| Abstract: |
| A reaction-diffusion population model with Dirichlet boundary condition and a directed movement oriented by a temporally distributed delay is proposed to describe the lasting memory of animals moving over space. The temporal kernel of the memory is taken to be the Gamma distribution function, in particular the weak kernel in which the animals can immediately acquire knowledge and memory decays over time and the strong kernel by which we assume that animals` memory undergoes learning and memory decay stages. It is shown that the population stabilizes to a positive steady state and aggregates in the interior of the territory when the delay kernel is the weak type; and in the strong kernel case, oscillatory patterns can arise and vanish when the mean delay value increases via two Hopf bifurcations, thus a stability switch phenomenon occurs and spatial-temporal patterns emerge for intermediate value of delays. |
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