Joint evolution of predator body size and prey-size preference

Abstract

We studied the joint evolution of predator body size and prey-size preference based on dynamic energy budget theory. The predators’ demography and their functional response are based on general eco-physiological principles involving the size of both predator and prey. While our model can account for qualitatively different predator types by adjusting parameter values, we mainly focused on ‘true’ predators that kill their prey. The resulting model explains various empirical observations, such as the triangular distribution of predator–prey size combinations, the island rule, and the difference in predator–prey size ratios between filter feeders and raptorial feeders. The model also reveals key factors for the evolution of predator–prey size ratios. Capture mechanisms turned out to have a large effect on this ratio, while prey-size availability and competition for resources only help explain variation in predator size, not variation in predator–prey size ratio. Predation among predators is identified as an important factor for deviations from the optimal predator–prey size ratio.

Appendix: Derivation of invasion fitness

In this appendix we show that for determining the invasion fitness of the DDE system Eq. 12 one can use an ODE formulation without delay. For this purpose, below we derive the invasion fitness of a mutant predator trying to invade a given resident population of predators. For determining the coexistence set a formulation without delay can be derived as well, which we will not demonstrate explicitly, since that derivation is very similar to the one presented below.

We start from the DDE system Eq. 12, consisting of a prey population x ₁ and a resident predator population x _A,r, and introduce a mutant predator population x _A,m according to Eq. 14. For the sake of clarity, we consider only a single prey population and leave out the tildes that denote scaled parameters in the main text. The derivation of the invasion fitness for multiple prey requires just a few adjustments, as is explained at the end of this appendix. The resulting full system is given by

$$\frac{\hbox{d}x_{1}(\tau)}{\hbox{d}\tau} = \left(x_{r,1}-x_{1}(\tau)\right)D -I_{1,{\rm r}} f_{1,{\rm r}}(\tau) x_{\rm A,r}(\tau)- I_{1,{\rm m}} f_{1,{\rm m}}(\tau)x_{\rm A,m}(\tau),$$

(21a)

$$ \frac{\hbox{d}x_{\rm A,r}(\tau)}{\hbox{d}\tau} = R_{\rm r}(\tau-a_{\rm b,r})\exp(-h_{\rm r} a_{\rm b,r}) x_{\rm A,r}(\tau-a_{\rm b,r})-h_{\rm r}x_{\rm A,r}(\tau),$$

(21b)

$$ \frac{\hbox{d}x_{\rm A,m}(\tau)}{\hbox{d}\tau} =R_{\rm m}(\tau-a_{\rm b,m})\exp(-h_{\rm m} a_{\rm b,m} )x_{\rm A,m}(\tau-a_{\rm b,m})-h_{\rm m} x_{\rm A,m}(\tau).$$

(21c)

We assume that there is a stable equilibrium of the prey-resident system, Eq. 21 for x _A,m(τ) = 0, at which the resident predator population has positive density. We confirmed this assumption numerically for the coexistence set shown in Fig. 1 using the default parameter values given in Table 2.

The subsequent analysis can be outlined as follows. In order to derive the mutant’s invasion fitness, we study the stability of the full system after the mutant has been introduced at the prey-resident equilibrium. The full prey-resident-mutant system above is then linearized around this equilibrium, and the characteristic equation of the resultant linear system is analyzed. When the real parts of all roots of this equation are negative, the resident is stable and the mutant cannot invade. By contrast, when the dominant root is positive, the resident is unstable and the mutant can invade. In particular, we will determine the combinations of trait values at which this stability changes.

