Bifurcation Analysis of a Prey-Predator Coevolution Model

Abstract

We show in this paper how numerical bifurcation analysis can be used to study the evolution of genetically transmitted phenotypic traits. For this, we consider the standard Rosenzweig-MacArthur prey-predator model [The American Naturalist, 97 (1963), pp. 209-223] and, following the so-called adaptive dynamics approach, we derive from it a second-order evolutionary model composed of two ODEs, one for the prey trait and one for the predator trait. Then, we perform a detailed bifurcation analysis of the evolutionary model with respect to various environmental and demographic parameters. Surprisingly, the evolutionary dynamics turn out to be much richer than the population dynamics. Up to three evolutionary attractors can be present, and the bifurcation diagrams contain numerous global bifurcations and codimension-2 bifurcation points. Interesting biological properties can be extracted from these bifurcation diagrams. In particular, one can conclude that evolution of the traits can be cyclic and easily promote prey species diversity.

Multilocus Genetics and the Coevolution of Quantitative Traits

“…Models using phenotypic approximations (Abrams 2001), such as adaptive dynamics (e.g., Dieckmann and Law 1996;Doebeli and Dieckmann 2000;Dercole et al 2003), game theory (e.g., Brown and Vincent 1992), and quantitative genetics (e.g., Saloniemi 1993;Abrams and Matsuda 1997;Gavrilets 1997a;Khibnik and Kondrashov 1997), describe the evolution of phenotypes directly, while skipping over the details of the underlying genetics. Explicit genetic models frequently consider only one locus (Gavrilets and Hastings 1998) or two loci (e.g., Bell and Maynard Smith 1987;Seger 1988;Preigel and Korol 1990;Kirzhner et al 1999) per species.…”

Evolutionary Dynamics Can Be Chaotic: A First Example

“…The number of possibilities is practically unlimited because even for well-identified species there are many meaningful options. To be consistent with the analysis of the ditrophic food chain carried out in [Dieckmann et al, 1995;Dercole et al, 2003], we assume that the parameters r, e i , and d i , i = 1, 2, are trait-independent, while resource intraspecific competition c is given by…”

“…In the present case intuition was based on the results obtained in [Dieckmann et al, 1995;Dercole et al, 2003], where it was shown that one way of obtaining the most complex evolutionary dynamics (cyclic dynamics in that case) of a ditrophic food chain composed of resource and consumer was to increase the mutational rate of the resource. We could then retain that message, and hope, on a purely intuitive ground, that the resource mutational rate could be an effective control parameter for transforming simple (i.e., stationary) into complex (i.e., chaotic) evolutionary dynamics in tritrophic food chains.…”

“…So far, the modeling effort has been limited to one or two evolving traits, for both application-specific arguments and mathematical tractability (see, e.g., [Abrams, 1992;Marrow et al, 1992;Dieckmann et al, 1995;Iwasa & Pomiankowski, 1995;Marrow et al, 1996;Iwasa & Pomiankowski, 1999;Abrams & Matsuda, 1997;Gavrilets, 1997;Khibnik & Kondrashov, 1997;Doebeli & Ruxton, 1997;Kisdi et al, 2001;Dercole et al, 2002;Dercole, 2003;Dercole et al, 2003Dercole et al, , 2006). The most relevant conceptual result emerging from these studies is that reciprocal adaptation under constant environmental conditions can entrain two coevolving species in a perpetual cycle of coevolutionary change-the celebrated "Red Queen" dynamics, after [Van Valen, 1973]-in contrast with an optimization view of evolution.…”

Remarks on branching-extinction evolutionary cycles

“…Evolutionary cycles have captured the attention of theoretical ecologists and geneticists in the last decades (see e.g. Abrams, 1992;Marrow et al, 1992;Dieckmann et al, 1995;Pomiankowski, 1995, 1999;Marrow et al, 1996;Abrams and Matsuda, 1997;Gavrilets, 1997;Dercole et al, 2003). In all the above cited works, the adaptive traits vary cyclically while the population densities track the equilibrium corresponding to the current trait values.…”