Present address: Shengqiang Liu, Department of Mathematics, Xiamen University, Xiamen, 361005, China. e-mail: email@example.com
Éva Kisdi, Rolf Nevanlinna Institute, Department of Mathematics and Statistics, FIN-00014 University of Helsinki, Finland. Tel.: +358 9 1915 1489; fax: +358 9 1915 1400; e-mail: firstname.lastname@example.org
Several consumers (predators) with Holling type II functional response may robustly coexist even if they utilize the same resource (prey), provided that the population exhibits nonequilibrium dynamics and the handling time of predators is sufficiently different. We investigate the evolution of handling time and, in particular, its effect on coexistence. Longer handling time is costly in terms of lost foraging time, but allows more nutrients to be extracted from a captured prey individual. Assuming a hyperbolically saturating relationship between handling time and the number of new predators produced per prey consumed, we obtain three results: (i) There is a globally evolutionarily stable handling time; (ii) At most two predator strategies can coexist in this model; (iii) When two predators coexist, a mutant with intermediate handling time can always invade. This implies that there is no evolutionarily stable coexistence, and the evolution of handling time eventually leads to a single evolutionarily stable predator. These results are proven analytically and are valid for arbitrary (not only small) mutations; they however depend on the relationship between handling time and offspring production and on the assumption that predators differ only in their prey handling strategy.
One limiting resource can support only one consumer species if population densities of the consumer and of the resource attain a stable point equilibrium (Volterra, 1927; MacArthur & Levins, 1964). In contrast, several consumer species may coexist utilizing a single resource if the populations undergo sustained oscillations, either because of the intrinsic dynamics of the system (Koch, 1974; McGehee & Armstrong, 1977; Levins, 1979; Armstrong & McGehee, 1980; Huisman & Weissing, 1999) or as a consequence of periodic or stochastic environmental effects (Levins, 1979; Kisdi & Meszéna, 1993; the relative nonlinearity mechanism of Chesson, 1994). Nonequilibrium coexistence is possible if at least one consumer's population growth rate is a nonlinear function of resource abundance. In the this case, long-term average population growth depends not only on the average resource abundance but also on the variance and higher moments of resource density; in effect, the variance and higher moments act as environmental feedback variables (effective resources) along with the average resource level (Levins, 1979; Meszéna & Metz, 1999).
Nonequilibrium coexistence is particularly well studied in simple predator–prey models with Holling type II functional response of the predator (Koch, 1974; Hsu et al., 1978a,b; Armstrong & McGehee, 1980; Muratori & Rinaldi, 1989; Abrams & Holt, 2002; Abrams et al., 2003; Liu et al., 2003; Wilson & Abrams, 2005). The saturating functional response introduces nonlinearity into the growth rate of the predator, and at the same time can cause stable limit cycles in the population dynamics. The point equilibrium of the predator and its prey is destabilized because saturated predators cannot regulate the prey, allowing high prey densities to grow further (i.e. until predator density also rises); and on the contrary, unsaturated predators capture prey efficiently, causing low prey densities to decrease further (i.e. until predator density also crashes). The resulting population cycles imply temporal variation in prey density. To see how this variation leads to coexistence, recall that predator saturation prevents fast predator growth when the prey is abundant, and therefore periods of high prey densities do not compensate for periods of low prey densities. Variance in prey density thus decreases the long-term average growth of the predator, which must be compensated by higher average prey density. This facilitates the invasion of a second predator species that needs relatively high prey density to grow, but has a less saturating functional response and hence is relatively insensitive to the variation in prey density. As the second predator spreads, the joint population dynamics is increasingly influenced by its less saturating functional response, and therefore the amplitude of the prey cycles diminishes. Less variance in prey density, in turn, favours the first predator. As a result, the two predators may coexist in a protected manner (Armstrong & McGehee, 1980).
