Theory of marine communities: competition, predation, and recruitment-dependent interaction strength.
Ever since the classic papers of Connell (1961), Paine (1966), and Dayton (1971), the rocky intertidal zone has been a model system for the experimental study of ecological communities. Much of this work has consisted of species removal experiments. In these studies, the ecological role of a species is inferred by comparing control plots to plots from which the species has been experimentally added or excluded. One lesson that can be learned from this experimental work is that the local impact of particular species on communities varies a great deal depending on the timing, micro-habitat, and geographical location of the study being conducted (see reviews in Underwood and Denley 1984, Menge and Farrell 1989, Menge 1991, Menge et al. 1994, Paine 1994).
Many of the animals that inhabit these communities have pelagic (oceangoing) larvae. Spatially and temporally variable recruitment of these species has emerged as an important cause of the variability in community dynamics found in experiments (e.g., Underwood et al. 1983, Underwood and Denley 1984, Connell 1985, Roughgarden 1986, Fairweather 1988, Menge 1991, Menge et al. 1994, Booth and Brosnan 1995, Robles 1997). As a result, models of benthic communities have been modified to account for the effect of recruitment variation on the intensity and importance of species interactions (e.g., Menge and Sutherland 1987, Gaines and Lafferty 1995). In most of these models, larval supply is assumed to be independent of benthic community dynamics. When explaining differences among nearby local communities, such as exposed and protected habitats, this assumption is probably appropriate. Larvae can spend weeks to months in the plankton, allowing for considerable alongshore mixing. Recruits to nearby sites are drawn from a common larval pool, so differences in rates of larval supply to nearby communities are not likely to be affected by between-site differences in larval production rates.
At the regional scale, however, the sizes of larval populations are determined in part by adult stock sizes. Stock sizes, in turn, are affected by benthic interactions, such as competition and predation. To explain community structure patterns at this scale, we need to take into account two-way coupling between planktonic and benthic communities. For example, Gaines and Lafferty (1995) modeled two-species competition for space in a local, open community. In their model, larvae of both species are continually supplied to the adult community from an external source. As a result, both species always coexist. In contrast, Iwasa and Roughgarden (1986) modeled a regional competition community in which larvae are produced by adults in local communities. Competition for space occurs exactly as in Gaines and Lafferty's model, but the species do not always coexist. Rather, each species must have a higher larval production to larval mortality ratio than the other, in at least one subpopulation. Thus, community structure at the regional scale may not be well characterized by models for open communities.
From this regional perspective, benthic community structure is affected by pelagic environmental conditions, because these conditions determine the probability that larvae will successfully recruit to the adult habitat. In particular, physical oceanographic processes may transport larvae great distances. Some of these processes carry larvae away from shore, inhibiting recruitment (Roughgarden et al. 1988, Gaines and Bertness 1992). Others transport larvae onshore, facilitating recruitment (Shanks 1983, Farrell et al. 1991, Pineda 1991). Can these transport processes influence benthic species interactions? If so, how? To answer questions like these, we need to unite larval dynamics and benthic species interactions in a common theoretical framework.
A CASE STUDY: THE NORTHEAST PACIFIC
The North American Pacific coast is one of the most extensively studied marine environments in the world, both oceanographically and ecologically. This makes it an excellent region for which to construct a framework linking oceanographic processes to benthic community dynamics. We focus on the California Current System, which extends from southern Canada to Baja California, Mexico (U.S. GLOBEC 1994, Mann and Lazier 1996). Cross-shelf circulation in this region, particularly in the spring and summer, is dominated by upwelling. Upwelling occurs when strong equatorward winds, in conjunction with the coriolis effect, cause a surface layer of water to move offshore. Nearshore, this water is replaced by cold, saline, nutrient-rich water that emerges from depth (Pond and Pickard 1995).
Meroplanktonic larvae become entrained in these offshore-moving surface currents and accumulate at fronts where this upwelled water meets the offshore, southward-flowing water of the California Current (Roughgarden et al. 1988, 1994, Wing et al. 1995, Grantham 1997). These larvae must return to the coast in order to recruit to adult populations. This occurs primarily during relaxation events, when upwelling-favorable winds weaken and the front moves onshore (Roughgarden et al. 1988, 1991, Farrell et al. 1991, Wing et al. 1995). Spatial variation in recruitment also depends on upwelling intensity. Higher recruitment tends to occur where offshore transport is weaker and fronts are more nearshore (Ebert and Russell 1988, Miller 1992, Wing et al. 1995, Grantham 1997).
Intertidal communities in the northern part of this region have been studied for decades. Clear geographical trends emerge from this work. On the outer coasts of Washington and Oregon, species removal experiments indicate that the allocation of space among mussels and barnacles is determined primarily by species interactions, particularly competition and predation (Paine 1966, 1974, Farrell 1989, 1991, Menge et al. 1994). In contrast, studies in central and northern California indicate that rates of larval supply largely determine adult abundances, whereas species interactions have smaller effects (Gaines and Roughgarden 1985, Wing et al. 1995, Grantham 1997; T. M. Farrell, unpublished data, J. S. Pearse, unpublished data).
These geographical differences in community structure correspond to marked differences in upwelling intensity. Upwelling off the coasts of Oregon and Washington is weak: the upwelling season is short, the offshore advection rate is low, fronts are nearshore, and relaxation events are frequent. In contrast, upwelling off central and northern California is strong: the upwelling season is long, the offshore advection rate is high, fronts are far from shore, and relaxation events are infrequent (Parrish et al. 1981, Huyer 1983, U.S. GLOBEC 1994). This association between oceanographic regime and community structure inspired the hypothesis that higher larval supply in Oregon and Washington, promoted by weaker upwelling, is responsible for the community structure differences (Roughgarden et al. 1988). High larval supply makes free space scarce, so competitive and predator-prey interactions regulating access to that space should be more important in Oregon and Washington than in central and northern California.
Our goal in this study is to characterize the link between larval transport processes and the strength of species interactions in the benthic habitat. In particular, we wish to determine whether there are theoretical grounds to support the hypothesis that there is a latitudinal gradient in interaction strength in the northeast Pacific rocky intertidal, and that it is caused by a latitudinal gradient in upwelling intensity (Roughgarden et al. 1988). Toward that end, we review a previously proposed model coupling hierarchical competition for space with oceanic larval transport (Connolly and Roughgarden 1998), and we develop a model coupling predation on space-limited prey with oceanic larval transport. These models can be used to determine the effects of changes in upwelling intensity on competitive and predator-prey interaction strengths. We investigate three definitions of interaction strength:
1) Per capita interaction strength. We define per capita interaction strength as the effect of an individual of one species on the per capita population growth rate of a competitor, prey, or predator.
2) Population interaction strength. We define population interaction strength as the net effect of a population of one species on the per capita population growth rate of another (cf. Navarrete and Menge 1996). As upwelling intensity changes, species abundances, as well as per capita interaction strengths, may change. Population interaction strength captures the combined effects of changes in abundance and changes in per capita interaction strength.
3) Local interaction strength. We define local interaction strength as the effect of a species' removal from a local, open community (such as an experimental plot) on the abundances of the remaining species (e.g., Paine 1980).
To determine per capita and population interaction strengths, Lotka-Volterra approximations to the models are developed. The terms in these approximations are functions of parameters in the original model, including larval transport parameters. These models can be written in the form
d[N.sub.i](t)/dt = [[r.sub.i] + [summation over j] [[Lambda].sub.ij][N.sub.j](t)][N.sub.i](t) (1)
where [r.sub.i] is the intrinsic rate of increase of species i, [N.sub.j] is the abundance of species j, and [[Lambda].sub.ij] is the per capita interaction strength of species j on species i. The population interaction strength of species j on i, then, is simply
[[Lambda].sub.ij] = [[Lambda].sub.ij][N.sub.j]. (2)
In this study, [N.sub.j] is evaluated at the interior equilibrium, i.e., the equilibrium at which the species coexist.