Below, a superscripted asterisk indicates that the considered variable is at equilibrium under constant environmental conditions. We now introduce new variables that denote displacements from this equilibrium,

$$ \begin{aligned} \xi_1&=x_1 - x^\ast_1, \\ \xi_{\rm A,r}&=x_{\rm A,r} - x^\ast_{\rm A,r}, \\ \xi_{\rm A,m}&=x_{\rm A,m} - x^\ast_{\rm A,m}. \end{aligned} $$

(22)

The linearized model at equilibrium then reads

$$ \begin{aligned} \frac{\hbox{d}\xi_{1}(\tau)}{\hbox{d}\tau} &= -\xi_1(\tau)D - I_{1,{\rm r}} \left(\xi_{1}(\tau)\frac{\hbox{d}f_{1,{\rm r}}}{\hbox{d}x_{1}}(x_{1}^\ast)x_{\rm A,r}^\ast+ f_{1,{\rm r}}(x_{1}^\ast)\xi_{\rm A,r}(\tau)\right)\\ &-I_{1,{\rm m}}\left(\xi_{1}(\tau)\frac{\hbox{d}f_{1,{\rm m}}}{\hbox{d}x_{1}}(x_{1}^\ast)x_{\rm A,m}^\ast + f_{1,{\rm m}}(x_{1}^\ast)\xi_{\rm A,m}(\tau)\right), \end{aligned}$$

(23a)

$$ \frac{\hbox{d}\xi_{\rm A,r}(\tau)}{\hbox{d}\tau} = \left(\xi_{1}(\tau-a_{\rm b,r})\frac{\hbox{d}R_{\rm r}}{\hbox{d}x_{1}}(x_{1}^\ast)x_{\rm A,r}^\ast+ R_{\rm r}(x_{1}^\ast)\xi_{\rm A,r}(\tau-a_{\rm b,r}) \right)\exp(-h_{\rm r} a_{\rm b,r})-h_{\rm r}\xi_{\rm A,r}(\tau),$$

(23b)

$$ \frac{\hbox{d}\xi_{\rm A,m}(\tau)}{\hbox{d}\tau} = \left(\xi_{1}(\tau-a_{b{\rm m}})\frac{\hbox{d}R_{\rm m}}{\hbox{d}x_{1}}(x_{1}^\ast)x_{\rm A,m}^\ast+ R_{\rm m}(x_{1}^\ast)\xi_{\rm A,m}(\tau-a_{\rm b,m})\right)\exp(-h_{\rm m} a_{\rm b,m} )-h_{\rm m}\xi_{\rm A,m}(\tau).$$

(23c)

In the following we use the shorthand notations R ^*_r  = R _r(x ^*₁ ) and R ^*_m  = R _m(x ^*₁ ).

Since we are interested in the invasion by a rare mutant population, we take x ^* _A,m = 0. Then the matrix ${\mathcal{P}},$ defined by

$$ \mathcal{P} \left(\begin{array}{l} \zeta_1\\ \zeta_{\rm A,r}\\ \zeta_{\rm A,m} \end{array}\right)=\left(\begin{array}{l} 0\\ 0\\ 0\\ \end{array}\right),$$

(24)

is obtained by substituting ξ _i in Eq. 23 by ξ _i  = ζ _i exp(λτ), i = 1, (A,r), (A,m) and division by exp(λτ) > 0,

The 2 × 2 matrix ${\mathcal{J}}_{1}$ is given by

$$ {\mathcal{J}}_{1}=\left(\begin{array}{ll} -(\lambda+h_{\rm r}) -I_{1,{\rm r}}\frac{\hbox{d}f_{r}}{\hbox{d}x_{1}}(x_{1}^\ast)x_{\rm A,r}^\ast & -I_{1,{\rm r}} f_{1,{\rm r}}(x_1^{\ast})\\\exp(-(\lambda+h_{\rm r}) a_{\rm b,r} )\frac{\hbox{d}R_{\rm r}}{\hbox{d}x_{1}}(x_{1}^\ast) x_{\rm A,r}^\ast & R_{\rm r}^\ast\exp(-(\lambda+h_{\rm m}) a_{\rm b,r})-(\lambda+h_{\rm m})\\ \end{array}\right) $$

(26)

and the 1 × 1 matrix ${\mathcal{J}}_2$ by

$$ \mathcal{J}_2=R_{\rm m}^{\ast}\exp(-(h_{\rm m}+\lambda)a_{\rm b,m})-(\lambda+h_{\rm m}). $$

(27)

The characteristic equation is obtained by the requirement that the determinant of the matrix ${\mathcal{P}}$ be equal to zero. Then the ζ _i play the role of eigenvector components and the complex number λ plays the role of eigenvalue, which is now a root of the characteristic equation.