The Armstrong–McGehee mechanism leads to robust coexistence of predators in a range of related models (Abrams & Holt, 2002). Population cycles are known to be widespread in nature (Kendall et al., 1998) and prey densities normally vary also due to environmental effects. This should facilitate nonequilibrium coexistence of predators. Despite the widespread opportunity, however, there is no empirical demonstration of coexistence due to the Armstrong–McGehee mechanism in natural systems (but see Nelson et al., 2005 for coexistence of artificially cultured lines).
This lack of evidence may well be attributed, at least in part, to lack of appropriate data (Abrams & Holt, 2002; Nelson et al., 2005). To assess how frequently we expect coexistence by a certain mechanism in nature, however, we should also consider how evolution changes the individual traits that determine coexistence.
The simplest mechanism leading to a nonlinear functional response is that predators need a certain time to handle (kill, consume and digest) their prey. Handling time is an individual trait that influences coexistence because it determines the shape of the resulting Holling type II functional response: the functional response is nearly linear if handling time is short, but saturates fast if handling time is long. The most important condition for coexistence is that the functional responses of the predators are sufficiently different (Abrams & Holt, 2002), and this will be fulfilled when the predators have sufficiently different handling times.
In this article, we begin to investigate the evolution of functional responses and the impact on nonequilibrium coexistence of predators by modelling the evolution of handling time. We assume that handling time is a genetically determined trait with no phenotypic plasticity. A predator with long handling time is at a disadvantage because it can capture only a limited number of prey when prey is abundant. On the contrary, long handling time allows the predator to extract more nutrients from an individual prey, e.g. by consuming also those body parts that are harder to access or by having slower gut passage for more efficient digestion. We thus assume that longer handling time results in a higher conversion factor between the number of prey consumed and the number of new predators produced, whereas the disadvantage of long handling appears in the saturated Holling type II functional response.
The functional relationship between handling time and the conversion factor could take many forms. In this article, we assume that the conversion factor is a hyperbolic function of handling time. This choice is biologically reasonable (see below) and makes algebraic derivations possible. We assume that all predators have a common trade-off relationship between handling time and the conversion factor, and they also share all characteristics that are unrelated to handling time (i.e. they have the same prey capture rates and death rates). These assumptions restrict the model to not-too-distantly related predators.
In the present model, we show analytically that evolution destroys any coexistence and leads to the establishment of a single predator with an evolutionarily stable handling time. Our results also suggest that the predator ESS is unique. To obtain these results, we use the modelling framework of adaptive dynamics (e.g. Geritz et al., 1998). We however do not assume that mutations cause only small changes in the phenotype; our results thus hold globally.
A companion paper (S.A.H Geritz, E. Kisdi and P. Yan, unpublished data) will investigate arbitrary functional relationships between handling time and the conversion factor, resorting to local and, in part, numerical analysis. It is then possible to construct examples where handling time undergoes evolutionary branching and different predators coexist in an evolutionarily stable system. For a class of biologically reasonable functions, however, evolutionarily stable coexistence of different handling strategies of otherwise similar predators is ruled out.
where X and Yi are, respectively, the population densities of the prey and of predator i, r and K are the parameters of logistic population growth, β is the capture rate per searching time and D is the death rate of predators. Predator strategies (or species) differ only in their handling time, Hi The conversion factor between prey consumed and new predators produced,Γ(Hi), is a function of handling time. By introducing the dimensionless variables x = X/K, yi = βYi/r and t = rT, eqn 1 can be written as
where the scaled parameters are hi = βKHi, γ(hi) = βKΓ(hi/βK)/r and d = D/r.
A predator strategy h is viable if γ(h) > d(h + 1). With a single predator, the system has a nontrivial point equilibrium at and . By the Rosenzweig & MacArthur (1963) criterion, the point equilibrium is stable for h ≤ 1; for h > 1, the predator–prey system has unique stable limit cycle when γ(h) exceeds d(h + 1)h/(h − 1), whereas the nontrivial equilibrium is stable when γ(h) falls between this value and the minimum for viability (Fig. 1). With two or more predator strategies, the population dynamics can exhibit complex behaviour including chaos (Abrams et al., 2003; see below).