To determine local interaction strengths, we develop submodels that simulate population dynamics in two types of plots: those from which competitors or predators have been excluded and those in which competitors or predators are present. Local interaction strength can be measured as the relative difference in abundances between these plots (Paine 1992):
[Mathematical Expression Omitted] (3)
where [Mathematical Expression Omitted] is the density of species i in experimental plots, and [Mathematical Expression Omitted] is density of species i in control plots. In this paper, experimental plots are considered to be those from which competitors or predators are excluded, and control plots are considered to be those in which species interactions are allowed to proceed normally. We derive the local interaction strength of competitive dominants on subordinates and of predators on prey. We do not derive the effect of prey on predators, because this would correspond to an experiment in which all prey were removed from experimental plots. Clearly, all predators in such plots would die before reaching adult size.
The hypothesis under investigation pertains specifically to the larval transport processes associated with upwelling (Roughgarden et al. 1988). Therefore, in this paper, the term upwelling refers specifically to offshore advection rates and front positions, and not to other upwelling-mediated environmental changes that might also affect benthic communities (e.g., water temperature or phytoplankton productivity). This study is not a comprehensive investigation of the effects of upwelling on benthic communities. Rather, it is a targeted study of a particular possible causal link between upwelling and benthic community structure.
To limit the number of figures presented, gradients in offshore advection rate are shown, but corresponding gradients in front position are not. However, the qualitative relationships hold with respect to both parameters: where a quantity increases with offshore advection rate, it also increases as the front moves offshore, etc. Because increasing upwelling is associated with increasingly offshore fronts as well as stronger offshore advection, the magnitude of gradients shown in figures will tend to be conservative with respect to the overall effect of upwelling.
The models extend a previously developed one-dimensional, single-species model for barnacles, coupling benthic population dynamics with oceanic larval transport (Alexander and Roughgarden 1996). In that model, larvae are distributed between the coast and an offshore front. At the coast, larvae are spawned by adults and recruit to free space in the adult habitat. Analysis of the model confirmed that the increasing offshore advection and further offshore fronts associated with increasing upwelling produce lower recruitment and adult abundances. Simulations of time-varying advection and front position were used to explore the effects of relaxation events on population dynamics. This produced temporal fluctuations in recruitment and adult abundances, but the overall relationships among upwelling, recruitment, and adult abundance were the same: the less frequent relaxation events associated with strong upwelling led to lower mean recruitment and adult abundances (Alexander and Roughgarden 1996). In this paper, we focus on the constant advection, stationary front case. This makes analytical derivations of interaction strengths possible. As a result, the response of interaction strength to upwelling can be characterized by simply plotting a function against advection and front position, rather than by running a series of lengthy numerical simulations. This allows us to explore a much wider range of parameter values. However, because relaxation events are not incorporated, this study probably underestimates the sensitivity of community structure to changes in upwelling intensity (Alexander and Roughgarden 1996).
Competition for space in the northeast Pacific rocky intertidal zone can be strongly hierarchical, particularly among mussels and barnacles. The strong shells, large size, and overgrowth ability of the California sea mussel, Mytilus californianus, enable it to outcompete barnacles, algae, and other mussels for primary space (Paine 1966, 1974). Among acorn barnacles, competitive dominants may settle on, overgrow, or undercut subordinates (Connell 1961, Farrell 1991). To determine the effect of upwelling on benthic competition, we analyze a model first formulated elsewhere (Connolly and Roughgarden 1998). The model approximates the interactions previously described by allowing the dominant to settle on, and replace, the subordinate (cf. Hastings 1980, Tilman 1994, Dial and Roughgarden 1998). In contrast, the subordinate can settle only on vacant space. Settlement is a mass-action process for both populations, an assumption that is probably more appropriate for acorn barnacles than for leaf barnacles or mussels (Seed 1976, Petersen 1984, Gaines and Roughgarden 1985, Satchell and Farrell 1993). Adult mortality occurs at a density-independent rate. This assumption may not hold when free space is scarce (Gaines and Roughgarden 1985), but the incorporation of density-dependent mortality does not appear to qualitatively change the results (see Discussion). With these assumptions, adult dynamics are
d[N.sub.1](t)/dt = [c.sub.1][L.sub.1](0, t)[F(t) + [a.sub.2][N.sub.2](t)] - [[Mu].sub.1][N.sub.1](t) (4a)
d[N.sub.2](t)/dt = [c.sub.2][L.sub.2](0, t)F(t) - [[Mu].sub.2][N.sub.2](t) - [c.sub.1][L.sub.1](0, t)[a.sub.2][N.sub.2](t) (4b)
where the subscript 1 denotes the dominant and 2 denotes the subordinate. [N.sub.i](t) is the adult population size per unit length of coastline at time t, [L.sub.i](0, t) is the abundance of larvae at the coast per unit surface area of the ocean at time t, F(t) is benthic free space per unit length of coastline, [c.sub.i] is the per capita rate at which larvae at the coast settle onto free space, [[Alpha].sub.i] is the basal area of an adult, and [[Mu].sub.i] is the adult mortality rate. Unoccupied space fluctuates according to
F(t) = A - [a.sub.1][N.sub.1](t) - [a.sub.2][N.sub.2](t) (4c)
where A is total benthic habitat area per unit length of coastline. Larvae move between the coast and an offshore front by offshore transport (advection) and turbulent mixing (diffusion). Advection is proportional to larval abundance, and diffusion is proportional to the spatial gradient in larval abundance (Okubo 1980, Holmes et al. 1994, Alexander and Roughgarden 1996). The larval mortality rate is independent of density (Alexander and Roughgarden 1996, Strathmann 1996). Thus, larval dynamics in space and time are as follows:
[Mathematical Expression Omitted]. (5)
[L.sub.i](x, t) is the larval abundance per unit surface area of ocean at offshore location x at time t, [u.sub.i] is the offshore larval advection rate, [k.sub.i] is the larval diffusion coefficient, and [v.sub.i] is the larval mortality rate. Values of x range from 0 at the coast to [x.sub.f] at the offshore front. The front is a reflecting boundary for both species:
[Mathematical Expression Omitted]. (6)
The flux of larvae through the coastal boundary is the difference between the rate at which larvae enter the larval population from the benthic habitat (larval production) and the rate at which they leave the larval population to enter the benthic habitat (settlement):
[Mathematical Expression Omitted] (7a)
[Mathematical Expression Omitted]. (7b)
Larvae are produced by adults at density-independent rate [m.sub.i], and leave the population by settling on available space in the adult habitat. Again, the subordinate can settle only on vacant space, whereas the dominant can also settle on space occupied by the subordinate.
