Dynamics of a diffusion-advection model with local perception of toxins

Spatial memory plays a critical role in animal movement modeling, yet explicitly modeling the learning processes underlying memory acquisition remains a significant challenge. This study focuses on the dynamics of a two-species model in a toxic environment, where both species are assumed to have a perceptual ability to sense toxins and actively avoid areas with high toxin concentrations to increase their survival chances. The models consist of three PDEs in composition with one ODE and the existence of globally bounded solutions is established in two dimensions by employing advanced coupled energy estimates and the smoothing properties of the Neumann semigroup. We then conduct a spectral analysis of the model and determine the stability of the steady-state solutions by analyzing the corresponding eigenvalue problems. Subsequently, we perform bifurcation analysis using spatial memory decay rate and perceptual diffusion rate as bifurcation parameters. The study reveals that both steady-state and Hopf bifurcations can occur in these systems, with bifurcation points identified to delineate stability regions. Moreover, these systems are capable of generating rich spatial and spatiotemporal patterns through various types of bifurcations. Our work introduces a novel approach for addressing hybrid PDE-ODE models and provides deeper insights into the cognitive movement-driven dynamics of consumer-resource interactions. This framework enhances the understanding of species behavior in response to environmental toxins and offers new perspectives on ecological stability and pattern formation.