Pattern formation in a system involving prey-predation, competition and commensalism

In this paper, pattern forming instabilities in a three species reaction-diffusion system involving prey-predation, competition and commensalism are explored. The system consists of two competing species, and the third species acts as a predator for one of the species and as a host for the other species. The equilibrium points of the model are determined. The conditions for existence of interior equilibrium point are derived. Bifurcation analysis of the model is done, and conditions for existence of Turing and non-Turing patterns are derived using Routh-Hurwitz criteria. A series of numerical simulation results are presented to show Turing as well as non-Turing patterns. Various types of patterns (e.g., spirals, spots, strips, mixture of spots and strips) are observed depending on the ecological parameters of the local system and diffusion coefficients. The existence of spatially homogeneous, inhomogeneous periodic and inhomogeneous aperiodic oscillations and chaotic oscillations is shown in the three species model.