We use the model in eqn 2 to investigate the evolution of handling time (h). Longer handling times give the advantage that more nutrients can be extracted from a captured prey individual, and therefore more new predators can be produced from a single capture. γ(h) must therefore be a nondecreasing function, and it must saturate at some level c that corresponds to the total resource content of a prey individual. γ(h) is obviously zero for no handling (h = 0), and may also be zero for short handling times (h ≤ h0) if some initial handling is necessary e.g. to kill the prey, during which time the predator does not acquire nutrients yet. In accordance with these biological properties, we assume that γ(h) increases as a hyperbolic function for h > h0 ≥ 0,
where θ determines how fast nutrient gain saturates with handling time.
Predators with very long handling times are not viable because they spend most of their time handling a carcass that has hardly any nutrients left. Obviously, very short handling times are not viable either (e.g. a predator with h ≤ h0 cannot reproduce at all). Viable handling times are between and , where and B = h0c/d + θ. Within the interval (hmin,hmax), there may be a range of handling times (η1,η2) where a single predator and the prey settle on a limit cycle (Fig. 1).
Evolution of handling time
If the predator–prey system attains a stable point equilibrium, then only one predator strategy can subsist on one prey. As the population growth rate of the predator is an increasing function of prey density, a new mutant strategy hmut is able to invade the resident population of strategy h if and only if . Assuming equilibrium population dynamics throughout, evolution leads to the optimal strategy hopt that minimizes the equilibrium density of prey, (MacArthur & Levins, 1964; Tilman, 1982; Mylius & Diekmann, 1995; Metz et al., 1996; Kisdi, 1998). With γ(h) given by eqn 3, the optimal handling time is .
The above argument, however, does not hold when the system has nonequilibrium dynamics. Assume that the population has settled on a limit cycle of length T, and let x(t) represent the time course of prey density along the limit cycle. The long-term invasion fitness of a rare predator with strategy hmut is then given by
(Armstrong & McGehee, 1980; Metz et al., 1992). This formulation of the invasion fitness can be used with any number of resident predator strategies. For the ease of presentation, we shall first assume that the resident population has a limit cycle with finite length T and discuss chaotic dynamics (which can occur only with more than one predator) later.
Mutants with positive invasion fitness may spread (unless lost to demographic stochasticity while rare); mutants with negative invasion fitness always die out. To obtain the invasion fitness of hmut in a population with given resident strategies hi, the limit cycle must be determined from eqn 2 and x(t) from this limit cycle must be substituted into eqn 4. Unfortunately, x(t) cannot be calculated analytically when the population dynamics are cycling, and thus the invasion fitness of a mutant cannot be obtained explicitly. Nevertheless, we can characterize the evolution of handling time almost completely. To do this, we shall use adaptive dynamics (Geritz et al., 1998), but in contrast to most models of adaptive dynamics that assume small mutations, we do not pose any restriction on which strategy can appear as a new mutant.
With a single resident predator, the model always has a singular strategy, h*, where the fitness gradient vanishes. If this were not true, then evolution by small mutations would lead to extinction either at hmin (if the fitness gradient is negative) or at hmax (if the fitness gradient is positive), with the equilibrium predator density gradually declining to zero; this kind of extinction is however excluded by Gyllenberg & Parvinen (2001). Therefore, there must be a singular strategy h* where the fitness gradient changes sign. Note that the optimal strategy hopt, found when the population dynamics have a stable equilibrium, is a special case of a singular strategy h* (cf. Meszéna et al., 2001).
In the Appendix, we prove the following three results.
1The singular strategy h* is always globally evolutionarily stable, i.e. no mutant predator can invade a population where handling time has evolved to h*. In particular, evolutionary branching (Geritz et al., 1998) is not possible in the present model.
2At most two handling strategies can coexist. (In general, three or more predator strategies may coexist in cycling populations (cf. Armstrong & McGehee, 1980), but this is possible only if γ(h) is not a hyperbolic function or if strategies differ also in some other traits, not only in handling time.)
