Abstract

We study the complex dynamics of a Monod–Haldane-type predator–prey interaction model that incorporates: (1) A constant time delay in the prey growth; and (2) diffusion in both prey and predator. We provide the rigorous results of our system including the asymptotic stability of a positive equilibrium; Hopf bifurcation; and the direction of Hopf bifurcation and the stability of bifurcated periodic solutions. We also perform numerical simulations on the effects of diffusion or/and delay when the corresponding ODE model has either a unique interior equilibrium with two interior attractors or two interior equilibria. Our theoretical and numerical results show that diffusion can either stabilize or destabilize the system; large delay could destabilize the system; and the combination of diffusion and delay could intensify the instability of the system. Moreover, when the corresponding ODE system has two interior equilibria, diffusion or time delay in prey or both could lead to the extinction of predator. Our results may provide us useful biological insights on population managements for prey–predator interaction systems.

Original languageEnglish (US)
Pages (from-to)1177-1214
Number of pages38
JournalJournal of Mathematical Analysis and Applications
Volume461
Issue number2
DOIs
StatePublished - May 15 2018

Keywords

  • Diffusion–reaction
  • Hopf bifurcation
  • Predator–prey system
  • Time delay
  • Turing instability

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics

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