Top predators induce the evolutionary diversification of intermediate predator species

We analyze the evolutionary branching phenomenon of intermediate predator species in a tritrophic food chain model by using adaptive dynamics theory. Specifically, we consider the adaptive diversification of an intermediate predator species that feeds on a prey species and is fed upon by a top predator species. We assume that the intermediate predator׳s ability to forage on the prey can adaptively improve, but this comes at the cost of decreased defense ability against the top predator. First, we identify the general properties of trade-off relationships that lead to a continuously stable strategy or to evolutionary branching in the intermediate predator species. We find that if there is an accelerating cost near the singular strategy, then that strategy is continuously stable. In contrast, if there is a mildly decelerating cost near the singular strategy, then that strategy may be an evolutionary branching point. Second, we find that after branching has occurred, depending on the specific shape and strength of the trade-off relationship, the intermediate predator species may reach an evolutionarily stable dimorphism or one of the two resultant predator lineages goes extinct.

Adaptive evolution of attack ability promotes the evolutionary branching of predator species

In this paper, with the methods of adaptive dynamics and critical function analysis, we investigate the evolutionary branching phenomenon of predator species. We assume that both the prey and predators are density-dependent and the predator's attack ability can adaptively evolve, but this has a cost in terms of its death rate. First, we identify the general properties of trade-off relationships that allow for a continuously stable strategy and evolutionary branching in the predator strategy. It is found that if the trade-off curve is weakly concave near the singular strategy, then the singular strategy may be an evolutionary branching point. Second, we find that after the branching has occurred in the predator strategy, if the trade-off curve is convex-concave-convex, the predator species will eventually evolve into two different types, which can stably coexist on the much longer evolutionary timescale and no further branching is possible.

FAST AND SLOW FEEDBACK IN SYSTEMS THEORY

Biological systems are complex systems with feedback. Very often the scales for the dynamics of the system and the dynamics of the feedback are very different. The mathematical tool used to deal with this different time scales is Tikhonov's theorem which permits to reduce the complexity of the system through suitable approximations. This paper presents a theory of two time scales feedback systems phrased in the language of Nonstandard Analysis (NSA), introduced in the sixties by A. Robinson. Our opinion is that this presentation is more understandable for a non-mathematicaly trained reader than the classical one. Th paper is entirely self contained. A short but comprehensive tutorial on NSA is provided and precise definitions for concepts from systems theory are given.

A PREY-PREDATOR-SUPERPREDATOR SYSTEM

We consider a prey-predator-superpredator model. Assuming that the growth rates of the three populations are highly diversified, we prove that under some conditions on the parameters of the model, the three populations can coexist through a cycle for which we give the period. Besides, we give the laws (entrance-exit functions) that govern the prey and predator regeneration when they tend to extinction.