Predator–prey model of Beddington–DeAngelis type with maturation and gestation delays

Abstract

A predator–prey model of Beddington–DeAngelis type with maturation and gestation delays is formulated and analyzed. This two-delay model is similar to the stage-structured model by Liu and Beretta [S. Liu, E. Beretta, Stage-structured predator–prey Model with the Beddington–DeAngelis functional response, SIAM J. Appl. Math. 66 (2006) 1101–1129] but contains an extra gestation delay term. Criteria for permanence and for predator extinction as well as the global attractiveness of the interior equilibrium are derived. The combined effects of the two delays and the degree of predator interference on the dynamical behaviors of the coexistence equilibrium are also studied both analytically and numerically. It is shown that complicated behaviors including chaotic and multi-periodic solutions may occur with the introduction of gestation delay, and that the predator interference can stabilize the system by simplifying the dynamical behaviors and enlarging the stability parameter fields.