{1}{[mu]}] s = [frac{1}{[sqrt{[lambda]}]}] (18)

where [mu] is the mean of the inverse Gaussian distribution and [lambda] is a shape parameter.

Now the survival probability density function is obtained by differentiating Eq. 15 with respect to age t to give

f(t) = (-k(1 - G(t)) - g(t))[e.sup.-kt] (19)

and the hazard function is

h(t) = [frac{f(t)}{S(t)}] = -k - [frac{g(t)}{1 - G(t)}] (20)

where the inverse Gaussian probability density function is

g(t) = [sqrt{[frac{[lambda]}{2[pi][t.sup.3]}]}]exp[lgroup]-[frac{[lambda][(t - [mu]).sup.2]}{2[[mu].sup.2]t}][rgroup]. (21)

Expected lifespan.--The expected age E(a) of organisms can be defined in terms of k and r. Begin with Eq. 13 and note the expected lifespan age is

E(a) = [[[integral of].sup.[infty]].sub.0] S(t) dt = [[[integral of].sup.[infty]].sub.0] [e.sup.-kt](1 - G(t)) dt

= -[frac{[e.sup.-kt]}{k}] - [[[integral of].sup.[infty]].sub.0] [e.sup.-kt]G(t) dt. (22)

The first term of the integrated equation is simply

-[frac{[e.sup.-kt]}{k}][[\.sup.[infty]].sub.0] = [frac{1}{k}] (23)

and the second term can be integrated by parts as follows. First,

[integral of] G(t)[e.sup.-kt] dt = [integral of] uv' dt = uv - [integral of] u'v dt (24)


u = G(t) v' = [e.sup.-kt] u' = g(t)

v = -[frac{[e.sup.-kt]}{k}] (25)


[[[integral of].sup.[infty]].sub.0] [e.sup.-kt]G(t) dt

= -G(t)[frac{[e.sup.-kt]}{k}][[\.sup.[infty]].sub.0] + [frac{1}{k}] [[[integral of].sup.[infty]].sub.0] [e.sup.-kt]g(t) dt. (26)

Note that G(0) = 0, G([infty]) = 1 and k [greater than] 0 so the first term in Eq. 26 is 0. Note also that g(t) is a probability density function of an inverse Gaussian and since the integration of a function, here exp(-kt), with a density function is the expected value of the function, then Eq. 26 reduces to

[[[integral of].sup.[infty]].sub.0] [e.sup.-kt]G(t) dt = [frac{1}{k}]E([e.sup.-kt]) (27)

where t is the discrete random variable with an inverse Gaussian distribution representing the lifespan age of organisms not subjected to accidental mortality. Now approximating the exponential as

[e.sup.-kt] [approx] [e.sup.-k[mu]] + (t - [mu])[frac{[de.sup.-k[mu]]}{d[mu]}] (28)

where the reference point [mu] is the mean value of the inverse Gaussian, then the expected value of the exponential function is approximated as

E([e.sup.-kt]) [approx] E([e.sup.-k[mu]][1 + k([mu] - t)])

= [e.sup.-k[mu]][1 + k[mu] - kE(t)] = [e.sup.-k[mu]]. (29)

Substituting Eq. 29 and Eq. 23 into Eq. 22 and noting [mu] = 1/r, the expected lifespan of the population is approximated by

E(t) [approx] [frac{1}{k}](1 - [e.sup.-ktr]). (30)

This approximation approaches the exact lifespan estimators when the survival is dominated by either the Poisson or the inverse Gaussian processes. If accidental death dominates mortality, so k [greater than] r, then Eq. 30 reduces to E(t) = 1/k, which is the lifespan for a Poisson process. If mortality is dominated by vitality-related causes, so r [greater than] k, then Eq. 30 reduces to E(t) = 1/k, which is the lifespan for a process based solely on the inverse Gaussian.

Maximum likelihood estimator.--For a maximum-likelihood estimator (MLE) (Klein and Moeschberger 1997) of the model parameters using interval-censored survival data such as can be constructed from survival curves selected from the literature, the incremental probability of survival is

[[hat{P}].sub.i] = -(1 - G(t))[e.sup.-kt][[\.sup.[t.sub.i+1]].sub.[t.sub.i]] (31)

and a maximum log-likelihood is defined as

L = min [[[sum].sup.T].sub.i=1] - [N.sub.i]log([P.sub.i]) (32)

where [N.sub.i] is the fraction of the observed population that survived between time [t.sub.i] and [t.sub.i+1].

The MLE parameters are determined numerically, which requires initial estimates of the model parameters. An initial estimate of k can be found from Eq. 19 by setting t = 0. Then,

[frac{[Delta]S(t)}{[Delta]t}][\.sub.r=0] = f(0) = -k (33)

since G(t) and g(t) are 0 and exp(-kt) is 1 at t = 0. To estimate r and s the survival curve can be transformed into an inverse Gaussian distribution,

J(t) = 1 - S(t)[e.sup.kt] (34)

from which the initial values of the model coefficients can be approximately defined. From J(t) we construct data [t.sub.j], [J.sub.j], j = 0, ... N and estimate a distribution of interval-censored probabilities:

[P.sub.j] = [J.sub.j] - [J.sub.j-1]. (35)

This distribution is approximately an inverse Gaussian distribution but it is a nonrandom sample because of its construction. Since inverse Gaussian parameters are defined in terms of the moments, the initial vitality model parameter estimates can be constructed from the distribution as:

[hat{[mu]}] = [[[sum].sup.N].sub.j=1] [P.sub.j][t.sub.j] = [frac{1}{[hat{r}]}] [hat{[lambda]}] = [[[sum].sup.N].sub.j=1] [P.sub.j][lgroup][frac{1}{[t.sub.j]}] - [frac{1}{[hat{[mu]}]}][rgroup] = [frac{1}{[[hat{S}].sup.2]}]. (36)

The accuracy of these initial estimators depends on the level of the accidental-mortality process, and the method is most successful if k is small. Therefore the survival distribution is more like the reliability function, with the exponential producing minor adjustments to the distribution.

The model parameters can also be estimated from a survivorship curve with standard nonlinear curve-fitting techniques. Using the Marquardt method the parameter space is searched to find the model parameters that minimize a [[chi].sup.2] (Press et al. 1988). Although the maximum-likelihood estimator produces unbiased estimates of the model parameters it requires more manipulation of the survivorship curves, converting the data into incremental survival probabilities, than estimates using a nonlinear search to fit the survivorship curve directly. Both techniques give essentially the same parameter estimates.

Dose--response formulation

Rate additivity hypothesis.--Identifying useful functional relationships for the rate terms r, s, and k to stressors becomes a second stage of the development of the vitality model and is akin to other mortality models that seek to find empirical equations relating stressors to the rate of mortality. Here the emphasis is on relating empirical equations to the vitality rate terms. The simplest hypothesis assumes that stressors alter one or more of the model parameters in a multiple-linear way. For example, assume the rate of loss of vitality is attributable to the additive contribution of a number of environmental and intrinsic factors or stressors. Mathematically this is akin to a superposition hypothesis and the response of each model parameter to the factors is assumed to have the form

[theta] = [sum] [f.sub.[theta],j]([x.sub.j]) (37)

where the rate parameters are [theta] = r, s, or k, the j factors can be quantified by exposure levels [x.sub.j], and the response of the rate coefficients to each factor is [f.sub.[theta],j]([x.sub.j]). This assumption of additivity implies that each factor contributes independently to one or more of the rate parameters r, s, or k.

Linearity hypothesis.--How the additive rate terms, [f.sub.[theta],j]([x.sub.j]), are related to the stressor or dose level must be ascribed next. For factors where the minimum impact occurs at a zero level, a linear approximation is reasonable to describe the impact, so that [f.sub.[theta],j]([x.sub.j]) = [a.sub.[theta],j][x.sub.j] where a is a response coefficient characterizing the effect of the factor on a rate term. Some factors, such as temperature, have an optimum level, and above or below the optimum survival decreases. In these conditions [f.sub.[theta],j]([x.sub.j]) is nonlinear with at least a single minimum at the optimum. However, a linear function may be suitable for expressing the parameter response away from the optimum level of the factor.

