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The evolution of floral longevity: resource allocation to maintenance versus construction of repeated parts in modular organisms.

Floral longevity (the length of time that flowers remain open and functional) is an important plant trait that influences the amount of pollen received and disseminated by a plant, and hence, its reproductive fitness (Primack 1985). There is considerable variation in floral longevity among angiosperm species. Some species have flowers that last a day or less (e.g., Ipomoea purpurea, Hemerocallis fulva, Oenothera spp., members of the family Dipterocarpaceae), whereas others have flowers that last from 1 to 3 wk (e.g., Trillium grandiflorum, Hepatica triloba, Lupinus spp.), and in some species up to 1 or 2 mo (e.g., members of the Orchidaceae) (Kerner von Marilaun 1895; Gori 1983; Primack 1985; Ashman and Schoen 1994). Primack (1985) reviewed the ecology of floral longevity, noting that variation in floral longevity is often associated with habitat, pollinator type, taxonomic class, and breeding system, and he outlined a model of floral longevity modified from an unpublished one originally developed by D. G. Lloyd (University of Canterbury, New Zealand). The model he outlined predicted whether flowers should be maintained on the plant either for a single day or for an indefinitely longer period, and thus placed the evolution of floral longevity into an analytical framework. Here, the approach developed by Primack and Lloyd is extended by examining how the accrual rates of male and female fitness, floral maintenance and construction costs, temporal delays in the start of male and female reproductive expression within the flower, and nonindependence among flowers in their attractiveness to pollinators combine, to determine the exact number of days that flowers should be optimally maintained on the plant.

The maintenance of flowers can be considered as a special case of the more general problem of optimal longevities of repeated structures in modular organisms. For example, plant species also show variation in the longevities of leaves, shoots, fruits, and branches. The study of senescence and shedding of plant parts has been approached primarily from the perspective of physiology (Kozlowski 1973). Here we assume that senescence and shedding of parts or structures in modular organisms have evolved as a type of allocation strategy akin to sex allocation in hermaphroditic organisms (Charnov 1982). In the case of the optimal longevity of a repeated structure, however, the allocation of limiting resources is not to male versus female gametes, but to the competing processes of construction versus maintenance of the structure in question. Such allocation problems are likely to be subject to natural selection. The longevity of repeated structures may thus be optimized within constraints posed by other factors that contribute to fitness.

From the information available in the literature, substantial interspecific variation clearly exists in ecological and physiological factors that may contribute to determining optimal floral longevity (e.g., the rate of pollen receipt and pollen dissemination, floral maintenance costs). For example, in Fushsia excorticata, Delph and Lively (1989) found that most plants had received and disseminated pollen within 4 d following floral opening. Likewise, in Clintonia borealis, pollen deposition on the stigma increased for up to 4 d following the opening of the flower (Galen et al. 1986), and for slightly longer in Silene acaulis (Shykoff 1992). In contrast, pollen receipt and deposition occur within a few hours of flower opening in Pontederia cordata and within the first day of flower life in Raphanus sativus (Wolfe and Barrett 1987; Harder and Barrett 1992; Ashman et al. 1994). At the opposite end of the continuum, the receipt and dissemination of pollinia has been viewed as a process requiring substantially longer periods of time in some orchids (Cole and Firmage 1984). Turning to floral maintenance, the largest costs are found in those species that produce floral nectar. Southwick (1985) estimated that 4%-37% of the photosynthate assimilated by plants of Asclepias syriaca in a day is allocated to the production of nectar. In Impatiens pallida and I. capensis, Schemske (1978) showed that nectar costs for single flowers amount to roughly one third of the total prefertilization resources invested in flowers. Respiration of the flower, especially the corolla, may also account for some of the cost of floral maintenance. We found that in Clarkia gracilis var. tracyi, 2.3% of the total floral maintenance cost was due to corolla respiration, and in T. grandiflorum, 32.7% (Ashman and Schoen 1994). Moreover, when water is a limiting resource, water loss through stomates of the flower may represent a further cost of floral maintenance (Nobel 1977). The way in which the time schedules of pollen and seed fitness accrual combine with floral maintenance and construction costs to determine optimal floral longevities are explored below in an analytical framework.

