Animal movements and their underlying mechanisms are an extremely important research area in biology and have been extensively studied for centuries. However, spatial memory and cognition, which is the most significant difference between animal movements and chemical movements, has been ignored in the modeling of animal movements. To incorporate cognition and memory of “clever” animals in the simplest and self-contained way, we propose a delayed diffusion model via a modified Fick’s law, whereas in literature the standard diffusion for chemical movements was applied to describe “drunk” animal movements. Our mathematical model is expressed by a reaction–diffusion equation with an additional delayed diffusion term, which makes rigorous analysis intriguing and challenging. We show the wellposedness and analyze the asymptotic stability of steady state in the spatial memory model. It is shown that for the three possible reaction schemes, the stability of a spatially homogeneous steady state fully depends on the relationship between the two diffusion coefficients but is independent of the time delay. Finally, we numerically illustrate possible spatialtemporal patterns when the system is divergent.

中文翻译：

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具有记忆的扩散空间运动

动物运动及其潜在机制是生物学中极为重要的研究领域，并且已经进行了数百年的广泛研究。然而，在动物运动的建模中，空间记忆和认知是动物运动与化学运动之间最显着的差异。为了以最简单，自成体系的方式整合“聪明”动物的认知和记忆，我们通过修正的菲克定律提出了一种延迟扩散模型，而在文献中，化学运动的标准扩散被用来描述“醉酒”动物的运动。我们的数学模型由反应扩散方程和附加的延迟扩散项来表示，这使得严格的分析有趣而富有挑战性。我们展示了适度性并分析了空间记忆模型中稳态的渐近稳定性。结果表明，对于三种可能的反应方案，空间均匀稳态的稳定性完全取决于两个扩散系数之间的关系，但与时间延迟无关。最后，我们以数字方式说明了系统发散时可能的时空模式。