Searching for spatial patterns in a pollinator-plant-herbivore mathematical model

This paper deals with the spatio-temporal dynamics of a pollinator-plant-herbivore mathematical model. The full model consists of three nonlinear reaction-diffusion-advection equations defined on a rectangular region. In view of analyzing the full model, we firstly consider the temporal dynamics of three homogeneous cases. The first one is a model for a mutualistic interaction (pollinator-plant), later on a sort of predator-prey (plant-herbivore) interaction model is studied. In both cases, the interaction term is described by a Holling response of type II. Finally, by considering that the plant population is the unique feeding source for the herbivores, a mathematical model for the three interacting populations is considered. By incorporating a constant diffusion term into the equations for the pollinators and herbivores, we numerically study the spatiotemporal dynamics of the first two mentioned models. For the full model, a constant diffusion and advection terms are included in the equation for the pollinators. For the resulting model, we sketch the proof of the existence, positiveness, and boundedness of solution for an initial and boundary values problem. In order to see the separated effect of the diffusion and advection terms on the final population distributions, a set of numerical simulations are included. We used homogeneous Dirichlet and Neumann boundary conditions.