R N EL n

dt 0E ^ 2 2E V 1 + hNj where E is the foraging effort, "P(E) represents predator fitness, dT + d2E is the predator death rate, and u is the additive genetic variance of the foraging effort. dE/dt is proportional to the derivative of the predator's fitness with respect to the foraging effort E. We assumed a saturated numerical response ^[bEN/(1 + hN)]. If we assume "P(E) = [—dT — d2E + bEN/(1 + hN)], there is no equilibrium, because dP/dt =— dTP when dE/dt = 0. Therefore, a nonlinear tradeoff between foraging effort and predator death rate is needed.

Why do predators not evolve a highly efficient capture ability that would result in prey extinction? In the late 1960s, Lawrence Slobodkin explained that predators avoid overexploitation of prey because prey extinction is not beneficial to predators themselves. This is called 'prudent predation'. There is no evolutionary mechanism by which evolution of a species avoids causing the extinction or reducing the population of another species. Avoidance of overexploitation evolves because excessive foraging by a predator does not benefit that predator.

The predator death rate increases as energetic costs increase or the risk of predation by a species at a higher trophic level increases. The optimal foraging effort is obtained by 0"P(E)/0E = 0, where "P(E) is given by the first equation in model [3]. Therefore, the optimal foraging effort is bN/4d22(1 + hN). The optimal foraging effort decreases as d2 increases. For a general form of the numerical response, the optimal foraging effort decreases as the number of enemies increases. In contrast, the optimal foraging effort can either increase or decrease with an increasing number of prey, depending on the form of the numerical response. In the case of model [3], foraging effort increases as the number of prey increases. The sensitivity of the prey density to the foraging effort is controversial.

Predator evolution has a stabilizing effect on the prey-predator dynamics (Figure 2). Without evolution (u = 0), the interior equilibrium is unstable. However, no empirical evidence matches this theoretical prediction. Even with predator evolution, a one-quarter period lag is generally obtained (Figure 2).

In conclusion, the evolution of antipredator traits by a prey species is more likely to cause instability than the evolution of foraging traits by the predator. When the evolution of prey traits destabilizes a prey-predator system, it produces cycles in both population size and trait values.

Prey can change antipredator traits. For the mathematically simplest case of prey-predator dynamics with prey evolution, dP /dt = P [ -d + bCN /(1 + hCN)]

dC /dt = v [0" (C, C)/0C] c= c = v[q- P/(1 + hCN)]

where "(C, C) is the logarithmic fitness of an individual prey with an antipredator effort of C and a population average antipredator effort of C, r from model [2] is replaced by R + qC as an increasing function of prey

Vulnerability C

Predator P

Vulnerability C

Predator P

Time, t

Time, t

or at 3

d re

Pr 2

/ \ j

f ^ i

\ /

1

600

800 1000

1.5 2 Prey, N

Figure 2

Prey-predator dynamics with the predator evolution of model [3]. Parameter values are: h = 0.5, b = 0.654, d1 = 0.18, r = 1, k = 0.3; u = 0.4 in panel (a) and u = 1.5 in panel (b).

vulnerability (the tradeoff between the antipredator effort and the growth rate of the prey population), v represents the rate of evolutionary change in trait value C.

In this quantitative genetic model, the rate of evolutionary change in trait value is proportional to the additive genetic variance and the selection differential of an individual with a slightly higher trait value than the average. Here we simply assume that v is a positive constant, although it is possible that the additive genetic variance may change with trait values.

We have assumed that w(C, C) = [R + qC - kN - CP/ (1 + hCN)]. Note that the numerator of the functional response, CP/(1 + hCN), is proportional to the individual trait value C, although the denominator is a function of the population average C. This is because a predator's handling time is affected by the average prey vulnerability, but each prey experiences a risk of predation dictated by its own vulnerability. This fitness therefore depends on both the population average and individual trait values. This is called frequency-dependent selection. Frequency-independent selection usually increases population abundance because no gap exists between individual optimization and group optimization. In contrast, frequency-dependent selection often decreases population abundance and can drive a population to extinction.

Model [4] exhibits effects of prey evolution on prey-predator cycles. Prey evolution often destabilizes prey-predator systems (Figure 3 a). In addition, there is a one-half period lag between prey and predator cycles. Some empirical evidence suggests that multiple prey population clones play an important role in prey-predator cycles and in the occurrence of the one-half period lag (Figure 1d). This occurs because prey availability represents the number of prey individuals that are not vigilant, CN. Therefore, prey availability oscillates with a one-quarter period lead on the predator's oscillation (Figure 3 a). Prey availability decreases with either

Vulnerability C Predator P

Vulnerability C Predator P

200 400 600 800 1000 Time, t

Figure 3 Prey-predator dynamics with the predator evolution of model [4]. R = 2.5, k = 1; v = 0.05 in panel (a) and v = 0.07 in panel (b).

200 400 600 800 1000 Time, t decreasing prey density or vulnerability. In Figure 3 a, prey density begins to decrease while population vulnerability is still increasing and when predator density is still small. Prey density decreases when the product of predator density and prey vulnerability, PC, is large. The rate of prey evolution determines the stability of the prey-predator system. A small rate of prey evolution stabilizes the equilibrium. With higher rates of prey evolution, the prey-predator system destabilizes and shows a stable limit cycle or chaos (Figure 3 b).

To evaluate contributions of evolutionary change and population dynamics to prey-predator systems using empirical data, we examined the relationship between mean population fitness, trait values, and density; namely w in model [4]. To avoid mathematical complexity, fitness is assumed to be a function of trait value and density, or frequency-independent selection. Mean fitness of the population w(C, N) changes with dw/dt = (8w/8C)(dC/ dt) + (8w/8N)(dN/dt). During empirical observations by Shertzer etal., temporal changes of density and trait values were obtained. If the functional form w(C, N) is given, the contribution of evolutionary change and that of population dynamics can be compared between (8w/8C)(dC/dt) and (8w/8N)(dN/dt). A few empirical studies have incorporated time series for both density and trait values, but very few empirical efforts have estimated the fitness function. In some chemostat systems, the magnitudes of these factors are comparable. During Darwin's fieldwork with finches in the Galapagos Islands, evolutionary change in prey size had a larger effect on fitness than did population dynamics.

The rate of prey evolution, v, is likely very slow when it is driven by genetic point mutations. The mutation rate of a single gene is usually 10~4 or smaller per generation per genome, and most mutants are lethal or selectively disadvantageous. However, the rate of evolution is comparable to the rate of population dynamics if

Prey, N

Prey, N

Parameter values are: h = 1, b = 1, q = 0.8, d = 0.5.

Figure 3 Prey-predator dynamics with the predator evolution of model [4]. R = 2.5, k = 1; v = 0.05 in panel (a) and v = 0.07 in panel (b).

the population mean trait value depends on the gene frequency of allele polymorphism or a polygenic quantitative trait. In Figure 1d, the prey trait value varies with the gene frequency of clones. In addition, the rate of evolution is much higher than the population dynamics if the prey's behavior changes due to phenotypic plasticity of a single strategy.

Evolutionary change due to either point mutation, genetic polymorphism, or behavioral plasticity is described by model [4] for a range of values for the rate of evolution, v. Evolutionary traits change to increase individual fitness, and the velocity of change is likely proportional to the gradient of the individual fitness with respect to its trait value, Sw(C, C)/SC in model [4]. Figure 1d suggests that evolutionary change may play an important role in the characteristics of prey-predator cycles.

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