Coevolution can reverse predator–prey cycles

Clonal Model Yields Clockwise Cycles.

In the clonal model 1, clockwise oscillations can occur between the total prey density (x=xl+xh) and the total predator density (y=yl+yh). These cycles arise when the defense of low-vulnerability prey is effective against low-offense predators and offense is costly. As an example, consider Fig. 3 with low- (dashed blue, Fig. 3A) and high-vulnerability (solid blue, Fig. 3A) prey clones and low- (dashed red, Fig. 3B) and high-offense (solid red, Fig. 3B) predator clones. In Fig. 3 A and B, the densities and frequencies of the clones fluctuate over time. Summing the clonal densities to obtain the total predator and prey population densities yields the predator–prey oscillations in Fig. 3C. These cycles have a clockwise orientation in the phase plane (Fig. 3D), the opposite of what is predicted by classical predator–prey models.

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Fig. 3.

Example of a clonal model exhibiting clockwise predator–prey cycles. (A) Low- (dashed blue) and high- (solid blue) vulnerability prey densities. (B) Low- (dashed red) and high- (solid red) offense predator densities. (C) Total prey (blue) and total predator (red) densities. (D) Total prey and predator densities exhibit clockwise cycles in the phase plane; arrows denote the flow of time. Blue and red rectangles denote the frequencies of the prey and predator clonal types along the cycle, respectively. In the rectangles, open areas correspond to the frequency of low-vulnerability or low-offense clones and filled areas correspond to the frequency of high-vulnerability or high-offense clones. Simulations are of clonal model 1 with logistic growth of the prey clones, Type II functional responses, and linear predator mortality rates; see SI Text, section D for equations and parameters.

Biologically, the clockwise cycles are driven by fluctuations in the densities and fitnesses of the clonal types. To understand how, consider a small prey population dominated by low-vulnerability clones interacting with a large predator population dominated by low-offense clones. The defended prey drive the predators to low abundance, allowing the prey to increase. Here, a prey peak follows the predator peak. The increase in low-vulnerability prey increases the fitness of high-offense predators, allowing high-offense predators to increase and begin to drive the prey population down. Due to high costs for offense, the abundance of high-offense predators remains low, resulting in a selective advantage for the high-vulnerability clone. As the prey population becomes dominated by high-vulnerability clones, the high-offense predators increase and the prey population continues to decrease. Since relatively few prey are present and low-offense predators pay a lesser cost, selection favors low-offense predators. The low-offense predators replace the high-offense predators, resulting in increased predator abundance and a further decrease in the prey population. With many predators present, selection favors low-vulnerability prey and the cycle repeats.

Continuous Trait Models Can Approximate Clonal Models.

The total prey and total predator population dynamics in Fig. 3C are quantitatively similar to the population dynamics in Fig. 2A. This suggests that the continuous trait model 2 can approximate the dynamics of the discrete trait clonal model 1. In SI Text, section A, we show that the continuous trait model can approximate the dynamics of the clonal model by focusing on the dynamics of the total prey density (x=xl+xh), the total predator density (y=yl+yh), the mean prey trait (α=([αlxl+αhxh]/x), and the mean predator trait (β=[βlyl+βhyh]/y). In SI Text, section A we also quantify the error that arises in using this approximation, identify cases in which the approximation is exact, and discuss extensions to systems with more than two clonal types per species. The time series in Figs. 3 and 2 A and B are examples of when the clonal and continuous trait models exhibit identical dynamics. In total, via this approximation, we can explore how clockwise cycles arise in systems with continuous or discrete traits by studying the continuous trait model 2.

Continuous Trait Model Yields Clockwise Cycles.

To understand when clockwise cycles arise in system 2, we focus on the dynamics where evolution is much faster than ecology (V≫1). In the fast evolution limit, the eco-evolutionary dynamics of the system decompose into fast evolutionary jumps and slow changes in population densities. These fast and slow dynamics can be understood, via fast–slow dynamical system theory (37) by studying the four 2D planes called critical manifolds, as shown in Fig. 4. Each critical manifold defines the population dynamics that occur when the traits are fixed at their lowest or highest values. The white regions of the planes are stable and the colored regions are unstable with respect to the evolutionary dynamics of the system. When a solution to system 2 is near a critical manifold, the populations slowly change as if stuck to that critical manifold (solid black curves in Fig. 4). During this time, the traits remain essentially fixed. After the population densities cross into a colored region of a plane, the solution jumps away and lands in the white region of another critical manifold (dashed gray lines in Fig. 4). During the jump, one trait evolves while the other trait and the population densities remain fixed. Repeating this process yields an eco-coevolutionary cycle. Concatenating the population dynamics on the critical manifolds results in the predator–prey cycle shown in the middle x,y plane of Fig. 4. See SI Text, section B for additional details.

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Fig. 4.

