Dynamical Behaviors of a Delayed Prey–Predator Model with Beddington–DeAngelis Functional Response: Stability and Periodicity

Abstract

This work investigates a prey–predator model with Beddington–DeAngelis functional response and discrete time delay in both theoretical and numerical ways. Firstly, we incorporate into the system a discrete time delay between the capture of the prey by the predator and its conversion to predator biomass. Moreover, by taking the delay as a bifurcation parameter, we analyze the stability of the positive equilibrium in the delayed system. We analytically prove that the local Hopf bifurcation critical values are neatly paired, and each pair is joined by a bounded global Hopf branch. Also, we show that the predator becomes extinct with an increase of the time delay. Finally, before the extinction of the predator, we find the abundance of dynamical complexity, such as supercritical Hopf bifurcation, using the numerical continuation package DDE-BIFTOOL.

Multiple dynamics in a delayed predator‐prey model with asymmetric functional and numerical responses

“…As the conditions in Case 4 are satisfied, this leads to existence of 𝜔 ± . The corresponding values of critical time delays are given from expression (18). As mentioned in Lemma 4.1, the sum of multiplicities of roots in C + will change only when a root appears on or crosses the imaginary axis.…”

“…Similar conclusion was derived by Rabago and Collera 17 in their delayed intraguild predator‐prey model. Zhang et al 18 have considered a predator‐prey model where time delay is due to the gestation of predator. They found that periodic solutions are possible when time delay is varied, i.e., predator coexists with the prey in an oscillatory mode.…”