Spatio-temporal dynamics in a diffusive Bazykin model: effects of group defense and prey-taxis

TL;DR

This paper models the spatial-temporal dynamics of prey-predator interactions incorporating group defense and prey-taxis, revealing complex bifurcations, pattern formations, and transient behaviors through analytical and numerical methods.

Contribution

It introduces a novel prey-predator model with non-monotonic response and prey-taxis, analyzing bifurcations, existence of patterns, and transient states in spatial-temporal populations.

Findings

Identification of local and global bifurcations in the temporal model

Existence of non-homogeneous stationary solutions near Turing thresholds

Observation of extinction, transient states, and oscillatory solutions within different parameter regions

Abstract

Mathematical modeling and analysis of spatial-temporal population distributions of interacting species have gained significant attention in biology and ecology in recent times. In this work, we investigate a Bazykin-type prey-predator model with a non-monotonic functional response to account for the group defense among the prey population. Various local and global bifurcations are identified in the temporal model. Depending on the parameter values and initial conditions, the temporal model can exhibit long stationary or oscillatory transient states due to the presence of a local saddle-node bifurcation or a global saddle-node bifurcation of limit cycles, respectively. We further incorporate the movement of the populations consisting of a diffusive flux modelling random motion and an advective flux modelling group defense-induced prey-taxis of the predator population. The global existence and boundedness of the spatio-temporal solutions are established using $L^p$-$L^q$ estimate. We also demonstrate the existence of a non-homogeneous stationary solution near the Turing thresholds using weakly nonlinear analysis. A few interesting phenomena, which include extinction inside the Turing region, long stationary transient state, and non-homogeneous oscillatory solutions inside the Hopf region, are also identified.