Dynamics and pattern formation of a diffusive predator - prey model in the subdiffusive regime in presence of toxicity

Motivated by the fact that the restrictive conditions for a Turing instability are relaxed in subdiffusive regime, we investigate the effects of subdiffusion in the predator - prey model with toxins under the homogeneous Neumann boundary condition. First, the stability analysis of the corresponding ordinary differential equation is carried out. From this analysis, it follows that stability is closely related to the coefficient of toxicity. In addition, the temporal fractional derivative does not systematically widen the range of parameters to maintain a point in the stability domain. Furthermore, we derive the condition which links the Turing instability to the coefficient of toxicity in the subdiffusive regime. System parameters are varied in order to test our mathematical predictions while comparing them to ecological literature. It turns out that the memory effects, linked to the transport process can, depending on the parameters, either stabilize an ecosystem or make a completely different configuration.