Pattern formation and delay-induced instability in a Leslie–Gower type prey–predator system with Smith growth function

In this article, we have investigated the spatiotemporal dynamics and delay-induced instability of a Leslie–Gower type prey–predator model under the influence of environmental toxicants with Smith growth function. This growth function is more realistic than logistic growth as it better describes the growth of the biological population. It has been used where the growth limitations are based on the proportion of available resources not utilized. A few works of Smith’s growth models are reported in the literature. Therefore, spatiotemporal dynamics and pattern formation with delay effect remain an exciting area of research, which motivates the present work. This work has studied two types of dynamical systems: (i) an ordinary differential temporal system with time delay and (ii) a reaction–diffusion system with time delay. The existence of equilibrium points and their stability conditions are discussed. Hopf bifurcation emerges in both proposed systems with respect to delay parameter. The stability and direction of Hopf bifurcation and delay–diffusion-driven instability have been investigated for the reaction–diffusion system. Numerical simulation is performed to support the analytical results and theorems. Moreover, the existence of Hopf and delay-induced instability are proved numerically. Interesting one-dimensional regular and irregular stripe patterns are obtained for increased values of the time delay parameter. Also, the presence of natural toxicants has a negative impact on the growth of prey–predator species.