On the dynamics of predator-prey models with the Beddington-DeAngelis functional response.

*(English)*Zbl 0991.34046The dynamics of predator-prey models with Holling-type II functional response is well known. So is the main dynamics of pure ratio-dependent predator-prey models with Holling-type II functional response. It is now known that these models pale in comparison to Beddington-DeAngelis-type predator-prey models when confronted with real data. Although the Beddington-DeAngelis-type predator-prey model has been around for over two decades, there has been little qualitative work done on it.

This paper provides a first attempt on the mathematical analysis of this model and its direct extension to a spatially heterogeneous setting (a reaction-diffusion model). The main results are criteria for permanence and for predator extinction. Recently, it is known (Tzy-Wei Hwang, personal communication (2002)) that the positive steady state of the ODE Beddington-DeAngelis-type predator-prey model is globally stable if it is locally stable.

This paper provides a first attempt on the mathematical analysis of this model and its direct extension to a spatially heterogeneous setting (a reaction-diffusion model). The main results are criteria for permanence and for predator extinction. Recently, it is known (Tzy-Wei Hwang, personal communication (2002)) that the positive steady state of the ODE Beddington-DeAngelis-type predator-prey model is globally stable if it is locally stable.

Reviewer: Kuang Yang (Tempe)

##### MSC:

34D23 | Global stability of solutions to ordinary differential equations |

92D25 | Population dynamics (general) |

34D05 | Asymptotic properties of solutions to ordinary differential equations |

35K57 | Reaction-diffusion equations |

##### Keywords:

predator-prey models; Beddington-DeAngelis functional response; permanence; predator extinction
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\textit{R. S. Cantrell} and \textit{C. Cosner}, J. Math. Anal. Appl. 257, No. 1, 206--222 (2001; Zbl 0991.34046)

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