MODELS OF OPTIMAL FLORAL LONGEVITY

General Considerations and Assumptions

First, it is assumed that at the time of flowering there exists a pool of limited resources that can be allocated either to the construction of new flowers or to their maintenance, that is, a resource-allocation trade-off (Charnov 1982; Goldman and Willson 1986; Charlesworth and Morgan 1991). Flowers that receive maintenance resources are, by definition, kept alive and functioning for longer periods of time compared with those that do not. The pool of resources used for floral construction and maintenance is considered separate from that used to provision fruits and seeds, as in several other models focusing primarily on the flowering stage (e.g., Charlesworth and Charlesworth 1987; Schoen and Dubuc 1990; Morgan 1992). The present model is, therefore, best applied to plants with a definite period of flowering, followed by a period of fruit and seed development. Modification of this assumption (e.g., to include overlap in resource use by flowers and developing fruits and seeds) does not change the qualitative behavior of the model.

It is further assumed that the daily cost of floral maintenance (e.g., nectar sugar production, floral respiration, transpiration) is constant. In the case of nectar production, there appears to be some evidence to support this, though accurate estimates of daily nectar production often are technically demanding to obtain (Plowright 1981; Zimmerman and Pyke 1986). The assumption of constant maintenance costs can be modified, but doing so increases the complexity of the model. One situation of particular interest pertains to the retention of postfunctional flowers (Gori 1983; Casper and La Pine 1984; Cruzan et al. 1988; Delph and Lively 1989; Weiss 1991) with reduced maintenance costs later in their lifetimes. This situation is treated as a special case below.

Next, it is assumed that each flower has an equal ability to contribute to the reproductive fitness of the plant, and that the realization of this fitness contribution is determined by the length of time over which the flower has been open and exposed to pollinating agents (biotic or abiotic). As the length of time that the flower has been open increases, its contribution to the fitness of the plant approaches a limit. This limit occurs when the maximum potential amount of pollen has been disseminated from the anthers and when the stigma has received enough pollen to ensure the maximum potential seed set per flower. The approach to the realization of the full pollen and seed fitness of a flower may be gradual or abrupt and will be influenced by extrinsic factors such as climate, pollinator activity, and pollinator behavior, as well as by intrinsic factors, such as the maturation schedules of stigmas and anthers. Moreover, pollen and seed fitnesses may accrue according to the same time schedule (e.g., when pollen receipt and deposition are tightly coupled during pollinator visitation) ([ILLUSTRATION FOR FIGURE 1A OMITTED]), or one fitness component may accrue at a more rapid rate than the other. For example, in some species, a few pollinator visits may suffice to fertilize all ovules, but significant amounts of pollen may remain on the anthers ([ILLUSTRATION FOR FIGURE 1B OMITTED]). Alternatively, in the case of pollen-harvesting insects, or in plants in which pollen is packaged into a few pollinia per flower, a single visit may suffice to remove all pollen, but many or all ovules may remain unfertilized. Differences between the time course of pollen dissemination and pollen receipt may also occur in species in which anther dehiscence occurs before or after stigma maturation ([ILLUSTRATION FOR FIGURE 1C OMITTED]). The assumption that the per-flower rates of pollen arrival and deposition depend only on the amount of time that the flower has been open does not imply that reproductive fitness accrual is independent of other plant and environmental features. For example, increases in flower size (and hence attractiveness) or increases in pollinator density may also influence pollen arrival and deposition, but below it is assumed that such features are held constant. This allows us to concentrate on the fitness consequences of variation in floral longevity.

The model also assumes that plants can evolve to control the longevity of their flowers. That is, floral senescence and abscission are assumed to be determined by a developmental program. This is undoubtedly an oversimplification for some species, as it is known that pollination itself can induce floral senescence (Gori 1983). However, the physiological changes to the flower resulting from pollination (or lack of pollination) are clearly not the only factors determining floral longevity. The flowers of many species (e.g., Ipomoea, Oenothera, Hibiscus, and Mirabilis) appear to be short-lived regardless of whether they have received pollen, whereas those of other species are maintained for several days after pollination (Lupinus, Lantana, and Lilium) (Gori 1983; Weiss 1991; Edwards and Jordon 1992). That species-specific variation in floral longevity exists, and that such variation in floral longevity is not determined only by pollination conditions, is further supported by the finding that the floral longevity in the absence of pollinators (maximum floral longevities) is correlated with the longevity under field conditions (realized floral longevity) (Ashman and Schoen 1994).