In general, for the rate of loss of vitality r, it is reasonable to assume that xenobiotic stressors increase the rate in a linear manner, but for natural factors that have an optimum level for survival a linear expression might be suitable if one endpoint of the relation is referenced to the optimum. Since s expresses the variability in the vitality loss rate, its functional form with x depends on whether or not individuals in the population are equally affected by the stressor or whether some individuals become stronger and others become weaker with exposure. Since k characterizes the average accidental-mortality rate over the lifespan of the organisms and is, in principle, independent of the state of the organisms, its relation to an environmental factor is unclear.

To explore the first-order response of the model variables to environmental factors assume the model parameters are related to factors or stressors by the linear equations

r(x) = [a.sub.r] + [b.sub.r]x (38)

s(x) = [a.sub.s] + [b.sub.s]x (39)

k(x) = [a.sub.k] + [b.sub.k]x (40)

where [a.sub.r], [a.sub.s], and [a.sub.k] express the levels of r, s, and k under a zero or reference value of stressor and [b.sub.r], [b.sub.s], and [b.sub.k] express the sign and magnitude of the responses to the stressor.

The first-order effects of x on the rate coefficients are quantified in terms of the b coefficients, but the significance of the a coefficients depends on the stressor. For xenobiotics, where low levels may be harmful, the a coefficients are approximations of the rate parameters for the base or natural conditions absent the stressor. For factors with optimum levels, such as temperature, the intercepts are artifacts of the stressor range explored in a particular study. In these cases a factor could be referenced to its optimum level. For example, temperature might be expressed as relative to the optimum temperature, such as x = T - [T.sub.opt].

Additionally, because the stochastic term in Eq. 2 expresses the variability in the vitality rate, in some circumstances it may be related to the mean rate r even though s and r individually are not clearly related to x. To explore this possibility a linear relationship between s and r is defined as

s(x) = [] + []r(x) (41)

where [] quantifies how the vitality rate variability increases with an increasing mean rate of vitality loss. The slope [] is a measure of how strongly the deterministic and stochastic rates are coupled.

The slopes of the regressions in Eq. 38, Eq. 39, Eq. 40, and Eq. 41 define, to a first order, the response of an organism to an environmental factor or stressor. The cumulative time-dependent effects of factors and stressors are characterized by [b.sub.r]. The range of susceptibility of individuals in a population is characterized by [b.sub.s], and time-independent impacts on the population are characterized by [b.sub.k].

Extrapolation across environments.--The model can, in principle, be used to extrapolate dose--response relationships from one environment, such as a laboratory, to another, such as the natural habitat of the organism. For this extrapolation, factors that control the rate parameters in the different environments must be expressed in a mathematically tractable form. From the additivity assumption, Eq. 37, the impact of a stressor in the second environment can be expressed in terms of the response to the stressor in the first environment plus the base survival in the second environment in the absence of the stressor. The rate equations for the second environment are:

[r.sub.2](x) = [r.sub.2](0) + [b.sub.r1]x [s.sub.2](x) = [s.sub.2](0) + [b.sub.s1]x

[k.sub.2](x) = [k.sub.2](0) + [b.sub.k1]x (42)

where subscripts 1 and 2 refer to first and second environments, respectively, the intercepts, [r.sub.2](0), [s.sub.2](0), and [k.sub.2](0), characterize the survival rate parameters in the second environment with zero dose, [r.sub.2](x), [s.sub.2](x), [k.sub.2](x) are the extrapolated rate parameters at dose level x in the second environment, and the coefficients [b.sub.[theta]1] are the responses to stressors obtained from dose--response studies in the first environment.

Effects of body Size

A considerable number of studies have identified cross species correlations between body mass and life-history, structural, physiological, and ecological characteristics of organisms. These include relationships such as skeleton mass, metabolic rate, running, swimming, and flying speeds, home range, population density, and lifespan (Calder 1996). Relationships between body mass and the vitality rate parameters should also exist.

Life-span.--To consider how body mass relates to r across species, note there is a well-established relationship between expected life-span and adult body mass that can then be related to r. A similar strategy was used to relate body mass to the Gompertz survival equation coefficients (Sacher 1978). The general relationship between mass and expected life-span ELS over several hundred species including mammals, birds, Anatidae, and Phasinanidae has the relationship

ELS [approx] [aM.sup.b] (43)

where M is body mass and a and b are the allometric coefficients that may be similar (especially the power term b) across different species (Calder 1996). The expected life-span in the vitality model is related to the vitality rate and the accidental-mortality rate and so equating Eq. 43 to Eq. 30 the model parameters and life-span age are approximately related as

ELS [approx] [aM.sup.b] [approx] [frac{1}{k}](1 - [e.sup.-k/r]). (44)

If the rate of accidental mortality is low so ELS is determined by vitality-related factors then the relationship of r to body size reduces to

r [approx] [frac{[M.sup.-b]}{a}]. (45)

This equation suggests that the rate of loss of vitality across species should decrease with body size by a power of -b, giving a gradual decrease in the rate of loss of vitality for increasing species adult body mass. A number of factors behind this correlation have been postulated (Calder 1996) including: (1) a genetically determined design margin to ensure animals survive to reproduce, (2) failure of crucial components after a number of cycle times, (3) the gradual accumulation of somatic mutations at the cellular level, (4) aging from toxic metabolites, and (5) brain size, which postulates that larger brains have better control of the physiological maintenance and homeostasis. Factors (2) through (5) have in common the idea that after a particular number of cycles a mortality event occurs and so the mortality scales with the frequency of cycles, which for a large number of physiological processes scale to the -- 1/4 power of body mass. In comparison, maximum lifespan scales to the 1/4 power of body mass.

Sensitivity to xenobiotic stressors.--The relationship between body mass at age and the sensitivity of an animal to a stressor may be very different from the relationship between adult body mass and life-span across species. While the cross-species relationships are likely associated with the physiological time scales, which have similar scalings with mass, the stressor-body mass relationship may be determined by the relationship of body mass to the specific organism structures affected by the stressor. To explore this hypothesis, assume that the body-mass effect is not on the rate parameter coefficient itself, for example [b.sub.r], but on the effective exposure level [x.sub.effective] and can be expressed as

[x.sub.effective] [approx] [M.sup.c]x (46)

where M is body mass and c is a coefficient relating how mass scales with the measured stressor level x. A universal scaling exponent for all dose-response cases is not expected if a specific stressor acts on a specific structure or metabolic process. These in turn are related to body mass by allometric relationships that are thought to be established by the principle of symmorphosis, in which the structural design of an organism is regulated to meet, but not exceed, the functional demand (Taylor and Weibel 1981). With this hypothesis, the effects of a xenobiotic acting on the brain could be significantly different from one acting on the liver since they have different relationships with body mass. In birds the brain mass scales to body mass by the 0.58 power while the liver scales by the 0.89 power. In pharmacokinetic studies dosage is typically scaled as basal metabolic rate scales with body mass (Dedrick 1974). This would give c [sim] 3/4 if exposure effectiveness is related to metabolism.


To explore survival dynamics through the vitality model we consider a number of case studies that represent typical problems and approaches in survival studies. First, the issue of classification of survival is considered. The traditional approach identifies three types of survival curves based on the shape of the log of survival over age. The vitality model provides a more mechanistic perspective that contributes to the understanding of the traditional classification. Second, a number of studies relating survival to natural and xenobiotic stressors are considered in order to explore the linearity hypothesis that the effects of stressors on model parameters can be expressed by linear functions. The third case study considers whether the linearity hypothesis is able to predict mortality over a range of exposures. Fourth, the rate-additivity hypothesis is used to extrapolate the effects of a toxin on fruit flies measured in one environment to the effects in a sub-optimal environment. The fifth case study explo res the problems in evaluating the effects of stressors in early life-history stages and provides a plausible explanation for the inability of the model to fit these situations. The sixth case study explores the relationship among body mass, lifespan, and the rate of loss of vitality. It illustrates that a cross-species relationship exists between r (normalized vitality rate) and body mass. The seventh case study explores the hypothesis that body mass affects the response to stressors through the relationship between mass and the structures and bioenergetics affected by the stressor. The final case illustrates how the vitality model may quantify a stressor's effect on natural selection through the characterization of how the mean and variability in the vitality rate change with stressor level.