The ochre seastar, Pisaster ochraceus, is the principal predator in the rocky intertidal communities of central California and the Pacific Northwest (Paine 1966, 1974, 1976, Gaines and Roughgarden 1985, Menge et al. 1994, Possingham et al. 1994, Navarrete and Menge 1996). Like barnacles, P. ochraceus has planktonic larvae (Strathmann 1987), so we assume that both predator and prey populations have a planktonic larval stage. Prey recruit only to free space, but predators recruit anywhere in the adult habitat. We have been unable to find studies that characterize the functional response of seastars. Therefore, we assume a linear functional response. This keeps the model simple and increases the comparability of the results with experimental estimates of interaction coefficients for intertidal predators, which also assume a linear functional response (Navarrete and Menge 1996). As in the competition model, adult mortality rates are density independent. With these assumptions, adult dynamics are
d[N.sub.1](t)/dt = [c.sub.1][L.sub.1](0, t)F(t) - [[Mu].sub.1][N.sub.1](t) - e[N.sub.1](t)[N.sub.2](t) (8a)
d[N.sub.2](t)/dt = [c.sub.2]A[L.sub.2](0, t) - [[Mu].sub.2][N.sub.2](t) (8b)
[TABULAR DATA FOR TABLE 1 OMITTED]
F(t) = A - [a.sub.1][N.sub.1](t). (8c)
The subscript 1 denotes the prey and 2 denotes the predator. Otherwise, [N.sub.i](t), [L.sub.i](0, t), F(t), [c.sub.i], A, [a.sub.i], and [[Mu].sub.i], have the same meanings as in Eq. 4. The rate at which predators encounter and consume prey is designated e. Because population sizes are in units of abundance per unit length coastline, the encounter rate is the proportion of habitat area in one unit length of coastline (A) that is searched per unit time. For instance, e = 0.1 means that, in one unit of time, a predator searches 10% of the benthic habitat in a unit length of coastline. Larval population dynamics of both species follow Eq. 5, with frontal boundary conditions given by Eq. 6. Coastal boundary conditions are
[Mathematical Expression Omitted] (9a)
[Mathematical Expression Omitted] (9b)
Here, [b.sub.2] is the number of predator larvae produced per prey consumed. Other parameters and state variables have the same meanings as in Eq. 7.
Parameter values used for the figures are presented in Table 1. We used oceanographic parameter values that are reasonable for the California Current System (Okubo 1971, Alexander and Roughgarden 1996), and we assumed that the larvae of all species are subject to the same transport processes, i.e., [u.sub.1] = [u.sub.2] = u and [k.sub.1] = [k.sub.2] = k. Where possible, biological parameters for the dominant competitor were based on field studies of the barnacle Balanus glandula, parameters for the subordinate were based on the barnacles Chthamalus dalli and Chthamalus fissus, and parameters for the predator were based on the seastar Pisaster ochraceus. Parameter values for the prey were intermediate between Balanus and Chthamalus.
Because a purely analytical study of the responses of interaction strengths to upwelling was not possible, we explored a range of values around our parameter estimates to insure that the results reported in this paper are not sensitive to the choice of a particular parameter set. We varied each parameter individually, holding the others at the baseline values reported in Table 1. We identified the largest and smallest parameter values within this range for which the interacting populations coexisted across the range of advection rates considered (50-150 m/h), and we verified that the results did not change qualitatively with these parameter values. Where we report that a quantity increases with, decreases with, or is insensitive to upwelling intensity, that relationship holds for all parameter sets considered. Similarly, the accuracy of the approximations illustrated in the figures is characteristic of all parameter sets that we explored.
To our knowledge, field estimates of larval mortality rates have not been made for these species. The only published estimates of barnacle larval mortality are for northern Atlantic species (Pyefinch 1948), an estimate (5% per day or 0.002/h) that has been used previously to model barnacle population dynamics (Possingham and Roughgarden 1990, Alexander and Roughgarden 1996). We know of no such field estimates for seastars. Therefore, we used 5% per day as a baseline for all species and considered a range of larval mortality rates from 1% to 10% per day (0.0005/h to 0.004/h).
The settlement coefficient of Balanus has been calculated from simultaneous field measurements of larval concentrations, benthic free space, and settlement rates (Gaines et al. 1985, Possingham and Roughgarden 1990). To our knowledge, no field measurements of settlement coefficients have been made for Chthamalus or Pisaster. Therefore, we used the settlement coefficient of Balanus (0.02/h) as a baseline for the dominant competitor, subordinate competitor, and prey, and we considered variation of an order of magnitude in either direction of this value. Pisaster larvae take 76-228 d to metamorphose in the laboratory (Strathmann 1978), whereas Balanus and Chthamalus take [approximately]15-20 days (Brown and Roughgarden 1985, Miller et al. 1989). This means that the fraction of the Pisaster larval population that is competent to settle, assuming 5% mortality per day, will be anywhere from 1/10 to 1/10000 as large as the fraction of the barnacle larval population that is competent. Therefore, we considered a wide range of predator settlement coefficients based on these fractions (0.002/h to 0.000002/h). We used an intermediate value (0.0001/h) for the figures presented in this paper.
Gaines and Roughgarden (1985) found that mortality rates for adult Balanus from sources other than seastar predation are [less than]20% per week, and average [approximately]5% per week, independent of age. Similarly, adult mortality rates estimated from life tables for Chthamalus fissus are [approximately equal to]5% per week (Hines 1976). Therefore, we considered a range of mortality rates from 2% to 20% per week (0.0001/h to 0.001/h) for the prey, dominant competitor, and subordinate competitor populations. Field data on Pisaster ochraceus (Menge 1975) indicate an annual adult mortality rate of only 5%, or [approximately]0.00001/h (Ebert 1996). We took this as a baseline and considered mortality rates from 1% to nearly 90% per year.
Possingham and Roughgarden (1990) describe a procedure for calculating average larval production rate, and estimate this from field data on Balanus glandula (Hines 1976). We used this estimate (0.024/h) for our dominant competitor, and we used the same procedure to estimate larval production rates for the subordinate (0.15/h) from the same study (Hines 1976). To determine whether differences in larval production rates between populations affected our results, we considered a range of larval production rates (0.01/h to l/h) for both species.
An age-averaged estimate of predator larval production per prey consumed, [b.sub.2], requires (1) the fraction of the predator population that is reproductively mature, (2) the average energy content of prey, (3) the proportion of that energy that a mature seastar converts into eggs, (4) the energy content of eggs, (5) fertilization success, and (6) survivorship from zygote to larval stage. We estimated these components as follows:
1) Assuming 95% annual survival (Ebert 1996) and reproductive maturity at 5 yr (Menge 1975), [approximately]75% of the postsettlement predator population is mature.
2) We assume that prey tissue contains 18.8 kJ/g dry mass, a reasonable value for barnacles (Harrold 1981). At a mortality rate of 0.0005/h (Gaines and Roughgarden 1985), the average prey is [approximately]2 mo old. Because neither Chthamalus nor Balanus matures before this age, upper bounds for age-averaged body mass can be taken as the smallest body mass for which brood masses are reported ([approximately]0.1 mg for Chthamalus and 0.8 mg for Balanus), yielding energy contents of [approximately]1.67 J and 14.65 J, respectively. We used an intermediate value, 4.19 J (one calorie), for our estimate of a baseline value for [b.sub.2].
3) Pisaster giganteus in central California allocates [approximately]10% of its annual energy intake to reproduction (Harrold 1981). However, the majority of species on which P. giganteus preys provide much more energy per time spent feeding than do barnacles (Harrold 1981). Because seastars may allocate less energy to reproduction as their energy intake declines (Nimitz 1971, Harrold 1981), we used a smaller value, 5%, for our estimate of [b.sub.2].
4) The energy content of predator eggs can be estimated by multiplying their volume, 2 x [10.sup.6] [[micro]meter].sup.3] for P. ochraceus (Strathmann and Vedder 1978), by their organic density, for which 400 [[micro]gram]/[mm.sup.3] is a reasonable value for echinoderms (Turner and Lawrence 1979). Assuming that the allocation of this organic matter to protein, lipid, and carbohydrate is 60%, 35%, and 5%, respectively (Turner and Lawrence 1979), the energy content of a predator egg is [approximately]0.02 J.
