Patterns in a Time-Fractional Predator–Prey System with Finite Interaction Range

Abstract

Diffusive predator–prey systems are well known to exhibit spatial patterns obtained by using the Turing instability mechanism. reaction–diffusion systems were already studied by replacing the time derivative with a fractional order derivative, finding the conditions under which spatial patterns could be formed in such systems. The recent interest in fractional operators is due to the fact that many biological, chemical, physical, engineering, and financial systems can be well described using these tools. This contribution presents a diffusive predator–prey model with a finite interaction scale between species and introduces temporal fractional derivatives associated with species behaviors. We show that the spatial scale of the species interaction affects the range of unstable modes in which patterns can appear. Additionally, the temporal fractional derivatives further modify the emergence of spatial patterns.

Keywords

• pattern formation
• predator–prey systems
• fractional derivatives