In this paper, we applied the well-known homotopy analysis methods (HAM), which is a semi-analytical method, perturbation method, to study a reaction–diffusion–advection model for the dynamics of populations under biological control. According to the predator–prey model, the advection expression represents the predator density movement in which the acceleration is proportional to the prey density gradient. The prey population reproduces logistically, and the interactions of prey population obey the Holling’s prey-dependent Type II functional response. The predation process splits into the following subdivided processes: random movement which is represented by diffusion, direct movement which is described by prey taxis, local prey interactions, and consumptions which are represented by the trophic function. In order to ensure a successful biological control, one should make the predator-pest population to stabilize at a very low level of pest density. One reason for this effect is the intermediate taxis activity. However, when the system loses stability, for example very intensive prey taxis destroys the stability, it leads to chaotic dynamics with pronounced outbreaks of pest density.
Turing instability in two-patch predator-prey population dynamics