5) Estimates of fertilization success in free-spawning marine invertebrates vary from nearly 100% to as low as 0.01% (Denny and Shibata 1989, Sewell and Levitan 1992). We used an intermediate value of 1%.
6) We have found no measurements of postfertilization mortality in the field, so we assume that this rate is comparable to that of larval mortality, 5% per day. If it takes 5 d for zygotes to develop into larvae (Strathmann 1987), then [approximately]75% of zygotes become larvae.
From these values, we obtain 0.06 predator larvae produced per prey consumed. Because of the uncertainties involved in estimating several of the described quantities, we considered a range of values from 0.0001 to 1 larva per prey.
Although we have been unable to find measurements of the rates at which Pisaster ochraceus locates and consumes intertidal barnacles, we can base an estimate on studies of other species. Pisaster giganteus consumes acorn barnacles (mostly Balanus aquila, B. nubilus, and B. crenatus) at a rate of about 5 barnacles/d (Harrold 1981). Adult sizes of these barnacles range from [approximately]10 to 130 mm diameter (Newman and Abott 1980). If we assume that the average prey basal area is 0.002 [m.sup.2] (i.e., 50 mm diameter), then this yields an encounter rate of 0.000002 [m.sup.2]/h or e = 0.00000002/h. This rate corresponds to the extreme case in which there is 100% cover, and the predator is eating continuously as it moves through the habitat. Because seastars may move faster through habitat in which prey are less abundant, we treated this value as a lower bound on the encounter rate and considered values up to six orders of magnitude higher, 2 [m.sup.2]/h or e = 0.02/h. We used an intermediate value, e = 0.00001/h, for the figures presented in this paper (Table 1).
Scaling argument. - The benthic dynamics can be characterized with a two-equation scaling approximation (Alexander and Roughgarden 1996, Connolly and Roughgarden 1998). This approximation takes advantage of the fact that larval dynamics occur on a faster time scale (days to weeks) than the growth and maturation of newly settled larvae (months). Because newly settled larvae are very small, they occupy a negligible amount of space in the benthic habitat and do not reproduce. Therefore, over the time scale at which larval populations equilibrate, free space and larval production by adults are relatively constant. These observations allow us to provisionally treat adult population sizes as constants and to solve Eqs. 5-7 for the larval equilibria-at the coast:
[Mathematical Expression Omitted] (11a)
[Mathematical Expression Omitted] (11b)
where [q.sub.1] and [q.sub.2] are the species-specific water column properties
[Mathematical Expression Omitted]. (12)
Here, [q.sub.i] represents the combined effects of larval mortality, offshore advection, and turbulent mixing on the abundance of larvae at the coast. Substituting Eq. 11 for [L.sub.1](0, t) and [L.sub.2](0, t) in Eq. 4, adult dynamics become
[Mathematical Expression Omitted] (13a)
[Mathematical Expression Omitted] (13b)
Although this approximation has been justified verbally and used previously (Alexander and Roughgarden 1996, Connolly and Roughgarden 1998), its quantitative accuracy has never been verified. To do this, we numerically analyzed the approximation (Eq. 13) and the full model (Eqs. 4-7), and we compared their dynamics (see Hirsch 1988 for an introduction to numerical analysis). The two models agree extremely closely [ILLUSTRATION FOR FIGURE 1 OMITTED], so we use the scaling approximation in lieu of the full model in all subsequent analyses. Hereafter, we refer to the scaling approximation as the "original model" to avoid confusion with the Lotka-Volterra approximation.
Lotka-Volterra approximation. - Because the model is now in the form of coupled ordinary differential equations, the benthic dynamics can be approximated with Lotka-Volterra competition equations (Eq. 1). This approximation, detailed in Appendix A, yields the following:
[r.sub.1] = ([m.sub.1] = [[Mu].sub.1])[p.sub.1]/[m.sub.1] (14a)
[Mathematical Expression Omitted] (14b)
[[Lambda].sub.11] = 2[c.sub.1][q.sub.1][a.sub.1][([m.sub.1] - [[Mu].sub.1]).sup.2]/[m.sub.1] (14c)
[Mathematical Expression Omitted] (14d)
[[Lambda].sub.12] = 0 (14e)
[Mathematical Expression Omitted] (14f)
where [p.sub.i] is the single-species positivity requirement from the scaling approximation,
[p.sub.i] = -[[Mu].sub.i] + 2A[c.sub.i]([m.sub.i] - [[Mu].sub.u])[q.sub.i] (15)
(Alexander and Roughgarden 1996). If p [less than or equal to] 0, the population cannot persist, even in the absence of its competitor.
Numerical analysis demonstrates that the Lotka-Volterra approximation agrees closely with the original model, even far from the interior equilibrium [ILLUSTRATION FOR FIGURE 2 OMITTED]. The approximation is accurate throughout the parameter space that we explored because the terms in the approximation are functions of parameters in the original model. Thus, the terms in Eq. 14 change as the parameters of the original model change.
Lotka-Volterra competition equations are often written in a form that includes terms for carrying capacities and competition coefficients rather than per capita interaction strengths. These terms can be obtained from Eq. 14. Carrying capacity is a population's equilibrium abundance in the absence of its competitor,
[K.sub.i] = [r.sub.i]/[[Lambda].sub.ii] (16)
and the competition coefficient is the intensity of interspecific competition relative to the intensity of intraspecific competition,
[[Alpha].sub.ij] = [[Lambda].sub.ij]/[[Lambda].sub.ii]. (17)
Fig. 3 plots intrinsic growth rates, carrying capacities, and competition coefficients as functions of offshore advection. Intrinsic growth rates and carrying capacities decrease as upwelling increases. Competition coefficients are comparatively insensitive to upwelling, although [[Alpha].sub.21] actually declines slightly in Fig. 3C.
Adult abundances. - Adult abundances equilibrate at
[Mathematical Expression Omitted] (18a)
[Mathematical Expression Omitted]. (18b)
As we have noted elsewhere, percent cover of the dominant decreases as upwelling intensity increases, but percent cover of the subordinate actually increases with upwelling intensity (Connolly and Roughgarden 1998). This is illustrated in Fig. 4.
Interaction strengths. - The per capita strengths of interspecific competition can be obtained directly from the Lotka-Volterra approximation according to Eq. 1. Obviously, because there is no effect of the subordinate on the dominant (Eq. 14e), [[Lambda].sub.12] is independent of upwelling intensity. However, the per capita interaction strength of the dominant on the subordinate (Eq. 14f) decreases in magnitude as upwelling increases [ILLUSTRATION FOR FIGURE 5A OMITTED].
Population interaction strengths (Eq. 2), evaluated at the interior equilibrium, are
[[Lambda].sub.12] = 0 (19a)
[[Lambda].sub.21] = -[[Lambda].sub.21]/[[Lambda].sub.11] [r.sub.1]. (19b)
Like the per capita interaction strengths, there is no effect of the subordinate on the dominant, but the population interaction strength of the dominant on the subordinate decreases in magnitude as upwelling increases [ILLUSTRATION FOR FIGURE 5B OMITTED].
