Abstract
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.
Highlights
Mutualism is one of the most important relationships in the real world
Hopf bifurcation theorem for functional differential equation [ ], we get the following result
If τ > and P O + P Q =, the positive equilibrium E∗ is asymptotically stable for < τ < τ, and it becomes unstable for τ staying in some right neighborhood of τ, with a Hopf bifurcation occurring when τ = τ
Summary
Mutualism is one of the most important relationships in the real world. For example, ants deter herbivores from feeding on plants [ ] and deter predators from feeding on aphids [ , ]. Some the predator-prey-mutualist models have been studied by several scholars [ – ]. Wu [ ] studies the positive periodic solutions for a mutualistic model with saturating term and the effects of toxic substance by using Mawhin’s continuation theorem of coincidence degree theory [ ].
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