3There is no evolutionarily stable coexistence. Because of result 2, we only need to consider the coexistence of two predator strategies. As we show in the Appendix, two coexisting strategies can always be invaded by a mutant with handling time in-between.
The first result means that an initially monomorphic predator population will stay monomorphic when handling time evolves by small mutations, although two coexisting predators can emerge by large mutation or by immigration of a different species if the resident population is not yet at the ESS. The last result, however, implies that any coexistence is eventually resolved and evolution leads to a single predator with a globally evolutionarily stable handling time.
In addition, our results suggest that the ESS handling time is unique. There are three lines of evidence supporting this. Firstly, inspection of a number of numerical examples (two of which are shown below) failed to turn up multiple ESSs. Secondly, numerical results obtained in a more general model (S.A.H Geritz, E. Kisdi and P. Yan, unpublished data) exclude the existence of more than one singular strategy whenever γ(h) is a concave function. Thirdly, we note that if there were more than one singular strategy, then at least one of them would be a repellor but all of them would be ESSs. In the neighbourhood of an ESS repellor, there must be pairs of strategies that mutually exclude each other (Geritz et al., 1998). Mutual exclusion (where neither strategy can invade the other's population) has not been observed in the predator–prey model of eqn 1.
We illustrate the above results with two examples in Figs 2 and 3. To construct these plots, we numerically determined the limit cycle of a one predator–prey system with resident handling time h, and used x(t) from this limit cycle to calculate the invasion fitness in eqn 4. When the system attains a point equilibrium instead of a limit cycle, then eqn 4 simplifies to , where .
The left panels of Figs 2 and 3 show two typical pairwise invasibility plots (PIPs; Geritz et al., 1998). In those parts where both strategies have stable population dynamics (both strategies are either smaller than η1 or greater than η2), the PIP is skew symmetric, i.e. areas with positive invasion fitness exactly overlap with areas of negative invasion fitness when mirrored on the main diagonal hmut = h. This is because the strategy with smaller equilibrium prey density invades and ousts the other one (see Metz et al., 1996). Mutual invasibility, which implies coexistence in cycling populations, is possible if at least one strategy is in-between η1 and η2 (right panels of Figs 2 and 3). Because of the Hopf bifurcations in the population dynamics when the limit cycle is born, the boundary line between areas with positive and negative invasion fitness is continuous but not differentiable (not ‘smooth’) when the resident strategy h equals η1 or η2.
In the case shown in Fig. 2, the population dynamics at the optimal handling strategy hopt have a stable fixed point. In the neighbourhood of hopt, the model obeys the optimization principle of minimizing the equilibrium prey density, and hence the PIP is locally skew symmetric. Coexistence is possible for certain combinations of handling times that are sufficiently different from one another and also sufficiently different from hopt. In Fig. 3, there is no optimal handling time in the intervals of stable population dynamics. There is nevertheless a singular strategy h*, which is convergence stable and globally evolutionarily stable. The qualitative difference between Figs 2 and 3 is that in the latter, mutual invasibility extends to the neighbourhood of h*.
As Figs 2 and 3 demonstrate, two predator strategies can robustly coexist as long as no mutations occur. Evolution by mutations and substitutions, however, drives the two strategies closer to one another (see result 3 above). In Fig. 2, evolution necessarily leads to the extinction of one strategy, and the remaining strategy evolves to hopt. In Fig. 3, two predator strategies may remain in the population, but the difference between them shrinks to zero as they both evolve to h*. The population may also alternate between a single predator and two different predator strategies, but in any case, handling times will evolve arbitrarily close to h*.
