Pattern Formation in a Diffusive Population-Toxicant System: Effects of Memory-Based Taxis and Nonlocal Accumulation

In this paper, we propose a spatiotemporal model incorporating memory-driven toxicant-taxis and nonlocal accumulation delay to investigate the interaction between the population and the toxicant in a polluted aquatic environment. Through linear stability theory and bifurcation analysis, we examine how the taxis coefficient, spatial memory delay, and average accumulation delay influence the stability of spatially homogeneous steady states and the formation of spatial patterns. Theoretical analysis reveals that the spatial memory delay does not affect the stability of the coexisting steady state, whereas the taxis coefficient and the average accumulation delay can destabilize it, triggering spatially heterogeneous patterns. We numerically validate these findings and illustrate how key toxicant-related parameters govern the distributions of the population and the toxicant. This study highlights the significant effects of memory-driven avoidance behavior and toxicant cumulative effects on shaping population distributions in polluted aquatic ecosystems.