Non-spatial Dynamics and Spatiotemporal Patterns Formation in a Predator-Prey Model with Double Allee and Dome-shaped Response Function.

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The extinction of species is a major threat to the biodiversity. Allee effects are strongly linked to population extinction vulnerability. Emerging ecological evidence from numerous ecosystems reveals that the Allee effect, which is brought on by two or more processes, can work on a single species concurrently. The cooperative behavior which raises Allee effect in low population density, can create group defence in species to protect themselves from predation. This article focuses on the dynamics of a predator-prey system with double Allee effect in prey growth and simplified Monod-Haldane form of dome-shaped response function to incorporate group defence ability of prey as time and space vary. The study obtains that, to some extent, group defence of prey plays a positive role for the stability of both the species, but on negative side, if defensive ability exceeds a threshold value then both the population can not survive simultaneously and predator population dies out. The Allee effect produces bi-stability (weak Allee) even tri-stability (strong Allee) in phase space reflecting that the system dynamics is very sensitive subject to initial population of the species. The combined impact of double Allee and group defence of prey leads in populations enduring stable periods punctuated by oscillations. The species' mobility based on only its own population is insufficient to this model for Turing instability. The presence of double Allee effect increases the instability regions that enhances the likelihood of various patterns. Whereas increasing group defence of prey decreases the instability region in spatial system. The species distribution stabilizes in forms of spots, stripes and mixture of both in heterogeneous environment. But for prey, gathering decreases with increasing growth rate and gathering increases with increasing Allee effect due to cross-diffusion which results paradox to temporal system. In contrast, populations in the Hopf and Hopf-Turing regions fluctuate (oscillatory) or their distribution becomes unpredictable (chaotic).