In each case study the model was fit to survival curves obtained from the literature yielding r, s (variability in r), and k (the rate of accidental mortality), for each dose or exposure level. Model parameters obtained by two techniques, the maximum-likelihood estimator and a Marquardt nonlinear search routine, yielded no significant difference in parameter estimates.

Survivorship-curve classification

The vitality-based survival model provides a dynamic way to classify survivorship curves. Traditionally, survivorship curves are classified into one of three types according to where the mortality occurs: early in life, later in life, or randomly throughout life (Hutchinson 1978). In the traditional description, type I survivorship curves are negatively skewed rectangular curves with most individuals living their full life-span and then dying during senescence. Type II curves are referred to as "diagonal curves," because a plot of the log of survival over time produces a diagonal line. In this type the chance of mortality is equal at all life stages. Type III curves in the classical sense exhibit greater mortality in the early life stages and have a convex shape on a log-survival plot. The vitality model produces these three curve types according to the relative significance of the model parameters r, s, and k. The type-r curves are generated when the mean rate of loss of vitality, parameterized by r, is the dominant factor driving mortality. These curves are equivalent to the classical type I survivorship curves. Type-k curves are dominated by accidental mortality as characterized by the parameter k. These curves are equivalent to the classical type II survivorship curves. Type-s curves are generated when the random vitality element, s, is the dominant parameter. These are a subset of the classical type III survivorship curves in that they exhibit higher mortality in the early life-history stages. The other subset of classical type III curves are produced by a distinct high-mortality period in early life history. In a vitality-based classification these are produced by a step-like change in r corresponding to a change between life-history stages. In these situations the assumption of constant model parameters is violated.

To compare them graphically, survivorship curve types can be normalized by an appropriate time scale such as [T.sub.50], the time to 50% mortality. This normalization removes the effect of the strength of mortality and reveals the differences in shape resulting from the dominant parameter. In Fig. 1 plots of the log of survival against age normalized to [T.sub.50] are illustrated for r-, s-, and k-dominated survival curves corresponding to humans, salmon, and limpets, respectively. The differences in the shapes of the curves are readily evident by the slopes at Time per [T.sub.50] = 1. The classification can be expressed through a measure of the rate terms themselves. To do this note that the expected life-span, akin to [T.sub.50], as approximated by Eq. 30, can be further reduced, noting that to a first order the expected life-span is 1/r. Then a suitable measure of curve type (CT) can be expressed by the measure

CT = [frac{[sqrt{r}] - [sqrt{k}]}{s}] (47)

where r-type (traditional type I) curves dominate when CT [greater than] 1, s-type curves (traditional type III) dominate when 0 [less than] CT [less than] 1, and k-type curves (traditional type II) denominate when CT [less than] 0. The transition between one type and another is gradual; for instance when CT = 0 all three factors may be equal.

An example of this classification scheme is illustrated in Table 1 for a range of species. The strongest r-type survivorship curve is for sheep, where mortality is principally from vitality-related causes late in life and the rate of accidental mortality is low. The strongest s-type survivorship curve is for sockeye salmon. Mortality occurs early in life because of a high random vitality loss rate. In conditions where vitality loss is not a significant factor, k-type curves occur and as such there is no clearly defined maximum age to such organisms. In Table 1 only the limpet has a clear k-type curve. Although species with indeterminate maximum life-spans are rare, Martinez (1998) suggests that hydra do not exhibit senescence. Therefore they would also be expected to have a k-type curve.

Environmental stressors

How the vitality model quantifies the effect of environmental stressors on survival is explored with studies on the effects of temperature, population density, and feeding interval (Table 2). Model parameters r, s, and k were estimated for each stressor level and the resulting estimates regressed against the stressor levels using Eqs. 38-41.

Temperature.--Temperature has a large effect on the survival of heterothermic organisms: a rotifer, two nematodes, and a fruit fly (Table 2). In general, in each case as temperature increased from an optimum level the median age of survival decreased, but the survivorship curves lessened their r-type character, moving towards an s-type shape. This response is illustrated for the rotifer Asplanchna brightwelli. At 15[degrees]C the negatively skewed curve was r type with CT = 2.7 as defined by Eq. 47. At the higher temperatures the curve assumed s-type shape with CT = 0.9 at 25[degrees]C (Fig. 2). The shift is quantified by the coefficients of the linear equations describing the change in model parameters with temperature (Fig. 3). The increase in mortality with temperature is characterized by an increase in the vitality rate as quantified by [b.sub.r] which is a quantitative measure of the temperature sensitivity of the species. In three of four species in Table 2 the regression of r with temperature is signi ficant at the P [less than] 0.05 level. A. brightwelli had greatest sensitivity to temperature while the nematode Rhabditis had the lowest sensitivity by a factor of eight. In all species, the variability of the vitality rate, as characterized by [b.sub.s], increased only slightly with temperature (Table 2). These studies did not explore survival at temperatures below the optimum, but the data for Asplanchna (Fig. 3) do suggest the rate of vitality loss increased below 17[degrees]C. Thus, an optimum temperature for survival can, in principle, be identified in terms of a minimum r. The linear regression for k is not significant at the P = 0.01 level for any species, indicating that a pattern in how accidental mortality rate changed with temperature could not be identified.

Plots of s against r characterize how the vitality-rate variability changes with the mean rate under a range of temperatures. Although the linear regression was not significant for the individual species, combining all data yields a linear relationship for the temperature stress (Table 2). This cross-species result does not exist for other parameters and suggests that the effect of temperature on the coupling of the vitality-rate mean and variability is similar across these heterothermic organisms. The potential implication of this in characterizing a species adaptation to temperature changes will be discussed below (see Within-population vitality differentiation).

Feeding interval.--The model can also quantify how food availability affects survival. A study by Verdone-Smith and Enesco (1982) indicated that rotifer survivorship decreased by increasing feeding interval from 12 to 72 h (Fig. 4). At short feeding intervals the survivorship curves were r type, with most animals alive after one week. Under reduced food availability, in which organisms were fed every 3 d, only half the animals survived one week and the curve took on a more s-type shape. This suggests that variability played an increasing role in population mortality at increasing stress levels. Although the three model parameters increased with longer feeding intervals, the individual regressions with interval were not significant. In contrast, variability in the vitality rate, s, with the mean rate r did have a significant linear relationship (Table 2). The slope of the regression, [] = 0.19, was the largest observed in any case study, suggesting the differences in vitality between individuals in th e population increased with starvation. This suggests that [] is potentially a measure of resource competition within a population. When [] is large, weaker individuals become progressively less able to compete under starvation, which in turn may give the strong individuals a competitive advantage.

Density dependence.--Population density impacts organism survival through a variety of processes. Density can affect survival through resource competition, as seen in the feeding-interval study (Table 2). Population density can also act on vitality through subtle physiological stress and directly through accidental mortality. Individual factors, although complex, can be partitioned and quantified with the vitality model. A study of survival of the water flea, Daphnia pulex, under different animal densities produced a complex pattern of survivorship curves (Frank et al. 1957). In the experiments an approximate constant density was maintained by successive transfers of animals to replace numbers lost in the culture beakers. The organisms introduced had come from the same density environments. Densities of 1, 2, 4, 8, 16, 24, and 32 Daphnia per milliliter were studied. The survival curves changed significantly between density levels in a complex manner and the growth rate decreased with increasing density, pres umably from limited food associated with increased density.

At low density (1 animal/mL) the survivorship curve was r type, with significant mortality not occurring until week 2 of the study. At an intermediate density (8 animals/mL) the survivorship curve was also r type, but most mortality was delayed a month. At a high density (32 animals/mL) the survivorship curve took on a k-type shape and mortality occurred equally throughout the Daphnia life-span (Fig. 5). The authors suggested two separable crowding effects. Density had positive effects before some limiting factor, probably food, was totally exploited. The second factor appeared at higher densities. The vitality model has a good fit to the data at all density levels and the change in model parameters with density (Fig. 6) quantifies the authors' two process hypotheses. The effects of crowding at low density are represented by a vitality rate that rapidly decreases with density and is essentially constant above 5 animals/mL. The second density factor is characterized by the accidental mortality rate, which is significant only at high densities.

