Rapid prey evolution and the dynamics of two-predator food webs

Abstract

Traits affecting ecological interactions can evolve on the same time scale as population and community dynamics, creating the potential for feedbacks between evolutionary and ecological dynamics. Theory and experiments have shown in particular that rapid evolution of traits conferring defense against predation can radically change the qualitative dynamics of a predator–prey food chain. Here, we ask whether such dramatic effects are likely to be seen in more complex food webs having two predators rather than one, or whether the greater complexity of the ecological interactions will mask any potential impacts of rapid evolution. If one prey genotype can be well-defended against both predators, the dynamics are like those of a predator–prey food chain. But if defense traits are predator-specific and incompatible, so that each genotype is vulnerable to attack by at least one predator, then rapid evolution produces distinctive behaviors at the population level: population typically oscillate in ways very different from either the food chain or a two-predator food web without rapid prey evolution. When many prey genotypes coexist, chaotic dynamics become likely. The effects of rapid evolution can still be detected by analyzing relationships between prey abundance and predator population growth rates using methods from functional data analysis.

Appendix

A Trait dynamics in clonal versus sexually reproducing prey

Here, we show that many of our results are also relevant to sexually reproducing prey because our clonal evolution model can be re-expressed as a gradient dynamics model for trait evolution in a sexual species.

Let x ₁ and x ₂ be the abundances of the two prey clones, X = x ₁ + x ₂, and let f = x ₂/(x ₁ + x ₂). We need to re-write the model in terms of X, the total prey population, and its “trait mean” f, rather than x ₁ and x ₂. In the predator and substrate equations, we just substitute x ₂ = fX, x ₁ = (1 − f)X, so that for example Q = ((1 − f)p ₁ + f p ₂)X. The dynamic equation for f is (omitting some algebra)

$$\begin{array}{rll} \dot f &=& \frac{d}{dt}{\left( \frac{x_2}{x_1+x_2} \right)} = f(1-f)\left[\frac{\dot x_2}{x_2} - \frac{\dot x_1}{x_1} \right]\\ &=& f(1-f)(W_2 - W_1)\end{array}$$

(6)

where W _i is the fitness of prey type i. Note that f is the population mean of the trait whose value is 1 for clone-2 individuals, and 0 for clone-1 individuals, and f(1 − f) is the population variance for this trait. The dynamic equation for X is

$$ \dot X = \dot x_1 + \dot x_2 = x_1 W_1 + x_2 W_2 = X\left[(1-f)W_1 + f W_2 \right]. $$

(7)

Comparing the last two equations, we see that the rate of change in f is the trait variance f(1 − f) multiplying the gradient with respect to f of prey fitness $\dot X/X$ with the predator’s grazing rate g/(k _y  + Q) held constant, which is the fitness gradient for a rare invader. Thus, our two-clone model is equivalent to a gradient-dynamics model for frequency-dependent evolution of the defense trait in a sexually reproducing species. Inherited from this model’s asexual origins are the form of the mean-variance relationship, and the relationship between mean trait value f and per-capita nutrient uptake rate in the quantitative trait model, namely $\frac{(1-f)m_1 S}{k_1 + S} + \frac{fm_2 S}{k_2 + S}.$ But mean-variance relationships like the one in (6) have often been assumed to model traits limited to a finite interval (such as fractional allocations), and while the nutrient uptake rate is not a standard model, it’s not any less reasonable than specifying m and k directly as a function of the mean defense level.

With three or more clones the correspondence between clonal and sexual models is no longer exact. Let f _i denote the population mean of the trait which has value 1 for clone i, for all other clones (i.e., the frequency of clone i). Then calculations similar to those above show that $\dot f_i = f_i(1-f_i)\frac{\partial}{\partial f_i}\frac{\dot X}{X}$, if the derivative with respect to f _i is taken by increasing f _i and decreasing all other clone frequencies by a constant fraction of their value while maintaining $\sum\limits_j{f_j} = 1$ and holding the predator grazing rate constant. So we get gradient dynamics, but not a standard model for correlated quantitative traits in a sexual species. It remains an open question whether the complex dynamics seen in our model with three clones would also be seen in a model for sexually reproducing prey.