Model Parameters Determining Floral Longevity and Number of Flowers per Plant

A constant pool of R resources is assumed available for floral construction and maintenance. An amount c(c [much less than] R) of these resources is required for the construction of a single flower. Following construction, the flower must be maintained to remain open and functional.

The daily cost of floral maintenance is expressed relative to the construction cost as the product mc (m [greater than] 0); for example, when m = 0.1, the daily cost of floral maintenance is one-tenth the cost of constructing a new flower. The parameter m may take on any positive value, but extreme maintenance costs (e.g., m [greater than] .25) appear to be uncommon in nature (Ashman and Schoen 1994). The total maintenance expenditure per flower (summed over the flower's lifetime) is influenced by the parameter t, the length of time (number of days) that the flower remains open, and is given by the product tmc. The number of flowers produced in a single flowering season by a plant having flowers that each remain open for t days can now be expressed in terms of the parameters described above as R/(c + tmc). This expression clearly indicates the direct trade-off assumed between flower number per plant and floral longevity; that is, as the cost of floral maintenance increases (caused by either large t or m), the number of flowers produced per plant decreases.

Floral maintenance cost could also determine the number of flowers per plant if floral maintenance costs are paid not from a standing pool of resources, but entirely from the products of daily photosynthesis - that is, there is no standing pool of resources. This may occur in some short-lived annuals. When floral maintenance costs are paid entirely from the photosynthate acquired on a daily basis, the number of flowers produced per plant would be constrained by the daily photosynthetic supply rate, and therefore, the formulation used above to express flower number as a function of maintenance and construction costs would not hold.

Basic Model of Floral Longevity (Homogamous Flowers)

The simplest situation for modeling floral longevity is in plants with flowers in which the start of stigma maturation and anther dehiscence occur simultaneously (homogamous flowers). Optimal floral longevity in plants where there is a delay within the flower in the start of one sex function relative to the other can be examined by modifying the basic model (see below).

To express the reproductive fitness of a plant having flowers each with longevities of t days, it is necessary to first derive an expression that quantifies how pollen and seed fitnesses accrue over the lifetime of the individual flower. For pollen, it is assumed that as the flower remains open for increasingly longer durations, pollinators will remove increasingly more pollen from the anthers, so that a proportion p (0 [less than or equal to] p [less than or equal to] 1) of the pollen contained in the anthers at the beginning of one day remains in the anthers at the end of that day, and after t days (assuming that anther dehiscence begins at t = 0) a proportion [p.sup.t] of the pollen in the anthers will remain. Thus, 1 - p is the fraction of pollen present in the anthers that is disseminated each day (hereafter referred to as the daily pollen fitness-accrual rate), and 1 - [p.sup.t] is the fraction disseminated after t days. Moreover, it is assumed that a constant proportion of the pollen that is removed by pollinators makes its way into a "pool" of pollen that competes with the pollen from other flowers for access to ovules (Lloyd 1984). Alternatively, 1 - p may be interpreted as the probability that on a given day the flower will be visited and all its pollen removed (Primack 1985). The assumption of a constant rate of addition of pollen to the pollen pool may not hold for some species; for example, those in which there has been selection for staggered release of pollen (Harder and Thomson 1989), or in species where pollen left in the anthers for an extended period of time loses viability (Bookman 1983; Thomson and Thomson 1992). In these cases, p could no longer be treated as a constant in the model, but rather would become a function of floral age. Doing this would increase the complexity of the fitness equations (and so in the interest of brevity, this case is not treated below), but such an extension is easily accommodated into the general model.

An analogous approach may be used for seed fitness. In this case, the parameter g (0 [less than or equal to] g [less than or equal to] 1) is the fraction of the flower's ovules not fertilized at the beginning of the day that remain unfertilized at the end of the day, and [g.sup.t] is the fraction that remain unfertilized after t days. Thus, on a given day, 1 - g is the fraction of unfertilized ovules that become fertilized (hereafter, the daily seed fitness-accrual rate), and 1 - [g.sup.t] is the fraction or unfertilized ovules that are fertilized after t days. It is assumed that the probability that an ovule matures into a seed once it is fertilized is not influenced by the day on which fertilization occurs (i.e., resource limitation of seed production does not vary over the course of the flowering season).

