Global Dynamics of Population–Toxin Systems with Nonlocal Usage of Memory Under Periodic Boundary Conditions

We study a reaction–diffusion model for population–toxin interactions on a two-dimensional torus, where avoidance is driven by a nonlocal average of toxin-related memory. The variables represent population density, toxin concentration, and a phenomenological information field generated by population–toxin encounters. This information diffuses, decays, and enters the taxis term through convolution with a perceptual kernel. We prove local well-posedness, global existence and uniform boundedness, and give sufficient conditions for exponential convergence to either the positive equilibrium or the toxin-only equilibrium. Numerical simulations with two perceptual radii yield stripe and spot patterns, illustrating that the sensing scale can alter spatial organization. These simulations do not establish an effect on population persistence and should be viewed as hypotheses for empirical study.