Localized patterns and their moving dynamics in a diffusive population-toxicant model with negative toxicant-taxis

Abstract We investigate the existence of aggregation patterns and their moving dynamics in the population-toxicant model proposed in [10], which incorporates the spatial avoidance response of a population to toxicants characterized by a negative (repulsive) toxicant-taxis term. In the limit of large toxicant-taxis sensitivity, we explicitly construct asymptotic equilibrium spike solutions using formal matched asymptotic expansions, and show that boundary spikes are linearly stable at leading order. In contrast, by deriving an explicit asymptotic formula for the translational eigenvalue, we show that interior spike solutions are linearly unstable to translations of the spike profile. We further derive a reduced system of ordinary differential equations describing the fast motion of the spike center and the slow motion of the spike height for both spatially homogeneous and inhomogeneous toxicant inputs. Our results reveal that spikes drift rapidly toward the nearest domain boundary in homogeneous environments, while they converge to a stable local minimum of the background toxicant concentration in strongly inhomogeneous environments. These asymptotic findings offer a new research direction for subsequent rigorous analytical work. 2020 MSC. 92D25, 35K57, 35K55, 35Q92