These data show that the interaction of population density with vitality can be complex and nonlinear, so that the linear equations relating the stressors to model parameters, as in the case of Daphnia, are only suitable to characterize the trend in the data. More complex functions would be needed to express the nonlinear relationships of the model parameters with density.

Response to xenobiotics

Three general types of survivorship curves were observed in response to toxins under laboratory conditions (Tables 3, 4, and 5). The most common response was an r-type survivorship curve with high initial survival followed by a sharp increase in mortality (Fig. 7: curves 1-4). A second response was the k-type survivorship curve in which mortality occurred at a uniform rate over time (Fig. 7: curve 5). In contrast to the natural stressors, which produced s-type curves with increasing stress, s-type curves were generally not observed with xenobiotics. The step-like curves were observed in some experiments with an episodic decline in survival followed by little change in survival throughout the remainder of the study (Fig. 7: curve 6). In all examples the vitality loss rate r increased with increasing dose level. In seven out of the nine experiments considered, the relationships fit a linear relationship at the 0.01 significance level. With the r-type response the main effect of the stressor was to change the v itality loss rate and this was characterized in the model by b, being the dominant model parameter. In a k-type curve both r and k changed with dose. In this case the response parameter [b.sub.k] was of the same order as [b.sub.r] (Table 4). Step-like survivorship curves (Table 5) typically had negative values of r at low doses and the variability s decreased with dose, as characterized by [b.sub.s] [less than] 0. In r and step-like curves, changes in accidental mortality rate were not significant, as characterized by [b.sub.r] [gg] [b.sub.k].

The r-type curves were observed in the exposures of organisms to a bacterial toxin, a heavy metal, pesticides, and a mixture of industrial effluent. Species included fruit flies, fish, birds, and isopods. The r- and k-type responses were observed in fish exposed to supersaturation and the step-like curves were observed in experiments involving early life stages of fish and birds exposed to a fungicide and pesticides. The interpretation of these responses and the application of the vitality model are discussed in the next three sections below.

Extrapolating response

The extrapolation of a toxin response for a single environment is straightforward and involves estimating model parameters in Eqs. 38-40. An example is illustrated with the study of pink salmon fry exposed to sulfite waste liquor (Table 3). Model parameters were obtained for each toxin level by fitting the vitality model to survivorship curves. The regression of the resulting values of r and k to the waste-liquor concentration x were significant at the P = 0.05 level while the regressions of s to x and s to r were not significant (Fig. 8). Even with this mixed ability to characterize the regression coefficients, the model was able to reasonably predict survival at a given dose.

A comparison of observed survival data, fitted survivorship curves, and extrapolated survivorship curves based on model parameters estimated using the dose level and the parameter coefficients is illustrated for three sulfite waste liquor levels (Fig. 9). In the zero dose example, the fitted and extrapolated survivorship curves fit the data well but the observation time was not long enough to characterize the curve at lower levels of survival (A in Fig. 9). At the low dose (1986 mg/L) ([B.sub.ext]) the extrapolated survivorship curve underestimates survival. It is evident from Fig. 8 that the regressions of the coefficients underestimate r and k for the zero dose and overestimated them for the 1986 mg/L dose. Since higher values of these parameters produced lower survival, the extrapolated survivorship curve is overestimated at the zero dose and underestimated at the low dose. At higher doses the regression curve fits the model parameters and the extrapolated survivorship curves fit closely to the observed d ata and the fitted curves (Fig. 9: C and D). Generally, the ability of the extrapolated curves to fit data are driven by how well the r vs. x regression fits the estimated model parameters at specific toxin levels (Fig. 8).

This comparison illustrates the ability of the vitality model to extrapolate a population's response to a dose, given a relationship between the vitality-model parameters and the dose level. In general, a good extrapolation can be achieved if the mean vitality rate is linear with dose. In the pink salmon example, a linear regression of r to x was highly significant, which gave well-defined extrapolations even though the linear regressions between dose and the other model parameters were insignificant. In general, fitting r-type survival curves only requires characterizing the response of r to dose.

Extrapolating to different environments

The applicability of the linear extrapolation and additivity hypotheses from which Eq. 42 is derived is demonstrated using a study enumerating the interactions of a toxin and foodstuffs on longevity of the fruit fly Drosophila (Van Herrewege and David 1970). Survivorship curves were determined for fruit flies exposed to chronic levels of toxin produced by Bacillus thuringiensis. Survivorship curves were determined in two environments: a complete food medium and one containing only sugar. The regression coefficients of model parameters against dose for this experiments are given in Table 4. In the complete medium (Fig. 10: curve C0) survival was high (57 d to 50% mortality) but survival decreased significantly (28 d to 50% mortality) with an 8-mg/L toxin exposure (Fig. 10: curve C8). With a zero dose in sugar medium survival was poor (Fig. 10: curve S0). The time to 50% mortality was 25 d, which was below the survival in the complete medium with 8 mg/L of toxin. In the sugar medium an 8 mg/L of toxin dose red uced the time to 50% mortality to 15 d (Fig. 10: curve S8). This was 13 d fewer than the time to 50% mortality in the complete medium. The extrapolation of the response to 8 mg/L toxin in the sugar medium using Eq. 42 gives a survival curve that predicts the time to 50% survival within 2 d of the observed time of 15 d.

The close fit to data of the survivorship curve derived from Eq. 42 (Fig. 10: curve S8) illustrates that the model can extrapolate the effect of a dose response from one environment to another. Furthermore, the model captures the survivorship-curve shape in the second environment. With this system the change in survival is characterized by a shift in the time to 50% mortality while the slope of the survivorship curve (at 50% mortality point) remained relatively constant (Fig. 10). That is, in this situation survivorship curves retained their r-type shape from one environment to another and with increasing dose. In comparison, in the experiments with natural population regulators (Table 2) increased stress caused a shift from r-type to s-type survivorship curves, which resulted in a change in both the time to 50% mortality and the slope at 50% mortality. This example supports the hypothesis that additive rates can be defined with one rate for the environment and another for the toxin.

Extrapolating to step-type survivorship curves and developmental change

Not all survivorship curves exhibit smooth change. Sometimes in both natural settings and laboratory experiments populations exhibit steps in the survivorship curve caused by a discontinuous changes in rate of loss of vitality. These stepped curves typically occur when animals have a distinct early life-history period of high mortality followed by lower mortality, with the change resulting from distinct morphological or behavioral changes in life history. In both natural and laboratory settings, a distinct change in the mortality rate violates the model assumption that mortality over the entire life-span can be represented by average rate processes, characterized by k and r, plus the high-frequency random component s. In step cases the life stages must be represented by different k and r values. Step-type curves are clearly distinguished in plots of r and s representing parameter values for different stressor levels. In typical curves r and s derived from a series of stressor levels cluster with a positive s lope (Fig. 11: line 1) while systems with step curves generated negative slopes of s with r and negative values of r at low-dose levels (Fig. 11: lines 2 and 3). The three cases given in Table 5 represent stepped systems with changing model parameters in an early life stage.

This step change is illustrated in the study of rainbow trout embryolarval to the fungicide Thiram (Van Leeuwen et al. 1986). Mortality was highest during the late gastrulation and early organogenesis and low afterwards. To fit these stepped curves over a range of fungicide [b.sub.s] and [] were negative, reflecting r and s decreasing with increasing dose. The negative values are likely an artifact of violating the model assumption of time-invariant r and s. In other xenobiotic systems studied, [b.sub.s] and [] were positive, reflecting the mean and variability in the vitality loss rate increasing together.

The trout response to fungicide (Fig. 12) illustrates the difficulty in identifying step-type curves from the model fit alone. The model fit the survival curve at low and high fungicide doses but not at an intermediate dose. The model fit the low dose (Fig. 12: curve A) through a, negative r, which mimics the step change from a high to a low r over time. At the intermediate dose (Fig 12: curve B) the model fit had a positive r, which only roughly approximated the average of r over the two life stages. At a high dose level (Fig. 12: curve C) the mortality occurred within a life stage so the assumption of a constant r was not violated and the model fit the survival curve well. Similar patterns were observed with the another fungicide, DIDT, (Van Leeuwen et al. 1986) and Bobwhite chicks exposed to the pesticide chlordimeform (Fleming et al. 1985) (Table 5).

