Indirect evolutionary rescue: prey adapts, predator avoids extinction

Recent studies have increasingly recognized evolutionary rescue (adaptive evolution that prevents extinction following environmental change) as an important process in evolutionary biology and conservation science. Researchers have concentrated on single species living in isolation, but populations in nature exist within communities of interacting species, so evolutionary rescue should also be investigated in a multispecies context. We argue that the persistence or extinction of a focal species can be determined solely by evolutionary change in an interacting species. We demonstrate that prey adaptive evolution can prevent predator extinction in two-species predator–prey models, and we derive the conditions under which this indirect evolutionary interaction is essential to prevent extinction following environmental change. A nonevolving predator can be rescued from extinction by adaptive evolution of its prey due to a trade-off for the prey between defense against predation and population growth rate. As prey typically have larger populations and shorter generations than their predators, prey evolution can be rapid and have profound effects on predator population dynamics. We suggest that this process, which we term ‘indirect evolutionary rescue’, has the potential to be critically important to the ecological and evolutionary responses of populations and communities to dramatic environmental change.

where N i is population density of species i, r i is the intrinsic rate of increase, and α ij is the competition coefficient. In the chemical allelopathy scenario, the coefficient α ij is a decreasing function of the difference between the traits u i and u j ( ∂α ij ∂u i < 0 and ∂α ij ∂u j > 0 ).
This implies that greater differences in trait values result in larger negative effects on the other species. The cost of allelopathy is incorporated by assuming that r i is a decreasing function of u i ( ∂r i ∂u i < 0 ). For adaptive evolution of u i , quantitative trait evolution model along the fitness gradient with the additive genetic variance, V i , is used (Abrams 2001). The evolutionary dynamics of the trait u i is then described by The equilibrium density of species i is N i = r i − r j α ij 1− α ij α ji . When the intrinsic rate of increase is represented by r i = r 0i f(u i ), where r 0i is the basal intrinsic rate of increase, decreasing r 0i by an abrupt environmental change causes extinction of species i when r 0i = r j α ij /f(u i ), which is an extinction threshold that we label as r 0i * . We examine whether evolution of the trait of interacting species, u j , can prevent extinction (i.e., it can decrease the extinction threshold, r 0i * ). Decreasing r 0i (and consequently N i ) causes u j to decrease from equation S2. Therefore, if the extinction threshold is an increasing function of u j , evolution of the interacting species can prevent extinction. Hence and indirect evolutionary rescue is possible when We show an example of indirect evolutionary rescue in competitive interactions in numerical simulations. In the following simulations, we assume a function ) for the competition coefficient so that it is always positive, where α 0i is the basal competition coefficient. The intrinsic rate of increase is represented by a linear function, r i = r 0i (1 -au i ), where r 0i is the basal intrinsic rate of increase and a is the defense cost coefficient. We assume that r 2 = r 20 -r 2e , where r 20 is the basic intrinsic rate of increase and r 2e is reduction in the intrinsic rate of increase due to an abrupt environmental change.
When the basal intrinsic rate of increase of species 2 (r 02 ) changed from 1 to 0.2 (r 2e = 0.8), species 2 goes extinct without evolution (Fig. S1A). Evolution of the trait of species 2 does not affect the outcome (Fig. S1B), but evolution of the trait of species 1 rescues species 2 from extinction (indirect evolutionary rescue). Coevolution of two species can also avoid extinction of species 2 (Fig. S1D). The bifurcation plots along r 2e indicate that adaptive chemical allelopathy of interacting species (species 1) can increase the maximum reduction in the intrinsic rate of increase at which species 2 can persist ( Fig. S2).

APPENDIX S2. AN EXAMPLE OF INDIRECT EVOLUTIONARY RESCUE BY ADAPTIVE FORAGING
We assume a diamond food web model (one-predator-two-prey-one-resource) model with adaptive foraging after Matsuda et al. (1996) and Kondoh (2003), where N i is population density of prey species i, P is predator population density, r i is the intrinsic rate of increase, α ij is the competition coefficient, G i is the maximum attack rate on prey i, b i is the conversion efficiency, and m is the predator mortality. The predator foraging effort allocated to prey i is e i , and is adaptively changing according to prey densities. Because there are only two prey species and e i i=1 2 ∑ = 1, the adaptive dynamics of the foraging trait e 1 is described by We show an example of indirect evolutionary rescue by adaptive foraging in the following analyses. We assume that r 2 = r 20 -r 2e , where r 20 is the basic intrinsic rate of increase and r 2e is reduction in the intrinsic rate of increase due to an abrupt environmental change. In the following simulations, we assume b 1 = b 2 = d = 1, r 1 = G 1 = 2, r 20 = G 2 = 1.5; therefore, species 1 is undefended prey and species 2 is defended prey. When the intrinsic rate of increase of species 2 (r 2 ) changes from 1.5 to 1 (r 2e = 0.5), this species goes extinct without evolution (Fig. S3A). Adaptive foraging of the predator rescues species 2 from extinction (indirect evolutionary rescue) and this occurs because of reduced predation effort on species 2 (Fig. S3B). The bifurcation plots along r 2e indicate that adaptive foraging can increase the maximum reduction in the intrinsic rate of increase at which defended prey can persist (Fig.   S4).   Reduction in growth rate, r 2e r 2e r 2e r 2e Figure S3. An example of indirect evolutionary rescue by adaptive foraging. The intrinsic rate of increase of defended prey (r 2 ) changed from 1.5 to 1 (r 2e = 0.5) at the timing of arrows.

Abrams
Gray solid lines: defended prey abundance, gray dashed lines: undefended prey abundance, black solid lines: predator abundance, black dotted lines: foraging effort to undefended prey. Reduction in growth rate, r 2e