Stability and spatiotemporal patterns in a diffusive predator-prey model with allee effect in predators and nonlocal delay

In this paper, we investigate the effect of nonlocal delay on stability and spatiotemporal dynamics of a diffusive predator-prey model with Allee effect in predators. For the system without nonlocal delay, the stability and Turing instability conditions are derived. For the system with nonlocal delay, all Turing bifurcation curves on (d₂, τ) plane are obtained, and the critical Turing bifurcation curve near the stable region is determined by the discussion of the properties of Turing bifurcation curves. It is found that the presence of nonlocal delay decreases the range of the occurrence for Turing bifurcation. The critical Hopf bifurcation curve is found by the geometric stability switch criterion. The stable region for the positive equilibrium is bounded by the critical Turing and Hopf bifurcation curves. On the boundaries of stable region, Hopf bifurcation, Turing bifurcation, Turing-Turing bifurcation and Turing-Hopf bifurcation may occur. To reveal the complex dynamic near Turing-Hopf bifurcation point, normal form for the system with nonlocal delay is derived, from which different types of bistability are found, including two stable spatially inhomogeneous periodic solutions and two stable spatially inhomogeneous steady states.