Branching Scenarios in Eco-evolutionary Prey-Predator Models

We show in this paper how simulations of ODEs and continuations of systems of algebraic equations can be combined to study the evolution of biodiversity in multi-species systems where phenotypic traits are genetically transmitted. We follow the Adaptive Dynamics (AD) approach, that provides a deterministic approximation of the evolutionary dynamics of stationary coexisting populations in terms of an ODE system, the so-called AD canonical equation. AD also provides explicit conditions to test whether a stable evolutionary equilibrium of the canonical equation is a branching point—resident and mutant morphs coexist and further differentiate thus increasing biodiversity—or not. We analyze a standard parameterized family of prey-predator communities, described by the most standard ecological model, and propose an iterative procedure to obtain the branching portrait, explaining the dependence of branching scenarios on two (demographic, environmental or control) parameters. Among a number of interesting results, in line with field studies and known ecological principles, we find that prey branching is induced by the predation pressure, and is favored when prey intraspecific competition is highly sensitive to the resident-mutant phenotypic mismatch; while predator branching is not possible when prey and predators are present in equal number of morphs. This results in alternate prey-predator branching sequences with possible phases of prey unilateral branching. The guidelines for deriving a general method for analyzing the evolution of biodiversity are also discussed. The indications that can be obtained typically have qualitative nature, but can be of help for the long-term conservation and management of biodiversity.