Functional Responses and Population Dynamics

Benefits and costs are rarely fixed attributes of species interactions, but rather vary with the abundance or population density of mutualistic partners. In other words, benefits and costs of mutualism exhibit functional responses. In its most general application, a functional response represents how the rate of change of one population varies with the density or abundance of individuals of another population. Historically, most models of mutualism simply used linear or saturating type 2 functional responses, with little consideration of the underlying biological mechanisms. Expressing functional responses of mutualism in terms of benefits and costs provides a mechanistic basis for understanding mutualism's influence on the intrinsic growth rate and population dynamics of interacting species.

The difference between benefit (B) and cost (C) functional responses equals the net effect (NE) functional response of mutualism on the rate of change in the size of a mutualist's population (i.e., NE = B - C). Many different scenarios are theoretically plausible for functional responses of benefits and costs, and hence for net effect functional responses (Figure 2). Figure 2a presents the scenario in which net effects to a mutualistic population, mutualist 2, increase linearly with the population density of its mutualistic partner, mutualist 1. That is, the more mutualists there are, the better and better a partner fares. It is this ever increasing net effect functional response that gives rise to the unrealistic result of unbounded population growth that was typical of early theoretical studies. In nature, however, many different limitations prevent net effect functional responses from increasing continually. Benefit and/or cost functional responses to mutualist 2 may saturate or diminish with increasing population density of mutualist 1, such that net effect functional responses saturate or diminish as well. Figures 2b-2f represent examples of how benefit and cost functional responses may vary with one another, and hence give rise to different net effect functional responses. Other shapes of benefit and cost functional response are feasible as well.

Population size or density of mutualist 1

Figure 2 (a-f) Examples of potential functional response curves in terms of benefits (B), costs (C), and net effects (NE) on the per capita rate of reproduction and/or survival of a mutualist population, mutualist 2, as a function of its partner's population size, mutualist 1. Benefits positively affect the rate of reproduction and/or survival, while costs negatively affect the rate of reproduction and/or survival, such that NE = B - C.

Population size or density of mutualist 1

Figure 2 (a-f) Examples of potential functional response curves in terms of benefits (B), costs (C), and net effects (NE) on the per capita rate of reproduction and/or survival of a mutualist population, mutualist 2, as a function of its partner's population size, mutualist 1. Benefits positively affect the rate of reproduction and/or survival, while costs negatively affect the rate of reproduction and/or survival, such that NE = B - C.

The well-known mutualism between yucca plants and yucca moths serves as one example of how benefit, cost, and net effect functional responses to one mutualist (the plant) vary with the population size or density of its pollinating insect partner (the moth). This example is graphically depicted by Figure 2c. Yucca moths both pollinate yucca flowers and lay their eggs into them. Eggs hatch to produce larvae that consume developing seeds. Thus, both benefits and costs ofyucca moths affect plant reproduction, via their effects on seed production: it is increased through moth pollination, and decreased through the consumption of developing seeds by the pollinator's offspring. If moth density is low, then the benefits of pollination to plant reproduction are small (B in Figure 2c). As moth density increases, more flowers are pollinated. However, at some point, moths are sufficiently abundant that all flowers become pollinated; that is, the benefit functional response of moths to plants saturates (B in Figure 2c). Further increases in moth density do not lead to greater seed production. Rates of oviposition, and hence costs ofseed consumption, follow a similar pattern, except that, given the biology of yucca moths, oviposition occurs at a lower rate than pollination. Ifmoth density is high, then costs oflarval seed consumption to plant reproduction are large, such that nearly all seeds are consumed by larvae of eggs laid in flowers (C in Figure 2c). Thus, the net effect functional response for plant reproduction via seed production, NE = B - C, is a unimodal function ofmoth density.

Two-species models that incorporate benefit and cost functional responses show that the dynamics and stability properties of populations involved in mutualisms can differ greatly depending upon the shapes of these functional response curves. As indicated above, the great diversity in natural history and in associated benefits and costs among mutualisms has hampered development of generalizations about these interactions. Expressing functional responses in terms of benefits and costs provides one general theory for mechanistic understanding ofhow mutualism influences the growth, dynamics, and ecological stability ofinteracting species. Nevertheless, as yet we know little about the shapes of these relationships in nature.

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