B Calculating phase lags

Here we explain our method for computing approximate phase lags, which we implemented using computer algebra. The oscillations in our model appear to always arise through a supercritical Hopf bifurcation, as in the one-predator model. We compute approximate phase lags for parameters just past that bifurcation. Solutions x(t) are then small-amplitude oscillations about an equilibrium with common period T, and x _j (t) is approximately proportional to sin(λt + ψ _j ) where λ = 2 π/T and ψ _j is the phase for component j.

To compute relative phases, we choose one solution component (say x ₀) as the reference and set the origin of time so that ψ ₀ = 0. The other components (which we still denote x(t)) then satisfy a forced nonlinear system $\dot{{\bf{x}}} = \mathbf{F}({\bf{x}},x_0(t))$ where x ₀(t) = εsin(λt) is regarded as an exogenous forcing term. The linearized system for deviations from equilibrium can be written as

$$ \dot{{\bf{x}}} = \mathbf{J} {\bf{x}}+{\bf{u}}(t) $$

(8)

where J is the Jacobian matrix of F at the equilibrium, x(t) is now the m-vector of deviations from the equilibrium, and ${\bf{u}}(t)=\varepsilon \sin(\lambda t) \frac{\partial \mathbf{F}}{\partial x_0}$ is the forcing. The asymptotic solutions to (8) are

$$\begin{array}{rll} x_i(t) & =& A_i \cos(\lambda t) + B_i \sin(\lambda t), \\ \mbox {with } u_i(t) & =& b_i \sin(\lambda t), \qquad b_i=\varepsilon \frac{\partial \mathbf{F_i}}{\partial x_0}. \end{array}$$

(9)

The unknown A’s and B’s, which determine the phase lags, can be found by writing (8) as a matrix equation. Differentiation of x is accomplished by transforming A _i → λB _i , B _i → − λA _i . Defining

$$\begin{array}{rll}{\bf{X}} &=& (B_1,B_2,\ldots,B_m,A_1,A_2,\ldots,A_m)\\ {\bf{U}} &=& (b_1,b_2,\ldots,b_m,0,0,\ldots,0) \end{array}$$

(10)

differentation of x is represented by the matrix $\mathbf{D} = \left[ {\begin{array}{*{20}c} 0 & { - \lambda \mathbf{I}} \\ {\lambda \mathbf{I}} & 0 \\ \end{array} } \right]$ and multiplication by J by the matrix $\mathbf{J}_2 = \left[ {\begin{array}{*{20}c} \mathbf{J} & 0 \\ 0 & \mathbf{J} \\ \end{array} } \right]$. The coefficient representation of (8) is therefore D X = J ₂ X + U with solution ${\bf{X}}=(\mathbf{D} - \mathbf{J}_2)^{-1} {\bf{U}}$. The blocks in D − J ₂ commute, so defining R = (J ² + λ ² I)^− 1, we have

$$ (\mathbf{D} - \mathbf{J}_2)^{-1} = \left[ {\begin{array}{*{20}c} { - \mathbf{R}\mathbf{J}} & {\lambda \mathbf{R}} \\ { - \lambda \mathbf{R}} & { - \mathbf{R}\mathbf{J}} \\ \end{array} } \right]. $$

(1)

The addition formula sin(λt + ψ) = sin(ψ) cos(λt) + cos(ψ) sin(λt) shows that the phase ψ _i of x _i (t) relative to sin(λt) has the properties

$$\begin{array}{rll} \sin(\psi_i) &=& A_i/\rho, \mbox{ where } \rho=\sqrt{A_i^2 + B_i^2} \\ \cos(\psi_i) &=& B_i/\rho \\ \tan(\psi_i) &=& A_i/B_i. \end{array}$$

(12)

The value of ψ _i is therefore uniquely determined by the signs and ratio of A _i and B _i , so we can compute any convenient positive multiple of (A _i , B _i ). As a visual aid for readers, Fig. 18 in the Electronic supplementary material shows how the scaled coefficients (A _i , B _i ) relate to phase differences with the forcing variable.