Local interaction strengths (Eq. 3) can be obtained by simulating small-scale species removal experiments. We can explore the effect of upwelling on these experiments by recognizing that small experimental and control plots are essentially open systems for species with planktonic larvae. Larval supply to these plots is determined at the regional scale, and is a function of larval transport processes (Eq. 11). Appendix B describes a submodel that simulates experimental removals of competitive dominants. The submodel includes small experimental and control plots nested within a regional system. In the experimental plot, dominants are removed and prevented from colonizing. In the control plot, dominants are present and are allowed to recruit normally. From subordinate abundances in these two plots, local interaction strength can be calculated according to Eq. 3. We can then determine how this quantity varies with characteristics of the regional system, such as upwelling intensity. At equilibrium, local interaction strength is evaluated as follows:
[Mathematical Expression Omitted]. (20)
This represents the combined effect of (1) the additional space made available to the subordinate by removal of the dominant and (2) the subordinate's ability to colonize that additional space (its recruitment rate). Fig. 5C illustrates that local interaction strength decreases as upwelling increases. In practice, experimentalists often cannot determine when or whether abundances in their plots have reached equilibrium. Therefore, we ran numerical simulations to verify that this trend holds when interaction strength is measured before the plots reach equilibrium. In the simulations, experiments were initiated at the same time in systems with differences in upwelling intensity, and interaction strength was plotted as a function of time until all plots equilibrated. The simulations confirm that interaction strengths are lower in regions of stronger upwelling, even when interaction strength is measured before any or all of the plots have reached equilibrium (results are not shown).
Scaling argument. - Following the procedure outlined for the competition model, benthic predator-prey dynamics can be approximated with
[Mathematical Expression Omitted] (21a)
[Mathematical Expression Omitted] (21b)
[Mathematical Expression Omitted] (22a)
[Mathematical Expression Omitted] (22b)
Here, [q.sub.1] and [q.sub.2] are the species-specific water column properties of Eq. 12. As in the competition model, numerical analysis indicates that the scaling approximation agrees closely with the full model [ILLUSTRATION FOR FIGURE 6 OMITTED], so we refer to the scaling approximation as the "original model" hereafter.
Lotka-Volterra approximation. - Approximating the benthic dynamics with Lotka-Volterra predator-prey equations (see Appendix A) yields
[Mathematical Expression Omitted] (23a)
[r.sub.2] = -[[Mu].sub.2] (23b)
[Mathematical Expression Omitted] (23c)
[[Lambda].sub.22] = 0 (23d)
[[Lambda].sub.12] = -e (23e)
[[Lambda].sub.21] = 2A[c.sub.2][q.sub.2]e[b.sub.2]/(2A[c.sub.2][q.sub.2] + 1). (23f)
As with the competition model, the approximation agrees extremely closely with the original model, even far from the interior equilibrium [ILLUSTRATION FOR FIGURE 7 OMITTED].
This Lotka-Volterra model is often written with prey carrying capacity and a predator production coefficient instead of the interaction strengths [[Lambda].sub.11] and [[Lambda].sub.21]. Prey carrying capacity is the prey equilibrium in the absence of the predator:
K = [r.sub.1]/[[Lambda].sub.11]. (24)
The predator production coefficient is the number of new adult predators produced per prey consumed:
[Beta] = - [[Lambda].sub.21]/[[Lambda].sub.12]. (25)
Fig. 8 plots the prey intrinsic growth rate, prey carrying capacity, and predator production coefficient as functions of offshore advection. All decrease as upwelling increases.
Adult abundances. - Adult populations equilibrate at
[Mathematical Expression Omitted] (26a)
[Mathematical Expression Omitted]. (26b)
Note that the prey equilibrium depends on the predator's life history parameters, but not its own. In particular, when the predator has planktonic larvae, the prey equilibrium depends on the larval transport parameters of the predator, because [[Lambda].sub.21] is a function of [q.sub.2] (Eq. 23f). As a result, prey percent cover actually increases as upwelling increases, because strong upwelling inhibits predator recruitment [ILLUSTRATION FOR FIGURE 9A OMITTED]. In contrast, predator abundance decreases as upwelling increases [ILLUSTRATION FOR FIGURE 9B OMITTED].
Interaction strengths. - The per capita strengths of predator-prey interactions can be obtained directly from the Lotka-Volterra approximation according to Eq. 1. The per capita effect of predators on prey (Eq. 23e) is independent of upwelling intensity. The per capita effect of prey on predators (Eq. 23f) decreases as upwelling increases [ILLUSTRATION FOR FIGURE 10A OMITTED].
Population interaction strengths (Eq. 2), evaluated at the interior equilibrium, are
[Mathematical Expression Omitted] (27a)
[Mathematical Expression Omitted]. (27b)
In contrast to the per capita effects, the population interaction strength of predators on prey decreases in magnitude as upwelling increases [ILLUSTRATION FOR FIGURE 10B OMITTED], but the population effect of prey on predators is independent of upwelling intensity.
As with the competition model, a submodel simulating predator removal experiments (Appendix B) provides local interaction strength at equilibrium:
[Mathematical Expression Omitted]. (28)
Fig. 10C illustrates that the local interaction strength of predators on prey decreases as upwelling increases. Again, numerical simulations indicate that this trend holds even when interaction strength is measured before prey abundances have equilibrated (results are not shown).
We have developed complex, process-based models coupling benthic adult and pelagic larval dynamics, and have approximated them with simpler, two-equation models for benthic dynamics. These simplified models were approximated with Lotka-Volterra competition and predator-prey equations whose terms are functions of the parameters from the original model. The Lotka-Volterra approximations provide per capita and population interaction strengths. In addition, a submodel that simulates species removal experiments was used to determine local interaction strength. Per capita, population, and local interaction strengths decrease in magnitude or remain constant as upwelling intensity increases (Table 2). We now turn to biological interpretations of these results.
Per capita interaction strength. - Competition for space occurs in two different ways in the competition model. The first effect is preemptive. Settlement is proportional to free space. Adults consume that space, thereby inhibiting the settlement of larvae adjacent to the coast. This effect is both interspecific and intraspecific. Secondly, larvae of the dominant can settle on subordinate adults. This second interaction is solely interspecific, so its effect must be incorporated into the competition coefficient [[Alpha].sub.21]. Because [[Alpha].sub.21] is relatively insensitive to upwelling [ILLUSTRATION FOR FIGURE 3C OMITTED], we infer that the gradient in per capita interaction strength [ILLUSTRATION FOR FIGURE 5A OMITTED] is driven primarily by the first interaction: inhibition of larval settlement by adults. As upwelling increases, an adult's offspring are advected farther offshore, leaving fewer at the coast to seek settlement sites. Therefore, an adult competitor inhibits the settlement of a smaller proportion of an individual's offspring where upwelling is strong, reducing per capita interaction strength. Note that this interpretation is not specific to offshore advection rate. Rather, it implies that a gradient in any transport process that influences larval supply to the benthic habitat (e.g., proximity of fronts to shore, frequency of relaxation events) should produce a gradient in per capita interaction strength.
Like the per capita competitive effects, the per capita effect of prey on predators (Eq. 23f) decreases as upwelling increases [ILLUSTRATION FOR FIGURE 10A OMITTED]. Prey provide the energy needed for predator larval production. Higher proportions of these larvae are swept away from the adult habitat where upwelling is stronger. Thus, the number of predator recruits produced per prey consumed decreases as upwelling increases. As with competitive interaction strength, this interpretation implies that any transport process influencing larval supply should produce a gradient in the per capita effect of prey on predators.
These gradients constrast strikingly with the per capita effect of predators on their prey. This term is independent of upwelling intensity (Eq. 23e). In the two other interactions, the effect of one adult on another's per capita growth rate is mediated through the larval pool. Competition occurs as adults inhibit the settlement of one another's larval offspring. Prey affect predators by influencing the rate of production of predator larvae. The effect of predators on prey, however, is not mediated through the larval pool. Predators do not directly inhibit or facilitate settlement, nor do they influence the prey's per capita rate of larval production. This difference allows the per capita effect of predators on prey to be independent of upwelling intensity.