For simplicity, so far we considered resident populations that settle on limit cycles; these cycles may be arbitrarily complicated but of finite length. With two resident predator strategies, however, the population may have chaotic dynamics (Abrams et al., 2003). In the example of Fig. 3, we found complex dynamics, including period doubling bifurcations of limit cycles and chaos, when the two resident strategies are considerably different (in Fig. 2, all populations exhibit simple limit cycles). In Fig. 3b, black dots denote points where the dominant Lyapunov exponent of the resident dynamics (eqn 2) is positive. Lyapunov exponents were obtained using the continuous orthonormalization procedure of Christiansen & Rugh (1997) over 50 000 time units (after allowing the transients to die out), starting from a single initial point. We note that the positive Lyapunov exponents were small (of the order of 10−3), and sometimes very difficult to calculate. When one of the predator strategies has a stable fixed point and the other one can barely invade, then it can take a very long time before the invading strategy becomes sufficiently common to destabilize the dynamics. Then in a fast burst of oscillations, the invading strategy crashes and the slow process of invasion starts over. These dynamics are similar to the asynchronous cycles described by Abrams et al. (2003), but the cycles can be much longer and need not repeat exactly. The bursts of oscillations are very important for the Lyapunov exponent, but they are few and far between.
In general, there is rather little known about adaptive dynamics in chaotic populations. The invasion fitness can be generalized by taking the limit T→∞ in eqn 4, and this limit exists under very mild conditions when computed along a specific trajectory of the resident population (Ferriere & Gatto, 1995). Fortunately, very little is needed to prove our results: we use only that the fitness of any resident strategy is zero, which is true because population densities are bounded.
With two predator strategies, the resident population may also have multiple attractors such as alternative stable limit cycles (Abrams et al., 2003). These pose no problem as our results hold for each attractor.
Nonequilibrium coexistence is a well-known mechanism whereby several predator species may coexist such that they are limited by the same prey. Our model exhibits nonequilibrium coexistence of predators with different handling times in their Holling type II functional responses (see the right panels of Figs 2 and 3). This coexistence is ecologically robust in the sense that small changes in handling times or in other parameters will not eliminate it. Coexistence is however not stable on an evolutionary time-scale: the evolution of handling time eventually destroys any initial coexistence in this model, and results in a single species with a globally evolutionarily stable strategy. This result holds under the assumptions that predators share a common trade-off function and their characteristics unrelated to handling time are the same, and moreover the trade-off between handling time and the conversion factor is given by a hyperbolic relationship.
There is a large body of literature that investigates the impact of evolution on cyclic behaviour in predator–prey systems (see Abrams, 2000 for review). These models, however, asked whether evolution leads to cycles or whether cyclic population dynamics itself is evolutionarily stable. In contrast, we focused on the question whether coexistence can be evolutionarily stable in cyclic populations. The ESS predator in our model may (Fig. 3) or may not (Fig. 2) exhibit cyclic population dynamics, but in any case evolution excludes coexistence.
Nonequilibrium coexistence depends on the shape of the functional response. We restricted this study to Holling type II functional responses because they have a mechanistic interpretation: the number of captured prey saturates because predators spend an increasingly long time by handling. Accordingly, the evolution of the shape of type II functional responses can be understood via the evolution of handling time, a well-defined individual trait. Recall that many ecological models investigate nonequilibrium coexistence assuming that one species has type I functional response whereas the other species has type II (e.g. Armstrong & McGehee, 1980; Abrams & Holt, 2002; Abrams et al., 2003; Wilson & Abrams, 2005). A type I functional response is possible only with zero handling time, which can obviously be only an approximation to reality; indeed, a short handling time inserted in a type II functional response yields an almost linear (type I) functional response. However, assuming that nutrient extraction at least initially increases with time, short handling times are not evolutionarily stable (very short times are not even viable). Thus evolution does not favour a type I functional response.
In contrast to most previous papers, we assumed a constant capture rate and investigated the evolution of handling time and the efficiency of conversion of prey into new predators. So far, this problem has largely been ignored: the review of Abrams (2000) identified this as a gap in existing theory. We are aware of only one previous model where an evolving trait influences handling time (Dercole et al., 2003). This model excluded cyclic population dynamics and hence excluded coexistence; moreover, handling time depended on both prey and predator traits while the same traits also affected the capture rate and the strength of intraspecific competition in the prey, but not the conversion efficiency.