The main limit to this method is that the expressions for A and B become very complicated as the number of state variables goes up, because they are symbolic solutions of a high-dimensional, non-sparse linear system. For scenarios with one or two prey genotypes we can make sense of the expressions for phase lags, at least in ecologically relevant limits of the parameter values, but with three genotypes it exceeds our abilities and we have to study the system numerically.

C LFC model without prey evolution

Here we consider LFC model (2) when there is only one genotype in the focal prey species x, assuming that all state variables have small oscillations around their mean values. The top predator only feeds on x, so it lags x by a quarter period (Bulmer 1975). The intermediate predator can behave differently because it is driven by both x and W. Writing $x=\bar x (1+ \varepsilon \sin(\lambda t))$ where λ = 2 π/T and T is the cycle period, the relevant subsystem is

$$\begin{array}{rll} \dot z & = & z(hx(t) + \phi w -1) \\ \dot w & = & M-w-\phi z w \end{array}$$

(13)

Without forcing (i.e. ε = 0), (13) has steady state

$$ \bar w = \frac{1-h\bar x}{\phi}, \bar z = \frac{M-\bar w}{\phi \bar w}. $$

(14)

This is a stable node whenever parameters are such that $\bar w$ and $\bar z$ are positive (the Jacobian has trace $-(1+\phi \bar z)<0$ and determinant $\phi^2 \bar z \bar w >0$), and numerical evidence suggests that it is globally stable (so far, no luck with the Bendixson–Dulac negative criterion). The forced system (ε > 0) is therefore “chasing” a moving equilibrium $(\bar z(t), \bar w(t))$ in which $\bar z(t)$ is a monotonically increasing function of x(t), hence z(t) exhibits oscillations that track x(t) but lag behind it.

To be more precise, the phase lag for small oscillations can be calculated by the method in Appendix B for the system (13) linearized about the steady-state $(\bar z, \bar w, \bar x)$ for ε = 0. These calculations yield that $z(t)-\bar z$ is proportional to sin(λt + ψ) where sin(ψ) < 0,cos(ψ) > 0 and

$$ \tan(\psi) = -\frac{\lambda^3+\left(\left(\phi \bar z+1\right)^2-\phi^2 \bar w \bar z \right)\lambda} {\phi^2 \bar w \bar z \left(\phi \bar z+1 \right)} $$

(15)

(note that because $\phi \bar w = 1- h \bar x < 1$, the numerator on the right-hand side of (15) is positive, hence tan(ψ) < 0). These properties imply that ψ ∈ ( − π/2, 0), so z lags behind x by an amount that approaches the upper limit of one quarter cycle period when tan(ψ) is very large in magnitude, and approaches zero (relative to T) when tan(ψ) is small.

Although the expression for tan(ψ) is complicated, some situations can be identified in which the lag is predicted to be short vs. long relative to T.

• Because λ → 0 as T →  ∞, the ratio between the lag and the cycle period goes to zero as the cycle period increases. That is, when the cycle period is very long, the dynamics of z and W become “slaved” to x(t), and z(t) is almost exactly at the value of $\bar z$ determined by the current value of x(t).

• The right-hand side of (15) can be written as

  $$ - \frac{\lambda \Omega + \lambda^3(1-h \bar x)^2}{\phi M (1-h\bar x)(\phi M + h\bar x -1)}. $$

  (16)

  where Ω > 0 and all terms in the denominator of (16) are positive so long as $\bar w, \bar z >0$. The lag therefore approaches the quarter-period limit when the denominator goes to 0. One way that this happens is when $h \bar x \to 1$, meaning that the intermediate predator is grazing heavily on the focal prey species and reaching high enough densities that its alternate food W is driven to very low levels ($\bar w \to 0$). The intermediate predator is then almost a specialist on the focal prey species, so it has the classical quarter-period lag behind the focal prey.

