Abstract
In this paper, we consider a diffusive predator–prey system with strong Allee effect and two delays. First, we explore the stability region of the positive constant steady state by calculating the stability switching curves. Then we derive the Hopf and double Hopf bifurcation theorem via the crossing directions of the stability switching curves. Moreover, we calculate the normal forms near the double Hopf singularities by taking two delays as parameters. We carry out some numerical simulations for illustrating the theoretical results. Both theoretical analysis and numerical simulation show that the system near double Hopf singularity has rich dynamics, including stable spatially homogeneous and inhomogeneous periodic solutions. Finally, we evaluate the influence of two parameters on the existence of double Hopf bifurcation.
Highlights
In nature, population growth is limited by environmental resources, this explains why the population cannot grow indefinitely
A diffusive predator–prey system with two delays and strong Allee effect is investigated in our paper
By applying the method of stability switching curves, we study the joint effect of the two delays on the stability of the positive constant steady state
Summary
Population growth is limited by environmental resources, this explains why the population cannot grow indefinitely. Population densities of many species are known to fluctuate nearly periodically over time [20], a phenomenon to which the delay may provide an explanation Notice that both the positive feedback of intraspecific cooperation and the negative feedback of intraspecific competition can have delay to the growth of the population. Chang et al [5] investigated the dynamics of a scalar population model with delayed Allee effect as follows:. Chen et al [6] proposed a diffusive predator–prey model with digestion delay They investigated the effect of the digestion delay on the system and obtained the stability of the equilibria and the existence of Hopf bifurcation. Du et al investigated a diffusive Leslie–Gower model with two delays [10], they performed the stability analysis and explored the existence of double Hopf bifurcation.
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