TABLE 2. Summary of responses of interaction strengths to increasing upwelling. Interactions Metric Competitive Predator-prey Prey-predator Per capita decreases constant decreases Population decreases decreases constant Local decreases decreases NA(*) * Not applicable.
Population interaction strength. - In the competition model, population interaction strength of the dominant on the subordinate (Eq. 19b) decreases in magnitude as upwelling increases [ILLUSTRATION FOR FIGURE 5B OMITTED]. This happens because both the per capita effect and the abundance of dominants decrease as upwelling increases [ILLUSTRATION FOR FIGURES 4 AND 5A OMITTED]. In the predator-prey model, population interaction strength of the predator on the prey (Eq. 27a) decreases in magnitude as upwelling increases [ILLUSTRATION FOR FIGURE 10B OMITTED]. Because the per capita effect is independent of upwelling intensity (Eq. 23e), this gradient is caused by the decreasing predator abundance with increasing upwelling [ILLUSTRATION FOR FIGURE 9B OMITTED]. The population interaction strength of prey on predators, however, is independent of upwelling intensity (Eq. 27b). This occurs because the decreasing per capita effect ([[Lambda].sub.21]) is canceled out by increasing prey abundance ([[Lambda].sub.21] in the denominator) (Eqs. 23f and 26a).
Local interaction strength. - In the competition model, recruitment of competitive dominants is lower where upwelling is stronger (Connolly and Roughgarden 1998). This has two consequences. First, the dominant occupies less space in control plots, so more space is available for subordinate settlement. Second, subordinate adults are less likely to be settled on and killed by dominants. These two effects drive the decrease in local interaction strength with increasing upwelling [ILLUSTRATION FOR FIGURE 5C OMITTED]. This occurs despite the fact that, in the model, subordinate recruitment is actually higher where upwelling is stronger (Connolly and Roughgarden 1998).
Predator-prey local interaction strength also decreases as upwelling increases [ILLUSTRATION FOR FIGURE 10C OMITTED]. Where upwelling is stronger, predator abundance in control plots is lower, so the difference in net prey mortality rate between plots with and without predators is smaller. This produces smaller differences in prey abundance between experimental and control plots where upwelling is stronger.
Other Lotka-Volterra parameters
Alexander and Roughgarden (1996) found that, for the single-species case, intrinsic growth rate and carrying capacity decrease as upwelling increases. The competition and predator-prey models confirm that these qualitative results do not change with the incorporation of a competitor or predator [ILLUSTRATION FOR FIGURES 3 AND 8 OMITTED]. Intrinsic growth rate is the per capita rate of population growth when density-dependent interactions ([[Sigma].sub.j] [[Lambda].sub.ij] [N.sub.j]) are negligible. For a space-limited population, this is the per capita growth rate when benthic free space is abundant, relative to the benthic free space available when the population is at equilibrium. As upwelling increases, the proportion of an adult's larval offspring that are at the coast, seeking settlement sites, decreases; thus, the per capita rate of growth of the benthic population decreases. Carrying capacity is the equilibrium abundance in the absence of the interacting population, and is equal to the ratio of intrinsic growth rate, [r.sub.i], to intraspecific (within-species) per capita interaction strength, [[Lambda].sub.ii] (Eqs. 16 and 24). Both intrinsic growth rates [ILLUSTRATION FOR FIGURES 3A AND 8A OMITTED] and intraspecific interaction strengths (not shown) decrease in magnitude as upwelling increases; carrying capacity decreases because the former are more sensitive to upwelling than the latter.
Where stronger upwelling should inhibit recruitment, the subordinate actually equilibrates at a higher percent cover [ILLUSTRATION FOR FIGURE 4 OMITTED]. This is most easily explained by rewriting Eq. 18b as follows:
[Mathematical Expression Omitted]. (29)
Part of the response of subordinate abundance to a gradient in upwelling results from a direct, population-dynamic effect. This is captured by the response of [K.sub.2] to changes in upwelling. As Eq. 29 makes clear, there is also a competition-mediated effect. This is captured by the response of [[Alpha].sub.21] [K.sub.1] to changes in upwelling. The competition-mediated effect has two components. Stronger upwelling leads to lower percent cover of the dominant and, therefore, more space on which subordinates can settle. It also leads to lower rates at which dominants settle on and kill subordinates. Because [[Alpha].sub.21] [greater than] 1 and the dominant's carrying capacity is more sensitive to upwelling than the subordinate's [ILLUSTRATION FOR FIGURE 3 OMITTED], changes in the competition-mediated effect as upwelling increases (i.e., the gradient in [[Alpha].sub.21], [K.sub.1]) outweigh changes in the population-dynamic effect (the gradient in [K.sub.2]). Consequently, the subordinate benefits from increasing upwelling intensity.
The predator-prey model exhibits similar behavior: prey abundance at equilibrium is lower where upwelling is weaker [ILLUSTRATION FOR FIGURE 9A OMITTED]. This occurs because prey abundance at equilibrium (Eq. 26a) depends on the predator's life history parameters, but not its own. In particular, it depends on the larval transport parameters of the predator. This is an example of the top-down control characteristic of Lotka-Volterra predator-prey models, in which increasing the prey's intrinsic growth rate or carrying capacity (in this case, by decreasing upwelling intensity) allows the predator, but not the prey, to increase in abundance (Oksanen et al. 1981).
Agreement with field data
The gradients in abundance predicted by the competition model match data for upper intertidal barnacles in the genera Balanus and Chthamalus between central California and northern Oregon. The competitively dominant Balanus is less abundant in California, where upwelling is strong, than in Oregon, where upwelling is weak. In contrast, the subordinate Chthamalus is more abundant in California than in Oregon (Connolly and Roughgarden 1998). The survey during which these data were collected included only the middle and high intertidal zones, above the upper limit of seastars, so those data cannot be used to test the predator-prey model's prediction that prey abundances are higher where upwelling is weaker. Similarly, we cannot confirm that seastars are more abundant in the Pacific Northwest than in central and northern California. The few reports of Pisaster ochraceus densities for central and northern California range from 0.5/[m.sup.2] to 4.5/[m.sup.2] (Feder 1956; J. S. Pearse, unpublished data) and are within the range of densities reported for Oregon and Washington (Paine 1974, 1976, Menge et al. 1994, Navarrete and Menge 1996). More data are needed to test predicted prey and predator abundance gradients.
Unfortunately, we know of no studies linking regional differences in per capita interaction strength to differences in larval transport processes. Navarrete and Menge (1996) provide a protocol for measuring the per capita interaction strength of predators on prey. By multiplying this term by predator density, population interaction strength can also be estimated. They use this protocol to measure interaction strength on clumps of mussels of fixed densities at different sites, whereas our interaction term corresponds to interaction strength on prey at average densities for a particular upwelling intensity. However, a similar approach could be used to estimate our interaction term. Application of this protocol in different regions of the northeast Pacific (e.g., Oregon and central California) would test two of the model's predictions: (1) between-region differences in per capita interaction strength of predators on prey, if they exist, should be attributable to causes other than larval transport processes; but (2) population interaction strength of predators on prey should tend to be smaller where offshore advection of larvae is stronger (because these regions will sustain lower predator densities). Per capita interaction strengths of prey on predators and of dominant competitors on subordinates depend on larval transport processes, so field estimates of these quantities will require information about the pelagic larval habitat as well as the benthic adult habitat.