• The other way for the denominator of (16) to go to zero is if ϕ decreases to $(1-h\bar x)/M$. This is the value of ϕ at which $\bar z =0$ when $x = \bar x$. The intermediate predator therefore increases only when the focal prey species is more abundant (above $\bar x$), and is heading for extinction whenever the focal prey species is less abundant. The intermediate predator therefore behaves as if the focal prey were its main food source, even if $h \bar x$ is small.

Numerical evaluations of (15) indicate that the effects of h and ϕ hold generally: any increase in h or decrease in ϕ causes the linearized system’s phase lag to increase. The accuracy of (16) for the nonlinear system (13) is very good for small fluctuations (Fig. 19a in the Electronic supplementary material) and remains fairly good for substantial fluctuations (19b, in which the cycle peak is roughly three times the trough). The linearized analysis generally under-predicts the lag by a bit, and is most accurate near the extremes of zero or quarter-period lag.

D IC model without prey evolution

As the second baseline case, we calculate here the approximate phase relations between x,y and z in the Intermediate Consumer model with only one prey type. It is simplest to take x(t) as the forcing function and determine the phase lags of y and z relative to x. The equations for this scenario are

$$\begin{array}{rll} \dot y & =& y\left[\frac{gx(t)}{k_y + x(t)} +\eta z - 1\right] \\ \dot z & =& z\left[hx(t) - \eta y - 1 \right] \end{array}$$

(17)

As before we set $x(t)=\bar x (1+ \varepsilon \sin(\lambda t))$, and assume small oscillations of y and z about $\bar y$ and $\bar z$, the equilibria of (17) when $x(t) \equiv \bar x$,

$$\begin{array}{lll} &&{\kern-6pt}\bar z = (1- G \bar x)/\eta, \quad \bar y = (h \bar x-1)/\eta \\ &&{\kern-6pt}\mbox{where} \ G=g/(k_y + \bar x). \end{array}$$

(18)

We must have $G \bar x <1, h\bar x >1$ for the equilibrium to exist. These inequalities correspond to z being the more effective predator, able to persist on x alone, while y depends on consuming z as well as x in order to persist.

The method of Appendix B can be applied to the linearization of (17). Note that these calculations are relevant to the full three-variable food web model at parameter values that generate a small-amplitude limit cycle, rather than to the dynamics of (17) on its own without any feedbacks from y and z to x. The results (up to a positive factor) are

$$ \begin{aligned} A_y &= -g k_y \lambda, & B_y &= -h(\bar x + k_y)^2 \eta \bar z \\ A_z &= -h(\bar x + k_y)^2 \lambda, & B_z &= g k_y \bar y. \end{aligned} $$

(19)

The signs of the A’s and B’s imply that ψ _y  ∈ ( − π, − π/2), meaning a lag between quarter and half the cycle period, while ψ _z  ∈ ( − π/2, 0), a phase lag of under a quarter period. These are exactly what the food web structure suggests: because y and z both feed on x and therefore should lag x by a quarter period, but at the same time y feeds on z so it should lag z by a quarter period. The outcome is a compromise between these two expectations determined by the relative importance of the direct and indirect paths between x and y. In particular, when either predator becomes rare, the other converges to a quarter period lag behind x (B = cos(ψ) → 0).

The “positive factor” in the previous paragraph requires some comment. Dividing out some strictly positive factors (such as $\bar x + k_y$) it is

$$ D = (\bar x+k_y)\lambda^2+ (g-1)h \bar x^2+ (1 -h k_y-g)\bar x+k_y. $$

(20)