When evolution is nearly instantaneous, eco-evolutionary cycles can be decomposed into slow ecological dynamics and fast evolutionary jumps. Slow population dynamics (solid black curves) occur on four critical manifolds (x,y planes labeled “Cij” in the corners). The critical manifolds are defined by fixing the mean prey (α) and mean predator (β) traits at their low (l) or high (h) values. White regions in the planes are evolutionary attractors and colored regions evolutionary repellers in the α-direction (red), the β-direction (blue), or both directions (purple). Fast evolutionary dynamics (dashed gray lines) occur as solutions jump between the critical manifolds. Solutions behave in the following way. A solution lands in the white region of a critical manifold (open circle) and then moves (solid black curves) toward the attracting equilibrium point on that manifold (filled circle). After crossing into a colored region of the manifold, the solution jumps to the white region of another critical manifold (dashed gray line). Repeating and concatenating the population dynamics on each critical manifold yields the eco-coevolutionary cycle in the center.

While clockwise cycles do not always arise in system 2, we find that clockwise cycles can arise when selection favors extreme trait values and intermediate trait values are never optimal (i.e., disruptive selection), the defense of the low-vulnerability prey is effective against low-offense predators, and offense is costly. These conditions come from our analysis of the general continuous trait model 2 in the fast evolution limit; see SI Text, section B.4 for details. The conditions and the species’ trade-offs ensure that the disadvantages of high vulnerability and low offense and the advantages of high offense and low vulnerability do not lead to evolutionary fixation. For a particular eco-coevolutionary model, these conditions will be realized through particular constraints on the parameter values of the model. In SI Text, section B.5, we provide a worked illustrative example of a coevolutionary Lotka–Volterra model and show that in the regions of parameter space where clockwise cycles arise, our general conditions also hold.

The sequence of slow ecological and fast evolutionary dynamics in Fig. 4 elucidates how clockwise cycles arise in the continuous trait model. Consider a small population of low-vulnerability prey and a large population of low-offense predators (Cll in Fig. 4). Due to the effective defense, the low-vulnerability prey drive the predators down while increasing in abundance. This results in selection for high-offense predators (Clh in Fig. 4). The high-offense predators decrease the prey population, however the predator population remains low due to high costs for offense. Low predation pressure selects for more competitive, more vulnerable prey (Chh in Fig. 4). Consequently, the predators increase and drive the prey population down further. Low-prey abundance selects for low-offense predators (Chl in Fig. 4). As predator density increases and prey density decreases, selection increases for low vulnerability and the cycle repeats.

While our conditions and analysis of clockwise cycles are general, we present three numerical examples in Figs. 2 and 5 where prey exhibit logistic growth and predation rates are Type II functional responses. In Fig. 2, peaks in predator abundance precede peaks in prey abundance. Since the dynamics of the continuous trait model in Fig. 2A and the clonal model in Fig. 3 are identical, the time series in Fig. 2A have a clockwise orientation when plotted in the predator–prey phase plane (Fig. 2D). Fig. 5 presents two additional examples of clockwise predator–prey cycles for varying evolutionary speeds. Note that the cycle orientation is preserved as the speed of evolution increases by a factor of 2 (Fig. 5 C and D) and 5 (Fig. 5 E and F).

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Fig. 5.

Two examples of clockwise predator–prey cycles. The speed of evolution is (A and B) as fast, (C and D) two times as fast, and (E and F) five times as fast as the ecological dynamics of the system. Simulations are of continuous trait model 2 where the prey exhibit logistic growth in the absence of predation and predation rates follow a Type II functional response. In A, C, and E, the predators have a linear death rate and in B, D, and F the predators have a nonlinear death rate; see SI Text, section D for equations and parameters.

Empirical Datasets Exhibiting Clockwise Cycles.

We have shown that predator–prey coevolution can, in theory, drive clockwise cycles. To what extent are clockwise cycles found in ecological time series data? We revisited long-term population time series and identified three cases of interest: phage–cholera chemostat experiments (38), gyrflacon–rock ptarmigan census data (39), and mink and muskrat trapping data from the Hudson’s Bay Company (40). In each case, we reexamined the time series and identified regions where predators (or analogs thereof) and their prey exhibit clockwise cycles; see SI Text, section C for a discussion of how cycle type was identified. In the thickened regions of the time series in Fig. 6 A, C, E, G, and I, peaks in predator density (or number) precede those of the prey. When the thickened segments are plotted in the predator–prey phase plane (Fig. 6 B, D, F, H, and J), the cycles have a clockwise orientation. These characteristics suggest that coevolution could be influencing the dynamics of those systems.

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Fig. 6.

Examples of clockwise predator–prey cycles from (A and B) phage–cholera chemostat experiments (38), (C–H) mink and muskrat trapping data from the Hudson’s Bay Company (40), and (I and J) gyrfalcon and rock ptarmigan census data (39). (A, C, E, G, and I) Predator (red) and prey (blue) time series. (B, D, F, H, and J) Thickened segments of time series plotted in the phase plane. Arrows denote the flow of time and green circles denote the first time point. Only the first thickened segments of A and G are plotted in B and H; see SI Text, section C for the second segments. CFU, colony-forming unit; PFU, plaque-forming unit. See SI Text, section C for data sources.

Additional evidence supporting this idea is available for the phage–cholera system. First, in Fig. 6A the predator peaks always occur with or before the prey peaks. Second, in the Wei et al. study (38), the authors observed that the host and phage populations were each composed of two types (but did not measure the density of each type). Our results suggest that the clockwise phage–cholera cycles in Fig. 6B are driven by changes in the frequencies of phage and cholera types, similar to the dynamics of the clonal model 1.