Allee Effect in a Generalist Predator Induces Nonstationary Patterns: A Detailed Study of Target and Spiral Patterns

The discovery of spatiotemporal pattern formation has unveiled a broad area of research focusing on exploring the dynamics of different interacting populations. This includes interactions such as the prey–predator relationships, competition between species, and mutualistic associations. The reaction–diffusion system plays an important role in explaining the pattern formation scenarios in a prey–predator model. In this paper, our aim is to study the dynamics of a spatiotemporal predator–prey model that incorporates the Allee effect in the predator’s growth rate with a Holling type-II functional response. Our investigation reveals the emergence of Turing patterns, including cold spots, hot spots, and a combination of spots and stripes. Additionally, our model demonstrates the appearance of non-Turing patterns, such as spirals and targets, for specific sets of parameter values. The amplitude equations for target and spiral patterns are established. Through numerical and analytical techniques, we confirmed the existence of the target pattern. Our analysis further shows that by increasing the Allee effect in conjunction with diffusion, the Turing patterns change from simple spot formations to a combination of spots and stripes.