Prey-taxis and time delay induced instabilities in a prey–predator system incorporating the smith growth function

In this work, prey-taxis and delay-induced instabilities have been investigated with the Smith growth function. A food-limited Holling–Tanner-type reaction–diffusion system is proposed and analyzed using various dynamic approaches. This is one of the earlier investigations of the Smith growth function with prey-taxis and time delay. Here, we have studied two mathematical models: (a) time-delayed prey–predator system and (b) prey-taxis reaction–diffusion system with time delay. The existence of equilibrium points and their stability conditions are derived for a nonspatial delayed system. Hopf bifurcation emerges in both proposed systems with respect to the delay parameter ([Formula: see text]). Hopf bifurcation exists even in the absence of prey-taxis in the reaction–diffusion system. Turing instability conditions are derived for a prey-taxis system. Numerical results are shown to support analytical findings for both repulsive prey-taxis ([Formula: see text]) and attractive prey-taxis ([Formula: see text]), with [Formula: see text] as prey-taxis sensitivity coefficient. Turing instability is verified numerically for different values of Smith’s constant parameter ([Formula: see text]). In order to achieve Turing instability, an increase in Smith’s parameter is required with an increased prey-taxis rate. Our proposed spatial system forms interesting regular vertical stripes and periodic oscillation patterns in a one-dimensional spatial domain.