Most field data on interaction strength in intertidal communities come from species removal experiments like those simulated here. Regional differences in the results of seastar removal experiments conducted in the northeast Pacific match those predicted by the model. When seastars are removed from the low intertidal zone, mussel beds expand downward relative to areas where seastars are not removed. This effect decreases in magnitude from northern Washington to central California. In northern Washington, Paine (1974) found a mean rate of expansion of [approximately]0.65 m/yr from 1970 to 1973. In central Oregon, Menge et al. (1994) found mean rates of expansion of [approximately]0.5 m/yr at one site and 0.25 m/yr at another from 1990 to 1993. In central California, J. S. Pearse (unpublished data) found a smaller effect, expansion at a mean rate of [less than]0.1 m/yr from 1979 to 1985. The effect of predator exclusion on barnacles shows the same trend. In central California, a predator exclusion experiment found that barnacle cover increased by [approximately]30% (from 40% to 70% cover) over 9 mo (T. M. Farrell, unpublished data). This contrasts starkly with Dayton's (1971) classic experiments in northern Washington, in which barnacle cover increased from nearly 0% to nearly 100% in the first 9 mo. Although these studies were not identical in design, were conducted at different times, and did not quantify the relative contributions of recruitment and individual growth to the results, the trends in both mussel and barnacle experiments match the model's prediction: smaller effects in the region of stronger upwelling.
Robustness of model predictions
To make the interpretation of results as unambiguous as possible, we used simple functions to characterize competition and predation and considered two well-mixed, interacting populations isolated from the rest of the community. The simplifying assumptions made in the process fall into two broad classes. The first class pertains to the form of the per capita interactions in the model. The second class mediates the effects of these interactions at the population level. The models presented in this paper provide a framework for thoroughly exploring these assumptions, because they can be modified, and reanalyzed, with the new results compared to those reported here. An extensive treatment of such modifications is beyond the scope of this paper. However, a preliminary investigation of both classes of assumptions suggests that the central result of this paper is robust: interaction strengths decrease or remain constant as upwelling increases. We now briefly summarize this investigation.
Functional form of per capita interactions. - Two assumptions about the form of the per capita interactions that may be relaxed in the northeast Pacific rocky intertidal zone are density-independent adult mortality and linearly increasing predation rate with prey density. The assumption that the adult mortality rate is independent of density is violated when free space is scarce (Gaines and Roughgarden 1985). To investigate the sensitivity of our results to this assumption, we modified the competition model to include a simple representation of density dependence, linearly increasing mortality with adult density. This increases per capita interaction strengths because individuals not only inhibit the settlement of larvae, but also directly increase the mortality rates of other adults. However, per capita, population, and local interaction strengths still decrease in magnitude as upwelling increases (results are not shown). For the predator-prey interaction, we cannot confirm the assumption that the predation rate increases linearly with prey density. Therefore, to explore the sensitivity of our results to predator satiation, we used
- e[N.sub.1](t)/D + [N.sub.1](t) (30)
to characterize predation rate instead of the - e[N.sub.1](t) of Eqs. 8-9 (Holling 1965). Here, e is the maximum predation rate per predator, and D is a constant set so that e/D is the slope of the functional response curve at [N.sub.1] = 0; i.e., e/D is the predation rate per predator at very low prey density. With this modification, the per capita effect of predators on prey decreases, rather than remaining constant, as upwelling increases, while all other trends are unchanged in direction (results are not shown).
Population-level processes. - The models presented here examine two well-mixed populations isolated from the rest of the community. As in the classical Lotka-Volterra predator-prey model, these models allow one population to strongly control the abundance of another along an environmental gradient (Oksanen et al. 1981). Neither the competitive subordinate nor the prey is able to benefit from weak upwelling intensity, because the direct, population-dynamic benefits are outweighted by increased abundance and recruitment of the dominant or predator. Previous work has shown that realistic modifications to two population models, such as additional species interactions, prey refuges, or nonequilibrium dynamics, can mediate this strong interspecific control of population abundances (see reviews in Pimm 1991, DeAngelis 1992, Polis and Winemiller 1996). Although the explicit incorporation of additional populations, refuges, and spatial and temporal stochasticity is beyond the scope of this paper, we can consider the sensitivity of our results to predicted abundance gradients by evaluating interaction strengths at prey and subordinate abundances other than their equilibrium values. As an extreme case, consider interaction strengths evaluated at prey and subordinate carrying capacities. In contrast to equilibrium abundances, these quantities decrease as upwelling increases. Thus, any gradient in interaction strength that is sensitive to strong top-down or competitive control of abundance should change direction when evaluated at carrying capacity. Nevertheless, we found that this modification does not change the direction of the gradients in per capita interaction strengths or local interaction strengths. Similarly, neither the population interaction strengths in the competition model nor the population interaction strength of predators on prey are changed in direction. However, the population interaction strength of prey on predators does change: when evaluated at prey carrying capacity, it decreases, rather than remaining constant, as upwelling increases (results are not shown).
This preliminary investigation suggests that the results that are likely to be sensitive to relaxation of model assumptions are those that are independent of upwelling in the original analysis: the per capita effect of predators on prey and the population effect of prey on predators. Interaction strengths that decrease as upwelling increases under the original analysis do not change direction. Note that this reinforces the overall tendency for interaction strengths to decrease as upwelling increases. Although we cannot guarantee that this tendency for species interactions to become weaker as upwelling increases will hold under any conceivable set of assumptions, this preliminary investigation indicates that it is not an artifact of the particular set of assumptions adopted for the models analyzed in this paper.
These results support the hypothesis that geographical gradients in larval transport processes produce gradients in interaction strength. These gradients can be driven by changes in per capita effects, changes in abundance, or both. The decreasing competitive interaction strength with increasing upwelling is driven by both decreasing per capita effects and decreasing abundance of the dominant. The decreasing population interaction strength of predators on prey with increasing upwelling is driven entirely by decreasing predator abundance. Conversely, the decreasing per capita effect of prey on predators with increasing upwelling is canceled out at the population level by increasing prey abundance.
A striking result of this study is the concordance between the responses of local and population interaction strengths to a gradient in upwelling intensity (Table 2). This indicates that a latitudinal gradient in upwelling may be responsible for geographical differences in the results of species removal experiments in the northeast Pacific (Roughgarden et al. 1988), and that these differences reflect population-level differences in the intensity of competition and predation. The qualitative agreement between local and population interaction strengths is encouraging, because much of our information about intertidal communities comes from local experiments. However, we caution that local interaction strengths are not identical to population interaction strengths, so the positive correlation between local and population interaction strength found here may not hold for all environmental gradients.
Considerable empirical work has been done to construct and analyze interaction webs for rocky intertidal communities (e.g., Paine 1980, Menge 1995, Wootton 1997). The assembly of these webs has been based on experimental manipulations on the scale of meters. For most benthic marine animals, communities at this scale are almost entirely open, in that virtually none of the recruits to experimental plots will have been produced by adults in those same plots. In contrast, most theory for interaction webs is formulated for closed systems (see reviews in Pimm 1982, 1991, DeAngelis 1992, Polis and Winemiller 1996). To compare community structure patterns across marine, terrestrial, and aquatic systems, we need a theoretical framework for marine communities that allows us to construct population-level webs for benthic marine communities. By demonstrating that per capita and population interaction strengths can be expressed in terms of measurable life history parameters, this paper provides a step in that direction.