Since the mutant is assumed to be rare, the determinant of ${\mathcal{P}}$ factorizes, being given by the product of det${\mathcal{J}}_1$ and det${\mathcal{J}}_2\,=\,{\mathcal{J}}_2,$ with these two factors corresponding to the two decoupled systems: the prey-resident system without the mutant (i.e., x ^*_A,m  = 0), described by ${\mathcal{J}}_{1},$ and the growth rate of the mutant population, described by ${\mathcal{J}}_2.$

The first factor yields the characteristic equation of the prey-resident system, $\det{\mathcal{J}}_{1}\,=\,0.$ This characteristic equation belongs to the eigenvalue problem for the set of one ODE and one DDE given by Eq. 23a, without the last term, and Eq. 23b, evaluated at the equilibrium of the prey-resident system.

The second factor yields the characteristic equation $\det{\mathcal J}_{2}\,=\,{\mathcal{J}}_2\,=\,0$ of the first-order linear homogeneous DDE (Eq. 23c) describing the specific growth rate of the mutant population. The expression for ${\mathcal{J}}_2$ in Eq. 27 is of a form discussed extensively by Diekmann et al. (1995, p. 312). For this case, the complex roots of the characteristic equation can be obtained analytically.

The function ${\mathcal{J}}_2(\lambda)$ with $\lambda \in {\mathbf{R}}$ is monotonically decreasing, $d{\mathcal{J}}_2/d\lambda\,<\,0.$ Therefore there is one unique real root λ₀. Since ${\mathcal J}_2(0)\,=\,R_{\rm m}^{\ast}\exp(-h_{\rm m}a_{\rm b,m})\,-\,h_{\rm m},$ the real eigenvalue equals zero, λ₀ = 0, if and only if R ^*_m exp(−h _m a _b,m) = h _m. Thus, Eq. 27 has exactly one positive real solution, λ₀ > 0, when R ^*_m exp(−h _m a _b,m) > h _m and exactly one negative real solution, λ₀ < 0, when R ^*_m exp(−h _m a _b,m) < h _m. Further, (Driver 1977, p. 321) showed that equations of this form have infinitely many complex roots. Let λ _k  = μ _k  + iω _k ; then substitution of this into the characteristic equation ${\mathcal J}_2\,=\,0$ and separately equating real and imaginary parts gives

$$\mu_k=\exp(-\mu_k a_{\rm b,m}) R_{\rm m}^\ast\exp(-h_{\rm m}a_{\rm b,m})\cos (a_{\rm b,m}\omega_k)-h_{\rm m},$$

(28)

$$\omega_k=-\exp(-\mu_k a_{\rm b,m})R_{\rm m}^\ast\exp(-h_{\rm m}a_{\rm b,m})\sin (a_{\rm b,m}\omega_k).$$

(29)

Clearly, if Eq. 29 holds for ω _k , it holds also for −ω _k , so the complex conjugate λ ^* _k  = μ _k  − iω _k is also a root of the characteristic equation. Furthermore, the unique real root λ₀ is the dominant eigenvalue, i.e., the real parts of all other roots are smaller than λ₀. This can be seen as follows. Comparison of Eq. 28 with the characteristic equation for λ₀ gives

$$ \lambda_0-\mu_k=\left(1-\exp((\lambda_0-\mu_k) a_{\rm b,m}) \cos (a_{\rm b,m}, \omega_k) \right)R_{\rm m}^\ast\exp(-h_{\rm m}a_{\rm b,m}). $$

(30)

Suppose that cos(a _b,mω _k ) = 1, then sin(a _b,mω _k ) = 0, and hence ω _k  = 0, which contradicts the fact that λ _k  = μ _k  + iω _k has non-zero imaginary part. We can thus conclude that cos(a _b,mω _k ) < 1. Now assume that λ₀ − μ _k  ≤ 0, then, with R ^*_m  > 0 (since R ^*_m  ≈  R ^*_r  > 0 due to small mutational steps), Eq. 30 implies 1 ≤ exp((λ₀ − μ _k ) a _b,m) cos(a _b,mω _k ), and also this leads to a contradiction. This shows that Re (λ _k ) < λ₀, k = 1, 2,…, or in other words: the real eigenvalue λ₀ is the dominant root of the characteristic equation $\det{\mathcal{J}}_2\,=\,0.$