Body-size effects

Life-span.--From the properties of the vitality model, various effects of body mass on model parameters can be postulated. Across iteroparous species we generally expect one relationship with body mass but within a species we may expect more varied relationships between body mass and the sensitivity of the model parameters to the stressor--depending on which structures the stressor affects.

From allometric arguments on body mass to life-span and a theoretical relationship of model coefficients to life-span across species r should be related to body mass as expressed by Eq. 44 and 45. The power relationship between maximum life-span and adult body mass, Eq. 43, is very consistent across a wide range of species. Eutherian mammals, birds, and Anatidae maximum life-spans are related to mass by the power of b = 0.2 (Calder 1996). The maximum life-span is a physiological upper limit of the most vital members of a population. This measure is not representative of the population as a whole and other measures yield different power relationships with mass. The scaling with life expectancy (time to 50% mortality) developed from data on 15 bird species (Calder 1996) gave a = 9.84, b = 0.46 with [r.sup.2] = 0.70. Calder also provided two estimates of the relationship between the time to 50% mortality and adult body mass in eutherian mammals. The results give a = 4.44 b = 0.32 using data from Sacher (1978) a nd a = 5.08, and b = 0.35 using data from Calder (1982 and 1983) with [r.sup.2] = 0.73 and 0.67, respectively. In comparison, the a and b parameters can be estimated from the relationship between body size and r and k using Eq. 44. For this estimation the information in Table 1 is used, excluding the plants because the adult size is not well defined, and the salmon because it is semelparous with a fixed life-span. A log-log regression of Eq. 44 gives a = 6.39 and b = 0.32, with [r.sup.2] = 0.84 and P = 0.0014. Using the simpler approximation of Eq. 45, excluding limpet since it is a k-type survival curve, a log-log regression gives a = 4.92 and b = 0.33, with [r.sup.2] = 0.94 and P = 0.0003 (Fig. 13).

The regression coefficients derived from these four different approaches are strikingly close. This is surprising since they include very different species. Sacher (1978) included the following nine species: short-tailed shrew, house mouse, cotton rate, rice rat, white-footed mouse, California mouse, beagle dog, thoroughbred mare, and U.S. white human female. In comparison, the regression with r to mass from Eq. 45 included the following seven species: rotifer, mosquito, fruitfly, lapwing, vole, sheep, and Australian human male, and with Eq. 44 a limpet was added. The coefficients from the two regressions are nearly identical, but the regression of body mass against r had much higher [r.sup.2] than the regression of life expectancy against body mass.

Sensitivity to stressors.--Addressing the second issue of body size as expressed by Eq. 46, the relationship of a measured stressor to the model parameters depends on which structural elements are affected by the stressor. If the effect of a stressor is simply per unit body mass then the scaling exponent is c = -1 so that a unit amount of xenobiotic stressor affects r inversely to body mass. If, on the other hand, the stressor affects a specific structure, the effective dose may be related to body mass, as the structure is allometrically related to body mass. To explore this further, consider fish exposed to supersaturated water, which causes small gas bubbles to form in their capillaries and eventually results in mortality. Large fish are more sensitive to this gas-bubble trauma than smaller fish (Dawley et al. 1976, Jensen et al. 1986) and so studies on supersaturation mortality provide a test of the hypothesis that the effective exposure scales with the affected organ.

Laboratory studies with subyearling chinook salmon (Oncorhynchus tshawytscha) (Table 3) showed survivorship changed significantly with increasing total dissolved gas (TDG) exposure (Dawley et al. 1976). The study used 40-mm juveniles exposed to supersaturation in tanks of 0.25 and 2.5 m depth. Because supersaturation decreases in a predictable manner with depth, experiments at the two depths were combined and adjusted to an equivalent gas exposure level at 0.25 m depth. Below a TDG saturation level of 105% (100% = saturation), fish did not exhibit signs of bubble formation, but above this level bubble formation induced mortality. The vitality model fit well with the survivorship curves at different TDG levels, and with increasing levels the curves shifted from a strong to a weak r-type form (Fig. 14). Of the three vitality rate parameters, only r had a significant linear relationship with TDG. The relationship of TDG vs. k was less significant and for TDG vs. s there was no relationship (Table 3).

To explore body-mass effects on TDG exposure, experiments on a variety of fish species were combined (Table 6). The individual survival curves were fit to generate the model parameters, and the effect of fish size on the relationships between TDG and r could be expressed by using Eq. 46 in Eq. 38 to give

r = [bM.sup.c](TDG - 105) (48)

where c is a coefficient expressing how the vitality rate changes with fish mass M, TDG is the total dissolved gas percentage supersaturation adjusted for fish depth, and 105 is the threshold saturation level for the onset of gas-bubble disease. To estimate b and c Eq. 48 was expressed as a log-linear regression.

log[frac{r}{TDG - 105}] = log b + c log M. (49)

The regression (Fig. 15) of r parameter estimates from 23 survival curves for brown trout, rainbow trout, and chinook salmon from the experiments of White et al. (1991) and Dawley et al. (1975, 1976) yields the parameters b = 0.0192 and c = 0.49 with [r.sup.2] = 0.83 and P [less than] 0.001.

In this example, for a fixed level of TDG, the vitality loss rate from gas-bubble disease increases at about the square root of fish mass. To explore a possible reason for this relationship note that gas-bubble trauma is a result of small bubbles forming in the capillaries of a fish. Since larger fish are more susceptible to the effects of gas-bubble disease it seems reasonable to look for allometric relationships for the circulation system that scale in a similar manner to the way r scales with body mass for TDG exposure. One possibility is that the susceptibility depends on how the total cross-sectional area of capillaries scales with body mass. From Calder (1996) capillary area scales with body mass as [M.sup.0.57]. A second possibility is that gas-bubble susceptibility scales to gill dimensions, such as the total filament length of a gill, which scales as [M.sup.04] in Torpedo marmorata, or the surface area, which scales as [M.sup.0.78 to 0.93] in salmon (Hughes 1984). A third possibility relates the sus ceptibility to a body-level gas flux that is directly proportional to the gill area, A, and inversely proportional to the gill thickness, h, and the residence time of blood in the fish, t, giving body-level flux = KA/ht, where K is a permeation coefficient and is independent of body mass (Hughes 1984). The other terms scale with body mass. On the basis of metabolism the blood residence time is the organism mass divided by the metabolic rate, giving a residence time scaling of [M.sup.1]/[M.sup.0.78] = [M.sup.0.22]. The gill-thickness measure, describing the distance between the blood and the water, has a weak scaling, giving h [sim] [M.sup.0.14] (Jones and Randall 1978). The resulting gas body flux is [M.sup.0.78 to 0.93]/[M.sup.0.22][M.sup.0.14] = [M.sup.0.42 to 0.57]. This equation suggests that the rate a fish supersaturates increases by the square root of mass.

The estimated effect of mass on the vitality loss rate from the experiments with salmonids exposed to supersaturation gives r = [M.sup.0.49]. The essential point is that the experimental estimate and the possible relationships all scale about the square root of mass. The body gas-flux scaling is the most appealing since it has a dynamic basis and its range includes the model-derived estimate. A final point is that we may expect a variety of relationships by which r, s, and k scale with a stressor level and body mass and such scalings are likely be related to the animal allometry.

Within-population vitality differentation

Because the model tracks the probability distribution of the vitality of individuals, it characterizes how a stressor changes the distribution of low and high vitality members in a population under stress. The effect of a stressor on the vitality distribution is quantified by the relationship between stressor level x and the rate parameters r and s. If r and s change in a proportional manner with x, such that the regression of r vs. s is significant and characterized by [] of Eq. 41, then this coefficient is potentially a measure of the ability of a population to produce stress-tolerant organisms. It is interesting to note that only the natural stressors in Tables 2 and 4 had regressions of r vs. s that were significant. The xenobiotic stressors in Table 3 had insignificant regressions; presumably populations would have less ability to become tolerant to xenobiotic stressors.

