Abstract
In this paper, we consider a diffusive predator–prey model with a time delay and prey toxicity. The effect of time delay on the stability of the positive equilibrium is studied by analyzing the eigenvalue spectrum. Delay-induced Hopf bifurcation is also investigated. By utilizing the normal form method and center manifold reduction for partial functional differential equations, the formulas for determining the property of Hopf bifurcation are given.
Highlights
Since the relationship of different biological species is very common in nature, many scholars have done a lot of works in this field [1,2,3,4]
In [18], the authors studied the Hopf–zero bifurcation in a delayed predator–prey model with dormancy of predators. These results all suggest that the time delay can enrich the dynamic properties of the predator–prey models
We considered a delayed diffusive predator–prey model with toxic substances released by prey
Summary
Since the relationship of different biological species is very common in nature, many scholars have done a lot of works in this field [1,2,3,4]. Some biological species can release toxic substances that can affect the growth of other species. In [6], Kar and Chaudhuri considered a two-species competing fish model with harvesting effect and toxic substances. They mainly studied the stability of the interior equilibrium. The functional response is of Holling II-type; β represents the efficiency toxic substance released by prey They mainly studied the stability of the constant positive steady states and existence of the nonconstant positive steady states. In [18], the authors studied the Hopf–zero bifurcation in a delayed predator–prey model with dormancy of predators These results all suggest that the time delay can enrich the dynamic properties of the predator–prey models.
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