Dynamics of a Pollinator-Plant-Herbivore Mathematical Model

In this paper, we carried out the analysis of a three-dimensional ODE nonlinear autonomous system which is derived with the aim of describing the interaction between three populations. These take the form of two mutualistic (pollinators and plants) and a third population (herbivores) is introduced. This one is feeded by consuming plants which, in turn, damages the pollinators population too by reducing the rate of visits (to plants) behalf of the pollinators. The specific type of interactions between the populations are described by two types of functional responses of type IV. One of these measures what we call ‘‘saciety rate’’ of consuming plants behalf of the herbivores. The main result contained in this paper is the proof of the existence of an attracting limit cycle for the ODE system. This emerges from a supercritical Hopf bifurcation. Its existence is proved by using the Hopf–Andronov bifurcation theorem, and its stability is proved by using the sign of the first Lyapunov coefficient. In addition to the analysis, a series of numerical simulations are carried out on the full ODE system. These show how the stability feature of an equilibrium point changes: from asymptotic locally stable to unstable hence the emergence of a stable limit cycle within the positive octant of the phase space.