The existence of a limit cycle in a pollinator–plant–herbivore mathematical model

Abstract This paper deals with the existence of a limit cycle in a nonlinear ODE mathematical model which describes the interaction between three homogeneous populations. These take the form of pollinators, plants and herbivores. The interaction between the pollinators and plants is of mutualistic type given by a functional response of type II; meanwhile that for the plants–herbivores, from the demographic point of view, it can be seen as a predator–prey interaction. The resulting coupled nonlinear ODE system contains several parameters which have an important ecological interpretation. In the study we present here we give necessary conditions on the parameter values for the emergence of an attracting limit cycle and other which is unstable. These come from a supercritical and a subcritical Hopf bifurcation, respectively. Because of the amount of the involved parameters, almost any calculation becomes really massive. Thus, part of the analysis we present here intensively uses the Mathematica symbolic software. With the same software, by using a theorem authored by Kuznetsov, we calculated the first Lyapunov coefficient whose sign gives us the type of stability the limit cycle has. A number of numerical simulations also are included with the aim of showing the limit cycle in the three dimensional phase space. This is done in two main cases: when the ODE system has one positive equilibrium and when such system has two positive equilibria. The main part of the paper ends by exploring other set of parameters values which allows us to recover the previous dynamics but also gives us other interesting behaviors.