BIFURCATION ANALYSIS OF A GENERAL ACTIVATOR-INHIBITOR MODEL WITH NONLOCAL DISPERSAL

In this paper, we are mainly concerned with the effect of nonlocal diffusion and dispersal spread on bifurcations of a general activator-inhibitor system in which the activator has a nonlocal dispersal. We find that spatially inhomogeneous patterns always exist if the dispersal rate of the activator is sufficiently small, while a larger dispersal spread and an increase of the activator diffusion inhibit the formation of spatial patterns. Compared with the spatial averaging nonlocal dispersal model, our model admits a larger parameter region supporting pattern formations, which is also true if compared with the local reaction-diffusion one when the dispersal spread is small. We also study the existence of nonconstant positive steady states through bifurcation theory and find that there could exist finite or infinite steady state bifurcation points of the inhibitor diffusion constant. As an example of our results, we study a water-biomass model with nonlocal dispersal of plants and show that the water and plant distributions could be inphase and antiphase.

Automated summary

This paper investigates the impact of nonlocal diffusion and dispersal spread on bifurcations of an activator-inhibitor system. It is found that spatially inhomogeneous patterns exist under certain conditions, and the model supports a larger parameter region for pattern formations compared to other models. Additionally, the study explores the existence of nonconstant positive steady states through bifurcation theory and identifies finite or infinite steady state bifurcation points in the system. As illustrated by a water-biomass model, the distributions of water and plants may exhibit inphase or antiphase patterns.