Existence of complex patterns in the Beddington–DeAngelis predator–prey model

The study of reaction–diffusion system constitutes some of the most fascinating developments of late twentieth century mathematics and biology. This article investigates complexity and chaos in the complex patterns dynamics of the original Beddington–DeAngelis predator–prey model which concerns the influence of intra species competition among predators. We investigate the emergence of complex patterns through reaction–diffusion equations in this system. We derive the conditions for the codimension-2 Turing-Hopf, Turing-Saddle-node, and Turing-Transcritical bifurcation, and the codimension-3 Turing-Takens-Bogdanov bifurcation. These bifurcations give rise to very complex patterns that have not been observed in previous predator–prey models. A large variety of different types of long-term behavior, including homogenous distributions and stationary spatial patterns are observed through extensive numerical simulations with experimentally-based parameter values. Finally, a discussion of the ecological implications of the analytical and numerical results concludes the paper.

► A case that is not dealt by the classical Turing theory.
► Complexity and chaos in the original Beddington–DeAngelis predator–prey model.
► Conditions for Turing-Hopf, Turing-Saddle-node, Turing- Transcritical bifurcation.
► Obtain conditions for the codimension-3 Turing-Takens-Bogdanov bifurcation.
► A general theorem is provided which gives the conditions for these bifurcations.