We assumed that handling time is a heritable trait of the predator, whereas the conversion factor, which is directly proportional to the amount of nutrients extracted from a prey individual, is a function of handling time. Accordingly, only the predator evolves in our model. It is certainly reasonable to assume that the predator has full control over how much time it will spend by handling the prey. The conversion factor could however be influenced by prey traits. If it is harder to extract the nutrients from the body of the prey, then the same handling time yields a smaller conversion factor, which changes selection on handling time. Assuming that captured prey cannot survive, changes in these prey traits are adaptive only if they give some indirect advantage to living prey. For example, they may correlate with the escape probability of the prey, or keeping the predator busy for a longer time may benefit related prey individuals via kin selection.
Predator strategies with different handling times can coexist when prey density oscillates because high and low prey densities favour respectively short and long handling times. In reality, predators may behaviourally adapt their handling strategy to the density of prey. Such phenotypic plasticity is not considered in our model. Notice, however, that a simple ‘ideal free’ form of phenotypic plasticity immediately excludes the evolutionarily stable coexistence of different handling strategies. Assume that predators know the density of prey at every instant, and can change their handling strategy with no delay and at no cost. Then the plastic strategy that chooses handling time h as a function of x(t) such that its instantaneous birth rate, γ(h)x(t)/(1 + hx(t)), is maximal at every instant will produce more offspring than any other fixed or plastic handling strategy, and therefore it will oust every other strategy (cf. eqn 2). To study the evolution of coexistence, we started from the opposite extreme of genetically determined handling strategies. A priori, this is more conducive to coexistence (although a posteriori evolutionarily stable coexistence is excluded in our model), and should also approximate the case when behavioural adaptation is too slow to be useful or too costly to be worthwhile.
The method of adaptive dynamics that we used assumes that evolution is mutation-limited, so that the fitness of a new mutant can be calculated assuming that the resident population has settled on its population dynamical attractor, and the mutant remains rare for long enough the attractor to be sampled sufficiently (cf. eqn 4). This amounts to a separation between the ecological (fast) and evolutionary (slow) time scales. In the case of extremely long limit cycles or chaotic resident populations, we need to assume at least that the right hand side of eqn 4 converges sufficiently fast with increasing T. The approach of adaptive dynamics gives the same result as the quantitative genetic equation that governs the evolution of continuous traits under frequency-dependent selection, provided that the genetic variance of the trait is small in each species and evolution is slow compared with population dynamics (Abrams et al., 1993). Fast evolution may change the dynamical properties of a predator–prey system considerably. For example, Abrams (1992) found that fast predator evolution is able drive joint cycles of population densities and a predator trait that increases the capture rate but is costly in terms of mortality, even though with slow (or no) evolution, the population would attain a stable equilibrium. It however seems unlikely that fast evolution would facilitate coexistence: quickly evolving handling times may approximate the case of instantaneous behavioural adaptation discussed above, and this excludes coexistence. Abrams (2003) demonstrates that fast evolution opposes the diversification of a trait that increases prey capture at the cost of increasing vulnerability to a top predator.
The present model is only a first step to understand the adaptive dynamics of handling time in general, and its consequences on biodiversity in particular. The strength of this model is that we can derive our results analytically, and, contrary to most models of adaptive dynamics, we can allow for mutations of arbitrary size. These benefits, however, come at a considerable cost: the results hold only for the hyperbolic trade-off function given in eqn 3. Other functions do not permit the algebraic derivation presented in the Appendix. A hyperbolic relationship between handling time and the conversion factor is biologically reasonable, but it is not the only reasonable choice.
In a companion paper (S.A.H Geritz, E. Kisdi and P. Yan, unpublished data), we investigate the evolution of handling time without imposing a priori a certain functional relationship with the conversion factor. This more general model can be analysed only locally (assuming small mutations) and in part only numerically. The results show that there is an evolutionarily stable handling time whenever the conversion factor is a sufficiently concave function of handling time (as in the present model). With less concave or slightly convex functions, however, evolutionarily stable coexistence of different predator strategies becomes possible. Moreover, handling time may undergo evolutionary branching, whereby a single predator strategy splits and evolves into two different strategies. In a class of biologically important models nevertheless, evolution of handling time destroys the coexistence of otherwise similar predators and leads to a single evolutionarily stable predator species.