The time course of male and female fitness accrual is, therefore, determined by the daily pollen and seed fitness-accrual rates (i.e., the parameters 1 - p and 1 - g). When the pollen and seed fitnesses accrued per day are large (1 p and 1 - g both near 1), the pattern of fitness accrual for both sexual functions exhibits rapidly diminishing returns with increasing floral longevity ([ILLUSTRATION FOR FIGURE 1A OMITTED]). Roughly linear fitness accrual over the flower's lifetime occurs when the fitness accrued per day is small (1 - p and 1 - g [much less than] 1) ([ILLUSTRATION FOR FIGURE 1C OMITTED]).

Using the Shaw-Mohler equation (Shaw and Moller 1953; Charnov 1982) together with the formulation above relating flower number per plant to floral maintenance and construction costs, the fitness of a rare mutant plant with floral longevity [t.sub.1] in a population of plants with floral longevity [t.sub.2] may be written as

w = (1/2) [R/(c + [t.sub.1]mc)/R/(c + [t.sub.2]mc)] (1 - [p.sup.[t.sub.1]]/1 - [p.sup.[t.sub.2]] + 1 - [g.sup.[t.sub.1]]/1 - [g.sup.[t.sub.2]]). (1)

From equation (1) it is clear that the size of the resource pool, R, and the absolute floral construction cost, c, cancel out in the fitness equation and thus do not enter directly into the optimal strategy. The floral construction cost does, however, come into play in the parameter m, the floral maintenance costs measured relative to the cost of floral construction. To find the optimal floral longevity, it is necessary to find solutions that satisfy [absolute value of]dw/[dt.sub.1] = 0 when [t.sub.1] = [t.sub.2] = t (and when [d.sup.2]w/[[dt.sub.1].sup.2] [less than] 0); that is, the floral longevity value that maximizes fitness. Numerical methods were used to do this, as no analytical solution could be found. These involved a computer algorithm to randomly search the parameter space for the combinations of p, g, m, and t that yielded maximum fitness, dw/dt [less than or equal to] [10.sup.-5], or in some cases, direct calculation of w to find the maximum relative fitness, [w.sub.max], when one floral longevity strategy was compared with another.

Because the model is multidimensional, it is convenient to display results for a portion of the parameter space at a time. This yields isoclines of maximum fitness for different floral longevities, that is, combinations of daily floral maintenance (m) together with daily pollen and seed fitness-accrual rates (1 - p and 1 - g) that result in an optimal floral longevities of t = 1, 2, . . . , T days ([ILLUSTRATION FOR FIGURE 2 OMITTED]). Short-lived flowers (t = 1 day) are favored when the daily cost of maintenance is high relative to the cost of construction of a new flower (large m), and when pollen and seed fitness-accrual rates are high (1 - p and 1 - g large). Longer-lived flowers become increasingly favored as m and the daily rate of male and female fitness accrual are reduced. When maintenance costs are low (e.g., m = 0.05), variation in the daily pollen and seed fitness-accrual rates has a larger effect in determining the optimal floral longevity, and longer-lived flowers are more likely to be selected. When maintenance costs are high (e.g., m [greater than or equal to] 0.25), however, most combinations of daily pollen and seed fitness-accrual rates select for short floral longevities. Low daily pollen fitness can be compensated for by a high daily seed fitness (and vice versa), but the trade-off is not linear, and hence the isoclines are curved. In general, it appears that variation in pollen and seed fitness-accrual rates has more impact in determining the optimal floral than does variation in floral maintenance cost. Although these general trends are illustrated in figure 2 only for m = 0.02, 0.05, and 0.25, the behavior of the model is the same for other values of m examined in the range of m = 0.01-1.0.