To illustrate how the different stressors may affect selection, compare Asplanchna brightwelli response to feeding interval ([] = 0.188) and temperature ([] = 0.039) (Table 2). Both stressors culled the less vital members of the population, but the shapes of the vitality distributions suggest more (Fig. 16). For both stressors the number of low-vitality individuals increased under increased stress, but for temperature the number of high-vitality individuals uniformly decreased under increased departure from optimum temperature, while for an increase in the feeding interval the number of higher vitality individuals actually increased with increasing time between feeding. That is, feeding stress widened the vitality distribution, reflecting more variability in the population and a wider range of vitality relative to the base case with lower stress. As a result, both the numbers of low- and high-vitality individuals, increased with food stress while the numbers around the mode of the vitality di stribution decreased.

A plausible explanation for this response is that under increased stress, stronger individuals have less competition from weaker members of the population than in a non-stressed condition in which weaker individuals are more competitive. This hypotheses suggests that when mortality is dependent on acquisition of resources through contests, limitation of the resource can decrease the ability of weaker individuals to compete, resulting in the stronger individuals winning more contests and becoming still stronger. This differentiation is exactly what happens in the vitality model. It has also been observed in a number of behavioral studies (Sutherland and Parker 1985). For example, in sticklebacks (Gasterosteus aculeatus) the least successful suffered considerably in the presence of others, while the more successful were shown to be little hindered by competition (Milinski 1982).


I have expanded the three-parameter vitality-based survival model (Anderson 1992) with four hypotheses into a model that relates survival to environmental factors, xenobiotic stressors, and allometric relationships of the animals. Before discussing how the model is extended it is worth considering what it offers in comparison to other survival models.

Basic model and comparisons

The basic vitality model has a time-dependent cumulative process (vitality), a time-independent process (accidental mortality), and fits virtually all survivorship curves (Anderson 1992). The three parameters are the vitality rate, r, the variability in the rate, s, and a rate of accidental mortality, k. This model is a combination of two well-established survival equations--the exponential equation and the inverse Gaussian equation. The exponential part describes the time-independent process and the inverse Gaussian describes the time-dependent vitality-related mortality. Although the statistical properties of the combined equation do not have a simple analytical form, an approximation for the expected life-span can be derived that illustrates the contribution of the model parameters to the statistical properties of survival.

If the three-parameter vitality model is to have utility it has to provide information beyond that provided by the well-known two-parameter models such as the Gompertz and the Weibull survival models. The Gompertz model with hazard function [h.sub.0]exp([gamma]t), like the vitality model, partitions mortality into two processes: an age-independent or initial mortality rate, [h.sub.0], and an age-dependent part, [gamma]. Both parameters are assumed to be constant and independent over the term of the experiment. Although the Gompertz model has been in service for over a century (Gompertz 1825) its validity is called into question as an accurate portrayal of survival curves. The parameters have only vague biological interpretations and the assumption of the constancy and independence of parameters across the lifespan have been questioned (Witten and Eakin 1997). The Weibull model with hazard function, [alpha][lambda][t.sup.[alpha]-1], suffers from similar weaknesses as the Gompertz model, although it has great flexibility in that it can accommodate both increasing and decreasing hazard rates as determined by the shape parameter, [alpha]. The strength of these models lay in their tractable mathematical properties. The ability to fit data with a two-parameter model can generally be improved by adding one or more parameters, but with the loss of mathematical tractability. For example, as with the vitality model the Weibull model could be multiplied by an exponential distribution to represent a time invariant mortality process. In fact, adding an exponential function to the Weibull distribution yields a function with similar properties to the vitality model (O.S. Hamel, unpublished manuscript). The utility of the vitality model as a basic three-parameter survival model must then lie in what it means to the biological understanding of survival.

Here the vitality model provides a unique approach compared to most survival models. Typically survival models address the survivorship curve or the hazard rate directly. That is, they seek to explain the rate of mortality itself as the biologically meaningful quantity. With the vitality model, mortality is not the underlying dynamic. It is the end result of the loss of vitality, which is an abstract measure that sums the positive and negative events leading up to mortality. The vitality model assumes that internal and external factors act on the rate of loss of vitality.

This distinction is most meaningful when trying to account for the plethora of factors that ultimately determine mortality. The approach in traditional survival models or in the vitality-based model is to correlate the factors with the model parameters. In any survival model this means explaining survival through equations that relate the model parameters to the factors. For example, Calder (1996) related the organism mass to survival with an allometric relationship between mass and the Gompertz parameters [h.sub.0] and [gamma]. The vitality model takes the same approach, relating factors to the vitality and accidental-mortality dynamics. This approach with traditional models that address mortality directly is problematic because parameters describing the shape of survivorship curves often have vague biological meanings. With the vitality approach, the connection is more transparent, since we seek to identify how environmental and intrinsic factors change the dynamics of vitality and accidental processes.

Another desirable trait of the vitality model is the implied initial state of the animal, expressed by the initial vitality. Although this factor becomes a normalizing constant and is not directly measurable (see Eq. 10), it does express that each species has its own initial state, or capacity for survival, and that over its life-span an animal uses up this capacity. A result of this gradual loss of capacity is the hysteresis and time dependency of losses--in that events that happen today affect the future. In the Gompertz model these general characteristic are represented with the mortality increasing exponentially with age, implying that mortality is an autocatalytic process such that the greater the mortality the greater its rate of change. This seems a strong condition to impose on mortality processes. In comparison the vitality model assumes an abstract measure that must be consumed before mortality, which seems closer to the nature of the process.

A third trait of the vitality model, which is missing in the traditional mortality-rate-focused models, is the characterization of the distribution of individual vitality within a population. With the vitality model organisms start out with equal vitality but they differentiate as the random processes of life favor some over others. A property of the vitality model is its characterization of the probability distribution of vitality within a population as a function of age. Furthermore, in its extended form relating stressors to the rate parameters, it characterizes how specific stressors differentiate the vitality of individuals over age. This allows for a quantitative description of how individuals compete with each other, in the sense of a continuum between scramble and contest competitions for resources as envisioned by Lomnicki (1985). In addition, by characterizing the distribution of vitality within a population the model provides a low-parameter approach to address questions of the impacts of various stressors on natural-selection processes. These ideas are missing in models that define the effects of stressors on the rate of mortality because, by its nature, the rate of mortality is an integrative measure of an entire population's survival experience.

Although the vitality model is more complex than traditional two-parameter models it is less complex than approaches that characterize the organism dynamics at the bioenergetic level. Kooijman's (1993, 1996) work represents a cumulation of studies characterizing the dynamic energy budgets of organisms. In his dynamic energy budget (DEB) model, aging and survival are not state variables and are of secondary importance since the model focuses on growth and reproduction. In dealing with mortality the DEB model equates aging to the accumulation of DNA damages. This approach integrates damage over time and is in the same spirit as the vitality model, which tracks not the accumulation of damage but the loss of vitality, which also is an integration. The two approaches diverge in how the time-cumulative measures, Kooijman's concentration of damage-inducing compounds, Q, vs. vitality, x, are related to mortality. The DEB model assumes that the traditional hazard rate is proportional to Q, while the vitality model as sumes that mortality occurs only when vitality is exhausted.

The DEB model has the desirable property that it connects growth to survival directly through two parameters, an aging-acceleration parameter and a growth-maintenance coefficient. The limitation of the DEB is that survival is expressed at a molecular level. The vitality model, although lacking the detail of any mechanism, admits a variety of possible mechanisms including behavioral, physiological, and molecular. An interesting approach would be to combine the DEB model and the vitality model. Kooijman's molecular damage, Q, instead of being proportional to the hazard rate, could be treated as an additional stressor related to the vitality rate. In this manner the formulations of the molecular DEB approach could be formulated with the non-molecular determinates of survival in the vitality model.

Although the vitality model has several desirable properties over other low-parameter survival models, its utility rests on its ability to fit survival data and add insight to survival processes. To explore these issues the model was applied to a number of case studies.