The sign of D is not obvious, and it depends on the cycle period T. Setting λ = 0 in D, the result is a negative multiple of $\bar z$. To determine the actual sign of D we therefore must estimate the cycle period of the full food chain. This was done by determining the period at the point where limit cycles arise through a Hopf bifurcation, for two cases. The first is the S,x,y system with z absent. This system reduces to two dimensions because asymptotically S + x + y ≡ 1. At the Hopf bifurcation point the eigenvalues of the Jacobian J are ±i λ, so $\lambda^2 = \det(\mathbf{J})$. Substituting the resulting value of λ ² into D, gives $D = \lambda^2 g k_y/(g-1)$ which is positive because g > 1 must hold for the predator to persist. The second case is the complete x,y,z food web but with exponential growth of the prey (i.e., S is absent, and $\dot x = rx -$ predation. In that case λ ² at the Hopf bifurcation equals the product of all Jacobian eigenvalues divided by their sum, which is c ₂/c ₀ where c _k is the k ^th order coefficient in the characteristic polynomial of J. The resulting value of (20) is the product of strictly positive terms and some of the c _j , which by the Routh–Hurwitz criterion are all positive at the Hopf bifurcation point, hence D is positive. Code for these calculations in Maxima (maxima.sourceforge.net) is available on request from SPE. The full S, x, y, z model has not yet succumbed to this approach, but numerical evidence supports the belief that D is positive whenever the system has a limit cycle.

E Two prey genotypes

In this Appendix we consider scenarios in which the ESC consists of two prey genotypes. In order for eco-evolutionary cycles to occur, one genotype must be defended against top predator y while the other is vulnerable to predation by y. For simplicity we assume that defense is 100% effective, though numerical solutions of the model assume that defense is imperfect. Also, we assume that λ ≪ 1 because eco-evolutionary cycles typically have a long period (Jones and Ellner 2007), and usually discard terms of O(λ ²) or higher in calculations.

E.1 One predator

The starting point is a system with only the top predator y, the scenario leading to “evolutionary cycles” in our previous studies. In the limit of vanishingly small cost for defense, we have shown (Jones and Ellner 2007) that the two prey genotypes oscillate almost exactly out of phase, while y (lagging the edible prey type by a quarter period) is out of phase with total prey abundance. However numerical solutions of the model (e.g., Yoshida et al. 2003) show the same pattern even when the defense cost is substantial. Here we confirm the numerical results, as a general approximation when the cycle period is long and prey population growth rate is sensitive to changes in limiting substrate concentration.

The persisting prey genotypes are x ₀ and x ₂ (Fig. 2). Because y feeds only on x ₂, it will lag x ₂ by a quarter period. To determine the other lags we can therefore regard S and x ₀ as forced by the oscillations in x ₂. The equations for that subsystem are

$$\begin{array}{rll} \dot S & =& 1- S - \frac{mS (x_2(t)+x_0)}{k_c + S} \\ \dot x_0 & =& x_0\left[\frac{m S}{k_c + S}- 1 \right] \end{array}$$

(21)

with $x_2(t)=\bar x_2 (1+ \varepsilon \sin(\lambda t))$.

The only nonlinear term is x ₀’s functional response f(S) = mS/(k _c  + S), which produces in entries containing $f'(\bar S)$ in the first column of the Jacobian for (21). Defining $b=f'(\bar S)$ the Jacobian is

$$\begin{array}{rll} \mathbf{J}&=& \begin{bmatrix} -1- b(\bar x_2 + \bar x_0) & - \dfrac{m\bar S}{k_c + \bar S} \\ b\bar x_0& 0 \end{bmatrix}\notag\\ &=& \begin{bmatrix} -1- b(\bar x_2 + \bar x_0) &-1 \\ b\bar x_0& 0 \end{bmatrix}. \end{array}$$

In the notation of Appendix B, the forcing coefficient vector U is proportional to ( − 1, 0). The procedure in Appendix B produces for x ₀ (up to a positive rescaling that depends on the forcing amplitude)

$$ A = \lambda \left(b^{-1} + \bar x_2 + \bar x_0\right), \quad B= \lambda^2 b^{-1} - \bar x_0. $$

(22)