Modifications for Dichogamous Species

Dichogamy (temporal displacement of anther and stigma maturation within the flower) is widespread among the angiosperms. Dichogamy may be selected as a means to reduce the level of self-pollination (Darwin 1876), and/or to reduce interference between the dissemination and receipt of pollen (Lloyd and Yates 1982; Lloyd and Webb 1986; Bertin 1993). Protogyny (stigma maturation occurring before anther dehiscence) and protandry (stigma maturation after anther dehiscence) can be considered as constraints around which the optimal floral longevity is selected, implying that the flower must be maintained at least as long as required for the delayed sexual function to commence. It is expected, therefore, that dichogamous flowers will have longer lifetimes than homogamous ones, all other things being equal.

To examine more precisely how dichogamy influences floral longevity, consider a protogynous flower, and let anther dehiscence begin d days (d [greater than] 0) after flower opening, that is, increasing the value of d corresponds to increasing the delay in the start of the male sexual phase relative to the start of the female sexual phase. For simplicity, it is assumed that the female sexual phase lasts throughout the entire life of the flower. Thus, the pattern of sexual expression within the flower is female followed by bisexual. The problem is to find the optimal floral longevity starting with d as a lower limit. That is, how many days beyond d is the flower optimally maintained? Let the amount of time elapsing after anther dehiscence be indexed by the variable t[prime] so that total floral longevity is given by t = d + t[prime] days. To find the optimal t[prime] the fitness formulation of the basic model (eq. 1) must be modified as follows:

w = (1/2) [R/[c + ([t[prime].sub.1] + d)mc]/R/[c + ([t[prime].sub.2] + mc]] (1 - [p.sup.[t[prime].sub.1]]/1 - [p.sup.[t[prime].sub.2]] + 1 - [g.sup.([t[prime].sub.1] + d)]/1 - [g.sup.([t[prime].sub.2] + d)]). (2)

Equation (2) makes explicit that the way in which the female fitness contribution of the flower accrues over a longer period of time (t[prime] + d) than does the male fitness contribution (t[prime]).

Results from this modification of the model were examined by searching directly for [w.sub.max] when different values of the model parameters were used. For instance, consider a daily floral maintenance cost of m = 0.05, together with daily pollen and seed accrual rates (1 - p, 1 - g) that would normally combine to select for 3-d homogamous flowers (i.e., the basic model above). With protogyny, it can be seen that the optimal floral longevity shifts in the direction of longer-lived flowers ([ILLUSTRATION FOR FIGURES 3A,B OMITTED]). A similar pattern of shift toward longer-lived flowers with protogyny is also seen for other combinations of m, d, 1 - p and 1 - g ([ILLUSTRATION FOR FIGURES 3C,D OMITTED]). Exactly how optimal floral longevity is influenced by dichogamy compared with homogamy depends on the specific values of 1 - p and 1 - g. When daily pollen fitness accrual is low, the shift toward longer-lived flowers is more pronounced than when it is high ([ILLUSTRATION FOR FIGURE 3 OMITTED]). This is because with protogyny and low pollen fitness-accrual rates, a comparatively longer-lived flower would be required to maximize fitness compared with the same situation and higher pollen fitness-accrual rates. Note that because of the trade-off between flower longevity and flower number assumed in the model (see above), dichogamy should lead to fewer flowers being borne per plant compared with homogamy. A parallel modification for plants with protandrous flowers is discussed elsewhere (Ashman and Schoen 1995).

The Inflorescence as the Unit of Attraction

In the models presented above, it was assumed that the number of open flowers on the plant does not influence the pollen and seed fitness-accrual rates of each individual flower. If, however, the attractiveness of the plant to pollinators (and therefore pollen and seed fitness-accrual rates) is correlated with the number of open flowers on the plant, equations (1) and (2) must be modified. To examine the effect of such between-flower interaction, assume for simplicity that flower number per plant combines additively with the length of flower life to determine the fitness-accrual schedules, and that all flowers are open simultaneously. The fitness of the mutant may now be written as

w = (1/2) [R/(c + [t.sub.1]mc)/R/(c + [t.sub.2]mc)] (1 - [p.sup.[e.sub.1]]/1 - [p.sup.[e.sub.2]] + 1 - [g.sup.[e.sub.1]]/1 - [g.sup.[e.sub.2]]), (3)

where [e.sub.1] = [t.sub.1] + kR/(c + [t.sub.1]mc), [e.sub.2] = [t.sub.2] + kR/(c + [t.sub.2]mc), and k is a constant that scales in a linear manner with the influence of flower number per plant on pollen and seed fitness-accrual rates. Note that in this model, the size of the resource pool (R) and the cost per flower (c) cannot be ignored, as they influence the number of flowers that are built, and hence, the attractiveness of the plant to pollinators.