Survival-curve classification.--The first-level analysis of survival has been to classify survival curves. The traditional approach classifies curves by their shape--either concave, convex, or linear when plotted in terms of the log survival vs. age. The vitality model provides a new classification scheme based on which of the three rate terms dominates the survival dynamics. Curves with mortality occurring late in life by senescence are dominated by the rate of loss of vitality. These r-type curves are equivalent to the classical type I survivorship curve. Examples include mammals and a variety of insects. Survivorship curves with early-age mortality are dominated by a high variability in the rate of loss of vitality. These s-dominated curves are one form of the classical type III survival curves. Examples of s-type curves include a species of bird, fish, and grass. Survivorship curves in which mortality occurs uniformly over life are dominated by accidental mortality. These k-type curves are equivalent to the classical type II survival curves. These curves are rare in nature; an intertidal limpet is given as an example. A fourth class of survivorship curves exhibit a step change in the model parameters corresponding to a lifestage change. These are often classified as type III curves but they are dynamically very different from s-type curves that also have early-age mortality. Examples of the step-type curves are taken from early life-history toxicology studies in which the organism's sensitivity to a toxicant changes with development. Finally, from the studies evaluated here, it appears that the natural stressors shift the survival curve from r to s type while the xenobiotic stressors maintain r-type curves with increasing stressor level.

The four hypotheses

A central focus of this paper is to relate internal and external factors to the vitality model. Four hypotheses were used. An "additive-rates" hypothesis assumes the rate processes, principally the rate of loss of vitality, can be characterized by additive contributions of different environmental and xenobiotic factors. A "linearity" hypothesis assumes that each additive rate term is linearly related to the level of its associated stressor. A third hypothesis assumes that adult body mass correlates with the intrinsic rate of loss of vitality. The fourth hypothesis assumes that the sensitivity of the rate parameters to a stressor depends on body mass, according to the allometric relationship between body mass and the animal structures and bioenergetics affected by the stressor. To the degree that these four assumptions are valid the vitality model becomes a multiple-dose-response model characterizing the effects of mixtures of internal and external factors on survival. The validity and suitability of these hy potheses are explored with a number of case studies involving different organisms and their responses to environmental factors, xenobiotic stressors, and body mass.

Linearity.--Fifteen stressor-organism systems were evaluated and from these examples several basic properties of survival dynamics emerge. For changes in natural ecological factors, the changes in model parameters may be nonlinear, and the optimum environmental conditions should correspond to minimum values of the parameters. Consequently, strong linear relationships between the environmental conditions and the model parameters were not expected, and few were found. In only one out of six cases (Table 2) was the relationship of stressor to r significant at the P = 0.01 level and none of the other parameters' regressions with stressors were significant. Examples include responses of a rotifer, nematodes, and a fruit fly to temperature, a rotifer response to the interval between feeding, and a water flea response to population density. In contrast, the linear regression between xenobiotic stressors and r was significant at the 0.01 level in 10 out of 12 cases and, again, regressions with s and k were generally not significant. Examples include a fruit fly response to a bacillus toxin (Table 4); fish responses to a heavy metal, sulfite waste liquors, and gas supersaturation (Table 3); and fish and bird responses to pesticides (Table 5).

The effects of population density on survival can operate through a variety of mechanisms, which may be represented and quantified by the vitality model. For example, an investigation of the impact of population density on water flea survival suggested two processes--one acting on the vitality rate and a second acting on the accidental-mortality rate. The two processes acted out of phase. The rate of loss of vitality, r, was highest at low densities while the accidental rate, k, was highest at high densities. The combination of these two factors produced the highest survival at an intermediate population density. A third density-related impact on survival is food availability. Fitting the model to a study on the effects of food deprivation in rotifers revealed that changes in the vitality rate mean and variability (Eq. 41) were significantly related whereas the rate parameters themselves were not significantly related to feeding interval. This finding may have implications for quantifying the selection of we ak over strong individuals with resource limitation, and is discussed below (see Selection).

Additivity.--The vitality-rate additivity hypothesis provides a framework in which to extrapolate the impacts of a stressor from one environment to another. It assumes that the vitality rate emerges from a combination of stressors, and that environmental conditions can be expressed as the sum of their individual contributions. To test the hypothesis, survivorship curves of fruit flies exposed to a toxin in one medium were used to characterize the model parameters and the resulting response was predicted in a second medium using the additive-rates hypothesis. Assuming the effect of the toxin is independent of the effect of the environment, then extrapolation to a second environment only requires knowing the toxin response in the first environment and the model parameters in the second environment in the absence of the toxin. The approach worked reasonably well and demonstrated the possibility of using the model to extrapolate laboratory dose-response studies to field conditions and for studying the combined e ffects of different toxins and environmental stressors.

Mass.--The third hypothesis assumes that adult body mass is related to the basic or intrinsic r. Since r is mathematically shown to be a primary determinant of life-span and allometric studies show that life-span is correlated with adult body mass, we expect and find a relationship between r and adult mass that contains the same parameters as the relationship between expected life-span and mass. The two coefficients obtained from a regression of log r vs. log mass were essentially equal to the values found from regressions of log of the expected life-span vs. log mass. The two regressions had very different selection of species.

Under the additivity hypothesis we expect an intrinsic value of r associated with body mass that characterizes the optimum survival curve for the population. To this base any additional components of r would constitute suboptimal conditions. For example, exposure to xenobiotics would constitute an additional component to r which could deviate the relationship between the estimated r and body mass. The intrinsic r should be highly correlated with body mass and deviations from the correlation may be a relative measure of the suboptimality of the environment. The hypothesis that the intrinsic r is correlated with body mass holds for iteroparous species with r- and s-dominated survival dynamics. This correlation did not hold for salmon, which are semelparous.

The fourth hypothesis assumes that body mass is related to the response of r to a stressor, as expressed by the linearity hypothesis, according to the allometric relationship of the animal's structures and bioenergetics affected by the stressor. The hypothesis was explored with survival curves for juvenile salmon exposed to supersaturated water. Larger fish are more sensitive to supersaturation, resulting in r increasing with the square root of fish mass. According to the hypothesis, since supersaturation affects the fish's circulatory system by forming bubbles in the capillaries, the affected systems should have a square-root relationship with body mass. Several elements of the circulatory system have this allometric relationship. At a structural level, the total capillary area of an animal goes up with the square root of body mass while the total surface area of gills increases with a power between 0.4 and 0.9 depending on the species and how gill area is defined. These allometric relationships could imply that larger fish are more susceptible because they have more structures capable of forming bubbles. The square-root relationship could also have a dynamic basis. A body flux of gas, scaling the rate at which a fish becomes supersaturated, can be defined by the ratio of the potential flux of gas across the gills to the transit time of blood through the body. The potential gill flux depends on gill area and thickness measures, and the blood transit time depends on the body volume and blood-pumping rate. The resulting gas body flux scales as the square root of mass such that larger fish become supersaturated more quickly than smaller fish. Since both the structural and dynamic relationships have the same square-root relationship with body mass, either one is a possible explanation for larger fish being more susceptible to supersaturation than smaller fish.

From the above discussion we expect that body mass affects both the intrinsic r and the susceptibility of r to stressors. The two measures are different though, with the intrinsic r inversely correlated with adult body mass, whereas for the effect of stressors r is directly correlated with body mass at age. For supersaturation across a range of fish species of differing adult body mass and mass at age we then expect an equation of the form

r = [[aM.sup.-1/3]] + [[bM.sup.1/2].sub.age] (50)

where [] is the adult mass, [M.sub.age] is the mass at the age of exposure to the supersaturation level x, and a and b are coefficients. This example illustrates that the response of animals to their environments may be complex, but to the degree that the hypothesis holds they are tractable.

Selection.--The vitality model has the potential of characterizing aspects of natural selection. The effect of the stressor on the relative change in r and s, as characterized by [], can be used to quantify the degree a stressor differentiates the vitality distribution within a population. If the high-vitality individuals then have a means to transfer the ability to attain higher vitality to their offspring, exposure to the stressor could result in natural selection of the population to the stressor. In this case the effect of the stressor on natural selection may in part be in characterized by [], and how the vitality of the parents couples to the vitality rate parameters of the offspring.

