A detailed study of the Beddington–DeAngelis predator–prey model

The present investigation accounts for the influence of intra-specific competition among predators in the original Beddington–DeAngelis predator–prey model. We offer a detailed mathematical analysis of the model to describe some of the significant results that may be expected to arise from the interplay of deterministic and stochastic biological phenomena and processes. In particular, stability (local and global) and bifurcation (Saddle-node, Transcritical, Hopf–Andronov, Bogdanov–Takens) analysis of this model are conducted. Corresponding results from previous well known predator–prey models are compared with the current findings. Nevertheless, we also allow this model in stochastic environment with the influences of both, uncorrelated “white” noise and correlated “coloured” noise. This showing that competition among the predator population is beneficial for a number of predator–prey models by keeping them stable around its positive interior equilibrium (i.e. when both populations co-exist), under environmental stochasticity. Comparisons of these findings with the results of some earlier related investigations allow the general conclusion that predator intra-species competition benefits the predator–prey system under both deterministic and stochastic environments. Finally, an extended discussion of the ecological implications of the analytical and numerical results concludes the paper.

► Complexity of the original Beddington-DeAngelis model.
► Twenty years controversy over the paradox of enrichment is resolved.
► Influence of predator intra-specific competition is established.
► This article finds a good model among all the models studied earlier.
► Find conditions for stability in both deterministic and stochastic environments for five models.