By examining [w.sub.max] when parameter values in equation (3) are changed, it can be seen that when separate flowers combine to increase the attractiveness of the plant to pollinators, there is selection for shortening of optimal flower life relative to the situation depicted in the basic model, in which fitness-accrual rates are independent of the number of open flowers per plant ([ILLUSTRATION FOR FIGURE 4 OMITTED]). That is, increasing the number of flowers per plant has an effect similar to that of increasing floral longevity, namely of allowing flowers to realize a greater proportion of their maximum contribution to the reproductive fitness of the plant, but in this case, over a smaller amount of elapsed time.

Reduction of Floral Maintenance Costs in Older Flowers

Floral maintenance costs may decline later in the flower's life (e.g., decreased nectar production) (Casper and La Pine 1984; Weiss 1991). The effect of such a decline can be examined by setting m in the basic model to a lower value when t exceeds some threshold, and examining how this influences the values of t at which [w.sub.max] is obtained. In general, doing so increases the optimal flower longevity. For example, with m = 0.05 and values of daily pollen and seed fitness-accrual rates that would normally select for 3-d flowers in plants with constant floral maintenance costs, reducing m to one fifth of its original value at the start of day 3 causes a shift in the optimal floral longevity from 3 to 5 d ([ILLUSTRATION FOR FIGURE 5 OMITTED]). Further reductions in m shift optimal floral longevities to still longer durations ([ILLUSTRATION FOR FIGURE 5 OMITTED]). Of course, there may be other costs to retaining flowers that are not represented in this simple model (e.g., increased floral predation, increased within-plant self-fertilization). Such costs, when present, would offset selection of longer-lived flowers.

DISCUSSION

Floral Longevity as a Specific Example of an Allocation Strategy

The evolution of floral longevity has been depicted above as an allocation strategy in which resources are expended either on the construction (of new) or the maintenance (of existing) flowers. Increasing the longevity of the flower is viewed as a direct consequence of increasing the allocation to floral maintenance. As in the case of other allocation strategies, the optimal allocations to the two competing functions are determined by how resource investment influences both the shape of gain relationship between investment in each function and fitness, as well as the overall opportunities available for fitness gain from each function. Lloyd (1985) demonstrated that when a and 1 - a represent proportions of a limiting resource spent on two competing functions (e.g., floral construction versus maintenance), the evolutionarily stable allocation strategy is given by a/(1 - a) = (y/z)([O.sub.a]/[O.sub.b]), where y and z in this instance are exponents that determine the shape of the fitness gain resulting from investment in floral construction and floral maintenance, respectively, and [O.sub.a] and [O.sub.b] are the respective number of opportunities for fitness arising from the two functions. By this approach, it can be seen that selection of increased floral longevity (i.e., greater overall allocation to maintenance of flowers) occurs when fitness gains from maintenance rise more rapidly and are large compared with those that would result from increased allocation to floral construction (z and/or [O.sub.b] larger than y and [O.sub.a]). This would occur when fitness-accrual rates are low and when the daily cost of floral maintenance is not excessively high, meaning that additional fitness gains may be realized by maintaining the flower compared with abandoning it and constructing a new flower (which itself would initially contribute little to fitness without additional maintenance). Continuing the analogy with other allocation strategies, a mixed strategy (Lloyd 1985) (i.e., the allocation of resources to both floral construction and floral maintenance, and leading to flowers with some appreciable life span), is likely to be common in most plants, because the two separate processes - building the flower and then maintaining it - each offer nonoverlapping opportunities for fitness gain.

Testing the Models

To test the models, it is necessary to accurately assess floral longevity. Because both pollination and environmental conditions may influence floral longevity, the most appropriate measure of floral longevity is for flowers protected from pollinators and grown under nonstressful conditions. These conditions would give an estimate of floral longevity that is not influenced by the intensity of pollination and subsequent postpollination changes (i.e., closer to the developmentally programmed floral longevity addressed by the model).