In this paper typically λ ≪ 1 because the cycles are long-period cycles driven by evolution, and b will be large because prey are potentially rapid-growing but limited by nutrient scarcity at equilibrium. Then to leading order (22) becomes $A = \lambda(\bar x_2 + \bar x_0), B = -\bar x_0$. This implies that to leading order in λ, the phase of x ₀ relative to x ₂ is $\psi = \arctan(A/B) = - \pi - \lambda(1 + \bar x_2/\bar x_0)$. Therefore

$$\begin{array}{rll} \sin(\lambda t + \psi) &\approx& \sin\left(\lambda t - \pi - \lambda(1+ \bar x_2/\bar x_0)\right)\\ &=& \sin\left(\lambda \left(t- 1 - \bar x_2/\bar x_0\right) - \pi\right). \end{array}$$

(23)

The defended prey type therefore lags the edible prey type by slightly more than half a cycle period, with the lag exceeding half the period by at least one unit of scaled time (which in dimensional time units is the inverse of the dilution rate). The defended prey type is only linked to S, so it lags S by a quarter period, hence S must lag the edible prey by a quarter period plus (to leading order) $1+\bar x_2/\bar x_0$ time units.

Figure 2 shows a numerical example, and as predicted the defended prey lags the edible prey by slightly more than half the cycle period. This small asymmetry has the important consequence that the peak in total prey abundance occurs in between a peak in defended prey and the subsequent peak in edible prey, and roughly halfway in between. Peaks therefore occur in the sequence

$$ \mbox{defended type} \rightarrow \mbox{total prey} \rightarrow \mbox{edible type} \rightarrow \mbox{predator} $$

with roughly a quarter-period lag between each. The slight deviation of the prey types from being perfectly out of phase is thus the underlying cause for the half-period lag between predator and total prey abundance, which is the defining feature of “evolutionary cycles” in the one-predator food chain.

E.2 Two predators

Now we add back the second predator z, still assuming that there are two prey genotypes, one highly vulnerable to y and the other well defended against y.

Firstly, suppose the defense against y is also effective against z, so the persisting prey clones are x ₂ and x ₀. Then (as discussed in the main text) the (x ₂, y, z) system is equivalent to the two-predator systems with uniform non-evolving prey x considered in Appendices C and D, and the results from those Appendices apply. x ₀ is linked to the (x ₂, y, z) system only through the substrate, so the results in Appendix E.1 apply, meaning that x ₀ will lag x ₂ by slightly more than half the cycle period. Figure 13 illustrates these for the IC model; for the LFC model see Fig. 7 in the main text.

Secondly, suppose that defenses against y and against z are incompatible, so the persisting prey clones are x _y and x _z . Then in the linked food chains model, the (y, x _y ) and the (z, x _z ) subsystems are both equivalent to a system of one predator feeding on a non-evolving prey type, so y lags x _y by a quarter period, and z lags x _z by a quarter period or less depending on how much it feeds on W. To determine the lag between the prey types we can take x _y as the forcing variable and consider the subsystem

$$\begin{array}{rll} \dot S &=& 1- S - x_z f_z(S) - x_y(t) f_y(S) \\ \dot x_z &=& x_z\left[f_z(S)- hz -1 \right] \\ \dot z &=& z\left[ h x_z + \phi W - 1 \right] \\ \dot W &=& M - W - \phi zW \end{array}$$

(24)

where f _y , f _z are the algal types’ functional responses. Applying Appendix B gives that the prey types are roughly out of phase: as λ → 0 we have A → 0, B < 0 for the phase lag of x _z relative to x _y , which implies a half-period lag. The O(λ) term in A is a complicated expression whose sign cannot be determined in general, but it is negative when z’s second food source is relatively unimportant ($\phi \bar w \ll 1$) implying that the lag is slightly under a half period.