A mathematical exploration of how the vitality distribution of one generation may affect the distribution in the next is beyond the scope of this paper, but the model suggests three possible pathways for this coupling. The first pathway assumes that the initial vitality of offspring is related to the vitality of parents, such that average parental vitality at reproductive age, [v.sub.ra], affects the initial offspring vitality, [v.sub.0]. Then from Eq. 10 the offspring rate terms r and s are inversely related to the vitality of the parents. More-vital parents would produce offspring with a lower rate of vitality loss, resulting in higher survival of the offspring. The second pathway assumes the coupling is through the initial vitalities of parent and offspring. Here a mathematical representation is more involved since we assume that the parent population has a distribution of [v.sub.0], which violates the assumption of a uniform [v.sub.0] in the population. A non-uniform [v.sub.0] will affect r and s. In par ticular, s would increase in relation to the variance in the distribution of [v.sub.0]. Although the model does not distinguish differences in initial vitality, the members of a population that survive to reproductive age are more likely to have started with a higher initial [v.sub.0]. Then, since stressors affect survival to reproductive age and the resulting distribution of vitality, the stressors may affect the distribution of [v.sub.0] in the reproducing population. Finally, if the [v.sub.0] of offspring are determined by the [v.sub.0] of the parents we have a pathway by which exposure of the parents to a stressor affects initial vitality of the offspring. In these two hypothetical pathways the parental vitality affects offspring r and s through [v.sub.0], and the mechanisms are likely to be genetic since [v.sub.0] is essentially an organism birth trait. For a third pathway assume that the vitality of parents affects offspring survival by altering [rho] and [sigma], which are the dimensional forms of r an d s. Since vitality is an abstract quantity, its dynamics can depend on the state of the environment, which in turn can be altered by the actions of the parents, and in turn the actions can be affected by the parents vitality. Parental care is a possible example of this pathway. A parent with high vitality would be expected to provide high-quality care to the offspring, which then may lower the variability in the rate of loss of vitality in the offspring. These three examples illustrate different pathways through which the vitality of the parents at reproduction can in principle affect the survival of offspring by directly changing r and s. Any coupling through [v.sub.0] is most likely to have a genetic mechanism since [v.sub.0] is essentially a birth trait, but coupling through r and s also could involve behavioral and environmental factors. The details of these pathways are beyond the scope of this paper.

Here the focus is on characterizing how stressors may alter the vitality distribution at some specific age, i.e., the age of reproduction. In general the slope of the relationship between r and s, as quantified by [], characterizes how a stressor increases the variability or differences between individual vitality within a population. Large positive values increase the differences in vitality, making for a greater distinction between high- and low-vitality individuals over age, and small positive or negative values decrease the variability. Thus [] may be useful in characterizing the degree to which a populations adapts to a stressor. For example, if reproducing individuals with higher vitality conveyed lower vitality loss rates to their offspring by one of the three pathways discussed above, then the fitness of these individuals would be preferentially increased over reproducing individuals with lower vitality. In this respect, populations that had a higher value of [] would better a dapt to a stressor than populations with a lower value. Or, conversely, a stressor that produces a low [] would be more difficult to adapt to than a stressor with a high [].

The possibility of quantifying through [] how stressors affect selection was considered by comparing the response of rotifers to food limitation and temperature. Food limitation has a higher value of [] than temperature, so that the vitality distribution within a population tends to spread more with food limitation than with temperature stress. The model suggests that under food stress the number of high-vitality individuals could actually increase compared to conditions with less food limitation. The possible mechanism is that under stress the difference between weak and strong is great, so the few strong individuals actually obtain more food because there is reduced competition from the weaker individuals. With temperature this differentiation does not occur and the vitality decreases with increasing temperature stress. In the cases studied, food limitation had the highest [], suggesting that selection could be greatest involving resource competition in which the success of acquisit ion alters the ability to compete.

The distribution of vitality at reproductive age could have other implications to population selection if the probability of reproduction and offspring survival is a function of the vitality of the parents. In this line of reasoning, the number alive at reproductive age is not a good indicator of the number of offspring produced. A few animals with high vitality could be the major contributors to the future generations, and in this case the distribution of vitality within the population may be more important than the total number alive. The vitality model, in coupling survival with the vitality distribution, is explicitly structured to deal with this issue.

Finally, since vitality is an abstract quantity involving all factors that accumulate over time and affect survival, parents can contribute to the offspring's vitality through genetic and behavioral factors as well as through parental alterations to the environment in which the offspring are born and live. A possible genetic link is evident by a relationship between adult body mass and r. Furthermore since r and s are rate and variance parameters normalized by the initial vitality [v.sub.0], the relationship to genetics may be through [v.sub.0], which by definition does not change with age. Thus parents can influence the r and s of the offspring through a variety of mechanisms. Parents could teach offspring survival behaviors or they could genetically influence the initial vitality of the offspring. Again the vitality model may offer a straightforward and low-parameter approach to investigate these issues and may provide new insights and approaches to life-history studies.


The vitality model, describing mortality in terms of an age-independent component and an age-dependent component through an abstract variable that tracks all age-dependent processes of mortality, offers a powerful and simplified way to characterize complex survival dynamics. As demonstrated with species from plants to humans, the vitality model captures the shapes of survival curves, which suggests that representing vitality loss as a random Brownian movement to a boundary captures the fundamental nature of ageing. This model is mathematically less tractable than classical approaches that describe the rate of mortality, i.e., the hazard rate, but what it loses in tractability it makes up for in its added realism.

To elaborate, vitality is a bounded property and has a clear connection to mortality. Zero vitality equals mortality. Treating the hazard rate as the underlying process is unsatisfying since rates are intangible and unbounded. The difference in the two approaches can be clarified through an analogy to an automobile journey. A hazard-rate approach to mortality, with the rate of change of mortality the fundamental process, is akin to describing an auto's journey through its velocity. Integrating velocity over time gives an accurate description of the journey but it is devoid of the mechanism and limitations of the auto. Describing the journey in terms of the amount of gas the automobile has in its tank and the rate at which it is consumed provides a more mechanistic description of the journey. The mortality rate is analogous to the auto's velocity and vitality is analogous to its fuel. When we model the dynamics of a journey, through life or along a road, viewing that journey through its fuel consumption bring s us closer to the process, and questions we ask about the trip in terms of fuel usage are likely to be more fundamental than if we only consider velocity. Considering survival in an analogous manner brings us closer to the organism, how it works, and its limitations. The vitality model, then, allows us to address issues of how an organism's vitality--its fuel--is used over its journey through life. The four hypotheses of the model provide a mathematically tractable description of how the external factors alter use of vitality. An additivity hypothesis says that the rate of loss of vitality can be attributed to individual factors. The linearity hypothesis says each individual rate is linear with its factor. One allometric hypothesis says an intrinsic rate of vitality loss is proportional to organism mass and a second allometric hypothesis says mass alters the rate of vitality loss from stressors. Mathematically this system of equations, which can be characterized with as few as four parameters, describes popu lation survival and the distribution of vitality within the population. With these capabilities we can potentially deal with problems of natural selection and ecotoxicology within a single framework.

The degree of success of the model will, to a large degree, depend on how biologically realistic and representative the model's assumptions and hypotheses are in reflecting fundamental aspects of nature. They have been tested in this paper with selected case studies, but if the fits obtained were simply the result of having enough parameters, which have little biological meaning, then the vitality model is just another mathematically challenging survival model. However, if the vitality model is a robust and simple description of mortality, then it should provide further ecological insights and synthesis. Most likely the model falls between these end points. In simple situations, describing the rate of vitality loss will be appropriate, but the approach will be inadequate for detailed endeavors where individual bioenergetics and behavior must be addressed. Because the predictions from the model are mathematically clear, the additivity, linearity, and allometric hypotheses can be readily tested in experiments.

Perhaps the most interesting feature of the vitality model, which has only been briefly addressed, is its description of how stressors differentiate the vitality of individuals within a population. The differentiation of strong and weak individuals is a fundamental concept of natural selection, and the vitality model--characterizing this in terms of the variability in the vitality loss rate--provides a tractable way to model and quantify the effect of natural and xenobiotic stressors on natural selection. These issues will be addressed in a future paper.


This work was supported by the Army Corps of Engineers. I wish to thank John M. Emlen for his stimulating conversations and constant perspicacity and I wish to thank David Salinger and Owen Hamel for their discussions and assistance.

(1.) E-mail:


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Publication:Ecological Monographs
Date:Aug 1, 2000

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