A complete test of the model would require information on all its parameters, but in the absence of some information, qualitative predictions may still be possible. For instance, when floral maintenance costs are unknown, it is still possible to discern general patterns relating optimal floral longevity to the other main parameters (fitness-accrual rates); for example, from the model results, it is expected that high rates of daily pollen and seed fitness-accrual will select for short-lived flowers across a large range of floral maintenance values ([ILLUSTRATION FOR FIGURE 2 OMITTED]). Likewise, very small rates of daily pollen and seed fitness-accrual should select for longer-lived flowers, again across a fairly broad range of m values (excluding very high m values). There appears to be some support for these predictions. For instance, species with flowers lasting several days or more, such as Asclepias tuberosa and the orchid Platanthera blephariglottis, are reported to have low to intermediate rates of pollinator visitation (Wyatt 1981; Cole and Firmage 1984), whereas tropical species pollinated by trap-lining (and presumably more reliable) euglossine bees, have shorter-lived flowers (Janzen 1971). Primack (1985) noted that alpine species generally had longer floral lifetimes than species growing nearby at lower elevations, as expected if pollination conditions are uniformly more reliable in the latter. We found additional support for the model by examining literature data on floral longevity and rates of pollination, for example, species with short floral longevities tended to be those characterized by high rates of pollinator visitation, whereas species with longer-maintained flowers tended to be those characterized by lower rates of pollinator visitation (Ashman and Schoen 1995).

To test the model in a more precise manner (i.e., to determine whether actual and predicted values of floral longevity correspond), data on pollen and seed fitness-accrual rates are required, together with data on floral maintenance and construction costs. Ideally, the time course of accrual of pollen and seed fitness should be examined under field conditions, and it is important to conduct such tests in habitats where there has been minimal disruption to the pollinator fauna; for example, plants growing in areas where native bees have been replaced by honeybees might exhibit schedules of pollen dissemination and receipt that are quite different from those under which the present-day floral longevity phenotype has evolved. If it is assumed that the disappearance over time of pollen from the anthers corresponds to the accrual of pollen fitness, whereas the appearance of pollen on the stigma corresponds to the accrual of seed fitness, then information on daily pollen loss and receipt can be equated to estimates of the daily pollen and seed fitness-accrual rates 1 - p and 1 - g. The accuracy of these assumptions should be verified, because pollen removal may be associated with pollen consumption by foraging insects, and pollen deposition may involve some inviable pollen grains or may occur on stigmas that are no longer receptive. The problems of estimating male fitness from pollen removal have recently been discussed by Snow and Lewis (1993).

When estimating floral maintenance and construction costs, decisions need to be made about which currency should be used to assess resource use. If carbon is assumed to represent the limiting resource, tests of the models require an assessment of the floral construction costs (in grams carbon, calories, and joules) together with daily respiration and nectar production rate measured in similar units. Moreover, if nectar is produced in the flower, it is important to distinguish between nectar present before the flower opens and begins to function (part of the construction cost) versus nectar that is produced after the flower opens, serving to replenish that lost to pollinators, nectar robbers, and evaporation (the maintenance nectar production).

Respiration costs are likely to be most significant for non-green flower parts, such as the corollas of many species. Studies in several species suggest that rates of photosynthesis in some floral parts (green sepals and ovaries) may roughly balance the respiration costs (Werk and Ehleringer 1983; Williams et al. 1985). Thus, the experimental effort to study floral respiration might best be spent on obtaining estimates from the corollas.

Also of relevance to quantitative tests of the models are cases in which a discrepancy is found between the observed and predicted floral longevities. These discrepancies can come about because floral longevity may show environmental variation and be influenced by other constraining factors that prevent the predicted optimum from being reached. When testing the model, it is worth noting that results such as depicted in figure 2, although indicating the particular combinations of model parameters for which different floral longevities are optimal, do not in themselves indicate to what extent a suboptimal floral longevity reduces fitness. For example, when floral maintenance values are low and either daily pollen or daily seed fitness-accrual rates are high, there is little variation in the relative fitnesses of a number of different intermediate floral longevity strategies (e.g., t = 3, 4, 5, and 6 days) ([ILLUSTRATION FOR FIGURE 2 OMITTED]). Direct information on relative fitnesses of different floral longevities, as in figures 3 to 5, is thus useful because it can reveal whether discrepancies between predicted and observed floral longevities provide strong grounds for rejecting the model. Rejection of the model would be most justifiable when the fitness of the observed relative to the predicted floral longevity strategy is much less than unity.