We thank Stefan Geritz for valuable discussions and Peter Abrams for comments on the manuscript. Our research was financially supported by the Finnish Academy of Sciences.
The three results listed in the main text are due to a particular property of the fitness function described by the following
Lemma: For hmut > h0, the invasion fitness ρ(hmut) can be written as , where h is the resident strategy or any of the resident strategies in a polymorphic population, and ϕ(hmut,h) is a strictly decreasing function of its first argument. In particular, ϕ has the following properties:
(i) ϕ(h1,h) = ϕ(h2,h) if and only if h1 = h2 and
(ii) ϕ(h1,h) − ϕ(h2,h) = −(h1 − h2)ψ(h1,h2,h), where ψ(h1,h2,h) is strictly positive for all arguments h1,h2,h ≥ h0.
First we prove the Lemma. Recall that 0 < x(t) < 1 denotes the time course of prey density along the stable limit cycle of the system with either one or several resident handling strategies of the predator, and T is the length of the limit cycle. (The Lemma holds also if the population dynamics have a stable fixed point; T > 0 can then be chosen arbitrarily. The results can be extended to chaotic populations by taking the limit T→∞.)
Let h be a resident strategy. As every resident strategy has zero invasion fitness, , we can rewrite the invasion fitness of an arbitrary mutant, hmut, as
By substituting from eqn 3 and after rearrangement, we obtain
which is the form stated in the Lemma with
The derivative of this function, , is negative whenever is not zero. This is obviously true for limit cycles, but also true if the dynamics are chaotic. To see this, note that x(t) must exceed d/[γ(h) − hd] for a positive fraction of time along the resident orbit, or else the resident strategy h could not have zero fitness, Hence, ϕ(hmut,h) is a strictly decreasing function of its first argument. The difference ϕ(h1,h) − ϕ(h2,h) can be simplified to
which is the form stated in part (ii) of the Lemma with
Now we can prove the main results.
1The singular strategy is always globally evolutionarily stable. To see this, first consider a single resident predator with an arbitrary strategy h. As ρ(hmut) is a continuous function of hmut and handling times shorter than hmin or longer than hmax have negative invasion fitness in any resident population, all mutants that can invade h must be strictly between h1 and h2 that satisfy ρ(h1) = ρ(h2) = 0. According to part (i) of the Lemma, ϕ(hmut,h) can be zero for only one mutant strategy, and therefore h1 can be different from h2 only if either h1 or h2 equals h itself. In other words, the mutants that can invade h are in one open interval with an end point hmut = h. It follows that if there is any mutant that can invade h, then mutants either infinitesimally smaller or infinitesimally greater than h can invade, and the fitness gradient [∂ρ(hmut)/ρhmut]hmut = h is not zero at h. At the singular strategy h*, the fitness gradient is zero by definition, which implies that no mutant can invade h*.
2At most two handling strategies can coexist. Suppose that a population contains three different resident strategies, h, h1 and h2. ρ(h1) = ρ(h2) = 0 implies ϕ(h1,h) = ϕ(h2,h) = 0 for h1,h2 ≠ h. Part (i) of the Lemma then gives h2 = h2, i.e. three (or more) different strategies cannot coexist.
3There is no evolutionarily stable coexistence. Consider a population with two resident strategies, h1 < h2. By the Lemma, ρ(h2) = 0 implies ϕ(h2,h1) = 0. The invasion fitness of a mutant, ρ(hmut) =, therefore can be rewritten as
where the last step follows from part (ii) of the Lemma. The invasion fitness (A6) is positive if and only if h1 < hmut < h2. Coexisting strategies therefore always can be invaded by a mutant in-between.