We had mentioned above that the prey-resident system has a positive stable equilibrium for the default parameter values given in Table 2. Under these circumstances, the real parts of the eigenvalues of ${\mathcal{J}}_{1}$ are strictly negative. Thus, the dominant eigenvalue λ₀ of ${\mathcal{J}}_{2}$ will also be the dominant eigenvalue of ${\mathcal{P}},$ if this λ₀ exceeds the largest real part of the eigenvalues of ${\mathcal{J}}_{1}.$ Hence, for $\det{\mathcal{J}}_2\,=\,0,$ that is, for R ^*_m  exp(−h _m a _b,m) = h _m, the dominant eigenvalue of ${\mathcal{P}}$ will equal λ₀ and thus zero. At trait values for which this holds, the prey-resident-mutant system changes stability, so that the prey-resident system becomes invadable by the mutant predator.

Now suppose that the real eigenvalue λ₀ is positive but small. Then the characteristic equation of ${\mathcal{J}}_{2}$ gives

$$ \begin{aligned} \lambda_0&=R_{\rm m}^\ast\exp(-(\lambda_0+h_{\rm m}) a_{\rm b,m} ) -h_{\rm m}\\ &=\left(1-\lambda_0a_{\rm b,m} +\frac{1}{2}\, \lambda_0^2a_{\rm b,m}^2+\cdots \right)R_{\rm m}^\ast\exp(-h_{\rm m} a_{\rm b,m} )-h_{\rm m}, \\ \end{aligned} $$

so that, for λ₀ a _b,m  ≪ 1, we have

$$ \lambda_0= R_{\rm m}^\ast\exp(-h_{\rm m} a_{\rm b,m} ) -h_{\rm m}. $$

(31)

Consequently, the rate λ₀ is the invasion fitness of the mutant predator at the equilibrium of the prey-resident system, (x ^*₁  > 0, x ^*_A,r  > 0, x ^*_A,m  = 0).

We have thus shown that, if λ₀ a _b,m ≪ 1, which holds for small mutational steps, the invasion fitness can be determined by a formulation without delay, corroborating our approach in the main text (Eq. 15). The rare mutant (x _A,m → 0) will be able to invade the stable prey-resident system if and only if R ^*_m exp(−h _m a _b,m) > h _m. The biological interpretation of this inequality is clear: the mutant’s effective birth rate has to exceed the dilution rate. After successful invasion, the mutant generally replaces the resident (Geritz et al. 2002); around evolutionary branching points they can coexist, leading to a dimorphic predator population (Metz et al. 1992, 1996; Geritz et al. 1997, 1998).

For the (ecological) stability at the boundaries of the coexistence set a formulation without delay can be derived as well. This derivation is very similar to that of the invasion fitness explained above, but simpler, as no mutants are considered but only the prey and resident predator populations. The resulting condition fulfilled at the boundaries of the coexistence set is given by R ^*exp(−ha _b) = h.

Considering multiple prey populations instead of a single one will affect system Eq. 21 in two ways. First, the dynamics of each prey population are described by a separate equation, such that for n prey populations, the full system will consist of n + 2 equations. Second, the prey populations will affect the growth rate R (Eq. 13) of the predators (both residents and mutants) through their functional response f (Eq. 6). The derivation of the invasion fitness itself, however, is largely analogous to that shown above. The determinant of the matrix ${\mathcal P}$ is still factorizable, and remains given by the product of $\det{\mathcal{J}}_{1}$ and $\det{\mathcal{J}}_{2}.$ ${\mathcal J}_{2}$ is still given by Eq. 27, adjusted for multiple prey through f in R. ${\mathcal{J}}_{1}$ will now be a (n + 1) × (n + 1) matrix, which corresponds to the system of the resident predator and the n prey populations without the mutant predator. Supported by the simulation results, we again assume that the system without the mutant is stable. It is then easy to see that the invasion fitness for multiple prey resembles that for a single prey (Eq. 31), adjusted through f in R.