Spatio-temporal pattern selection in a prey-predator model with hunting cooperation and Allee effect in prey✩

In this work, we have studied the temporal as well as spatio-temporal dynamics of a prey-predator model with additive Allee effect in prey growth and hunting cooperation among the specialist predators. Hunting cooperation impacts both the stability and the existence scenario of the coexistence equilibrium of the temporal model. The temporal system is capable of exhibiting a wide range of local bifurcations such as transcritical, saddle-node, Hopf, Bogdanov-Takens bifurcations and a global Homoclinic bifurcation. For the diffusive system, the well-posedness is first proved. Then Turing instability is investigated to understand the relationship between the diffusion and the hunting cooperation behind the stationary pattern formation. By employing weakly nonlinear analysis, the amplitude equation of stationary solution is derived, and using the stability of amplitude equation, the pattern selection of several stationary patterns such as spot, stripe, and their combinations have been discussed under various parameter regimes. Extensive numerical simulations are carried out to verify the analytical results. Main contribution of this work is the identification of all possible stationary patterns depending upon the signs of the coefficients involved in the amplitude equation. In addition, some time-dependent nonhomogeneous irregular solutions are observed for parameter values in the Hopf domain or Turing-Hopf domain. With the help of time-series analysis of the obtained numerical results, we have established that the time and space-varying solution is spatio-temporal chaos.