Stability of Hopf-bifurcating limit cycles in a diffusion-driven prey-predator system with Allee effect and time delay.

Abstract

In this paper, we investigate the effect of the gestation delay on the spatiotemporal pattern formation in a prey-predator system with monotonic functional response with saturation and intraspecific competition among the predator population in presence of the additive Allee effect in prey growth. In this regard, we present rigorous analytical results for determining the delay-induced Hopf bifurcation threshold and the associated properties of the Hopf-bifurcating periodic solutions and verify them with the help of numerical simulations. We derive analytically the delay-induced Hopf bifurcation threshold by employing the linear stability analysis about the unique spatially uniform coexistence steady state. Also, we provide the expressions for determining the direction and stability of the Hopfbifurcating periodic solutions by using the normal form theory and center manifold reduction. The numerical simulation results reveal that the Hopf bifurcation can potentially lead to spatially homogeneous periodic in time distribution of the populations which will eventually settles to chaotic in space and time distribution for sufficiently large value of the time delay. Further, our numerical investigations reveal that the time delay can change one stationary pattern to another through the loss of monotonicity property of the spatially averaged densities.