We thank L. V. Lucas and J. Koseff for assistance with the numerical analysis. We are also grateful to J. S. Pearse and T. M. Farrell for sharing the results of unpublished species removal experiments in central California. J. Huisman, J. S. Dukes, and E. K. Nelson provided helpful comments on early drafts of the manuscript. We thank B. A. Menge, H. Caswell, and an anonymous reviewer for helpful comments, criticisms, and suggestions. The first author was supported by a National Science Foundation Graduate Fellowship and the Department of Biological Sciences at Stanford University. with additional support provided by Phi Beta Kappa of Northern California.
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The Lotka-Volterra equations for two species can be written in the form
d[N.sub.1]/dt = [[r.sub.1] + [[Lambda].sub.11][N.sub.1] + [[Lambda].sub.12][N.sub.2]][N.sub.1] (A.1a)
d[N.sub.2]/dt = [[r.sub.2] + [[Lambda].sub.21][N.sub.1] + [[Lambda].sub.22][N.sub.2]][N.sub.2]. (A.1b)
We wish to obtain a model of this form that approximates the model of Eq. 13. Therefore, throughout this appendix, d[N.sub.1]/dt and d[N.sub.2]/dt refer to Eqs. 13a and 13b, respectively. To insure that the approximation has the same interior equilibrium as the original model, we evaluate the approximation at this equilibrium. Thus, the interior equilibrium of the original model,
[Mathematical Expression Omitted] (A.2a)
[Mathematical Expression Omitted] (A.2b)
(Connolly and Roughgarden 1998), will be equal to the interior equilibrium of the Lotka-Volterra model (Eq. 18).
First, we differentiate Eqs. 13a and A.1a with respect to [N.sub.2]:
[Delta]/[Delta][N.sub.2](d[N.sub.1]/dt) = [[Lambda].sub.12][N.sub.1]. (A.3)
Solving this expression for [[Lambda].sub.12] and evaluating, we get Eq. 14e. Next, differentiating Eqs. 13a and A.1a with respect to [N.sub.1], we obtain
[Mathematical Expression Omitted]. (A.4)
Evaluating both sides at their respective interior equilibria, [[Lambda].sub.11] drops out and we can obtain [r.sub.1] (Eq. 14a). We then set Eq. A.1a equal to zero and solve for [[Lambda].sub.11]:
[Mathematical Expression Omitted]. (A.5)
Evaluating at the interior equilibrium, we get Eq. 14c. Differentiating Eqs. 13b and A.1b with respect to [N.sub.1], we get
[Delta]/[Delta][N.sub.1](d[N.sub.2]/dt) = [[Lambda].sub.21][N.sub.2]. (A.6)
Solving this expression for [[Lambda].sub.21] and evaluating at the interior equilibrium, we get Eq. 14f. Now, differentiating Eqs. 13b and A.1b with respect to [N.sub.2], we obtain
[Mathematical Expression Omitted]. (A.7)
Evaluating both sides of this expression at their respective interior equilibria, [[Lambda].sub.22] vanishes and we can solve for [r.sub.2] (Eq. 14b). We can then set Eq. A.1b equal to zero, solve for [[Lambda].sub.22] and evaluate at the interior equilibrium of the original model, yielding Eq. 14d.
Note that [r.sub.1] and [[Lambda].sub.11] correspond to r and [Lambda] for a single-species model, because competition is hierarchical. In addition, as the dominant approaches its extinction threshold ([p.sub.1] [approaches] 0), [r.sub.2] and [[Lambda].sub.22] also approach the single-species expressions.
Again, we write the Lotka-Volterra equations in the form of Eq. A.1, where d[N.sub.1]/dt and d[N.sub.2]/dt refer to the appropriate equations from the scaling approximation (Eq. 21). The interior equilibrium of the original model is
[Mathematical Expression Omitted] (A.8a)
[Mathematical Expression Omitted]. (A.8b)
Because there is no effect of predators on one another, [[Lambda].sub.22] = 0, and the Lotka-Volterra equilibrium is given by Eq. 26.
First, we differentiate Eqs. 21a and A.1a with respect to [N.sub.2] to obtain
[Delta]/[Delta][N.sub.2](d[N.sub.1]/dt) = [[Lambda].sub.12][N.sub.1]. (A.9)
Solving this expression for [[Lambda].sub.12], we get Eq. 23e. Similarly, from
[Delta]/[Delta][N.sub.1](d[N.sub.2]/dt) = [[Lambda].sub.21][N.sub.2] (A.10)
we can solve for [[Lambda].sub.21] (Eq. 23f). Differentiating Eqs. 21b and A.1b with respect to [N.sub.2], we obtain
[Delta]/[Delta][N.sub.2](d[N.sub.2]/dt) = [r.sub.2] + [[Lambda].sub.21][N.sub.1]. (A.11)
Solving for [r.sub.2] and evaluating at the interior equilibrium of the original model, we get Eq. 23b. Next, we differentiate Eqs. 21a and A.1a with respect to [N.sub.1]:
[Delta]/[Delta][N.sub.1](d[N.sub.1]/dt) = [r.sub.1] + 2[[Lambda].sub.11][N.sub.1] + [[Lambda].sub.12][N.sub.2]. (A.12)
Evaluating each side at its respective interior equilibrium, the unknown [r.sub.1] drops out, and we can solve for [[Lambda].sub.11] (Eq. 23c). Finally, we set Eq. A.1a equal to zero, solve for [r.sub.1], and evaluate at the interior equilibrium of the original model, yielding Eq. 23a.
DERIVATION OF LOCAL INTERACTION STRENGTH
The submodel used to derive local interaction strength is based on two observations about species removal experiments. First, abundances in experimentally manipulated plots are small relative to the sizes of the regional breeding populations whose larvae recruit to the plots. Second, the dominant and subordinate coexist in the system as a whole. If these assumptions hold, larval abundances adjacent to both experimental and control plots are simply [L.sub.1](0, [N.sub.1]) and [L.sub.2](0, [N.sub.1], [N.sub.2]) (Eq. 11). Here, we assume that the regional system is at equilibrium and evaluate larval abundances accordingly. Population dynamics in the control (dominant present) plot follow
[Mathematical Expression Omitted] (B.1a)
[Mathematical Expression Omitted] (B.1b)
[Mathematical Expression Omitted], (B.1c)
and subordinate dynamics in the experimental (dominant removed) plot follow
[Mathematical Expression Omitted] (B.2a)
[Mathematical Expression Omitted]. (B.2b)
The superscript C refers to the control plot, and E refers to the experimental plot. Local interaction strength can be measured by comparing abundances in both plots according to Eq. 3. At equilibrium, subordinate abundance in the control plot is given by Eq. A.2b, and subordinate abundance in the experimental plot is
[Mathematical Expression Omitted]. (B.3)
Local interaction strength evaluates to Eq. 20.
Following the procedure just outlined, prey dynamics in predator removal experiments can be simulated. Population dynamics in the control (predator present) plot follow
[Mathematical Expression Omitted] (B.4a)
[Mathematical Expression Omitted] (B.4b)
[Mathematical Expression Omitted]. (B.4c)
Prey dynamics in experimental (predator-removed) plots follow
[Mathematical Expression Omitted] (B.5a)
[Mathematical Expression Omitted]. (B.5b)
At equilibrium, prey abundance in the control plot is given by Eq. A.8a, and prey abundance in the experimental plot is
[Mathematical Expression Omitted] (B.6)
Local interaction strength evaluates to Eq. 28.
Scripts for the simulations reported in this paper are available in ESA's Electronic Data Archive: Ecological Archives M069-004.
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|Author:||Connolly, Sean R.; Roughgarden, Joan|
|Date:||Aug 1, 1999|
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