Examination of floral longevities in species with unisexual flowers (monecious or dioecious species) offers the opportunity to conduct both quantitative and comparative tests of the model. Such a test would involve the separate assessment of floral longevities in the male and female flowers and would provide a case where there is a possible contrast between the floral sexes in important model parameters (e.g., maintenance costs and fitness-accrual rates), while controlling for other aspects of the ecology and pollination biology of the species in question. Primack (1985) reported that in most species with unisexual flowers, the female flowers were longer-lived. According to the models above, this would suggest that seed fitness accrues more slowly than does pollen fitness, possibly because of the time delay between pollen deposition on the stigma and fertilization (Primack 1985).

Some Model Extensions

A number of extensions to the present model are possible, but in the interest of brevity, are not explored here. One extension would be to allow overlap in the resources used for flower and seed production. As in the case of other models in which the resource pool is allocated to more than two components (e.g., allocation of resources to male gametes, female gametes, and attractive structures) (Charnov and Bull 1986; Charlesworth and Charlesworth 1987), there are likely to be interactions among components in how they contribute to overall fitness. For example, the number of seeds produced is not only a function of resource availability at the time of seed maturation, but also of how many flowers have been produced (allocation of resources to floral construction) and how many ovules have been fertilized (allocation to floral maintenance). Geber and Charnov (1986) have modeled sex allocation in hermaphrodites when there is partial overlap in resource use by flowers and seeds.

Another possible extension pertaining to resource use, but relevant only to perennial plants and trees, would be to allow the floral longevity strategy to influence the energy reserves remaining at the end of the season. Also of interest is the possible effect of the breeding System on floral longevity. In selfing species, where pollen dissemination and receipt are likely to occur early on in the flower's life (i.e., daily pollen and seed fitness-accrual rates are high), one might expect that floral longevities would be reduced relative to outcrossing species. This appears to be the case in cleistogamous flowers. Finally, optimal floral longevity, when the continuation of flower maintenance is adjusted facultatively in response to pollination, is expected to be shorter on average than when floral maintenance is a fixed (nonadjustable) strategy, as assumed here. An extension of the model to cover such facultative responses to pollination may be relevant to some species.

Longevity of Other Repeated Structures in Plants

As noted above, other structures, apart from flowers (e.g., leaves, shoots, roots, and fruits), show variation in longevity (Kozlowski 1973; Thimann 1980; Chabot and Hicks 1982). For example, leaf longevity varies from several weeks to several years among different higher plant species (Chabot and Hicks 1982). In some plants, leaf senescence may follow a particular spatial pattern, for example, leaves lower on the shoot system may senesce first, followed by leaves higher on the shoot system (Leopold 1961). Such programmed senescence of leaves can be understood in terms of the principles outlined above. For instance, leaves contribute to overall plant fitness (through photosynthetic activity), but they also require maintenance (respiration) for continued functioning. When the plant first begins to grow, the initial daily rate of fitness contribution for leaves on the lower portions of the shoot system may be relatively high. But as the plant increases in size, lower leaves may become shaded by other leaves. The fitness-accrual rate for the leaf, therefore, declines while maintenance remains more or less constant. Eventually it may be more profitable to invest resources in building new (higher-positioned) leaves than maintaining the existing lower leaves. This viewpoint, which could be extended to provide a quantitative model of optimal leaf longevity, shares features with the cost-benefit approach to leaf carbon economy advocated by Chabot and Hicks (1982). It may be profitable to explore the optimal longevities of nonreproductive organs in modular organisms using the general principles outlined above for floral longevity.

ACKNOWLEDGMENTS

This work was supported by an Operating Grant to D.J.S. from the Natural Sciences and Engineering Research Council of Canada (NSERC), an International Post-doctoral Fellowship to T.L.A. from NSERC, and by a grant from the Fonds pour la Formation de Chercheurs de l'Aide a la Recherche (FCAR-Quebec). We thank M. Johnston, L. Harder, M. Morgan, S. Salant, and J. Thomson for suggestions and comments on the models and manuscript.

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