In the IC model, again considering the other variables as forced by x _y , the equations are

$$\begin{array}{rll} \dot S &=& 1- S - x_z f_z(S) - x_y(t) f_y(S) \\ \dot x_z & =& x_z\left[f_z(S)- hz -1 \right] \\ \dot z & =& z\left[ hx_z - \eta y - 1 \right] \\ \dot y & =& y\left[\frac{gx_y(t)}{k_y + x_y(t)} + \eta z - 1\right] \end{array}$$

(25)

Applying Appendix B, we obtain the following results for λ ≪ 1. In the limit h = η = 0 the food web (25) is equivalent to the one-predator evolutionary cycles scenario, so x _z lags x _y by a half period plus an amount of O(λ). With h and η increasing from 0 in constant proportion (η = dh), the derivative of tan(ψ) is O(λ) > 0, indicating a decrease in lag, and tan(ψ) = O(λ/h) < 0 as h → ∞, so the lag converges to exactly a half period. The h → ∞ limit is not actually relevant, because the x _y → y link has to be dominant for cycles to occur, but the limit indicates the direction of change as the x _z → z → y pathway increases in importance. y lags x _y by a quarter period for h = η = 0 (A = 0(1) < 0,B = O(λ)), and the lag increases to a half period as h →  ∞ with η increasing in proportion (A = O(λ/h) > 0, B = O(1) < 0). z is only linked to x _z when η = 0, so it then lags x _z by a quarter period. As the x _z → z →y path is strengthened, the linearized analysis predicts an increase in lag to a half period within O(λ), meaning that z’s lag behind x _z shrinks.

Figure 14 illustrates these predictions, for η = 0.5,h = 4 meaning that the $z \leftrightarrow y$ link is weaker than the $z \leftrightarrow x$ link, but not vanishingly small. As predicted by the linear analysis, peaks in y lag peaks in x _y by slightly more than a quarter period (30%), while peaks in z lag peaks in x _z by slightly less (18%).

The final scenario with two prey types is coexistence of x _z and x ₂, the types with no defense against z (Fig. 10). For the LFC model, with x ₂ as the forcing function the equations for the forced variables are

$$\begin{array}{rll} \dot S &=& 1- S - x_z f_z(S) - x_2(t) f_2(S) \\ \dot x_z &=& x_z\left[f_z(S)- hz -1 \right] \\ \dot y &=& y\left[\frac{gx_2(t)}{k_y + x_2(t)} - 1\right] \\ \dot z &=& z\left[ h (x_z + x_2(t)) + \phi W - 1 \right] \\ \dot W &=& M - W - \phi zW \end{array}$$

(26)

where f _i (S) are the prey functional responses. The method in Appendix B yields that the lag of x _z relative to x ₂ when λ ≪ 1 is a half-period plus an amount of order λ (specifically, tan(ψ) = A/B = O(λ) with A = O(λ) > 0, B = O(1) < 0 for λ small). Everything else follows from the food web structure and previous results, as described in the main text.

For the IC model, the equations for the forced variables are the first two lines of (26) plus

$$\begin{array}{rll} \dot y &=& y\left[\frac{gx_2(t)}{k_y + x_2(t)} + \eta z - 1\right] \\ \dot z &=& z\left[ h (x_z + x_2(t)) -\eta y - 1 \right] \end{array}$$

(27)

For h, η → 0 the (y, x ₂, x _z ) subsystem reduces to one-predator evolutionary cycles, so x _z and x ₂ are nearly out of phase, y is out of phase with total prey and z lags total prey by a quarter period. When the intermediate predator becomes important (increasing h and η in constant proportion), x _z and x ₂ remain out of phase (A = O(λ/h) > 0, B = O(1) < 0), but the predator lags are completely different: y becomes synchronized with x ₂ (A = O(λ) > 0, B = O(1) > 0), while z is out of phase with x ₂ (A = O(λ/h) > 0, B = O(1) < 0), and therefore synchronized within O(λ) with x _z . The result is that the two predators are out of phase with each other (Fig. 11). Because the two prey types become almost exactly synchronous, two things occur: the cycles of total prey abundance become “cryptic” (Yoshida et al. 2007), and peaks in total prey coincide with peaks in one of the two prey types (at some time when $\dot{x}_z = \dot{x}_2 = 0$). As a result, one predator is in phase with total prey abundance, and the other is out of phase with total prey abundance.