Hopf bifurcation and periodic solution of a delayed predator-prey-mutualist system. (English) Zbl 1418.92100
Summary: In this paper, we study a predator-prey-mutualist system with digestion delay. First, we calculate the threshold value of delay and prove that the positive equilibrium is locally asymptotically stable when the delay is less than the threshold value and the system undergoes a Hopf bifurcation at the positive equilibrium when the delay is equal to the threshold value. Second, by applying the normal form method and center manifold theorem, we investigate the properties of Hopf bifurcation, such as the direction and stability. Finally, some numerical simulations are carried out to verify the main theoretical conclusions.
MSC:
| 92D25 | Population dynamics (general) |
| 37N25 | Dynamical systems in biology |
| 34C25 | Periodic solutions to ordinary differential equations |
| 34C23 | Bifurcation theory for ordinary differential equations |
Keywords:
predator-prey-mutualist system; digestion delay; stability; Hopf bifurcation; periodic solutionReferences:
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