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<p>In this paper, to describe the cognitive process, local perception and the simple learning mechanism were introduced into the resource-consumer models. Their effects on the spatial concentration distribution of the resources and consumers were investigated, and non-constant positive steady states were found to exist. Due to the presence of the convective effect, by using the theory of fixed point index in positive cones, the existence results of non-constant positive steady states were established. Further, their asymptotic profiles, as cognition process was extremely calmed or chaotic, were presented. When the diffusion rate of the cognition was tending to zero or infinity, with the fixed diffusion rates of consumers and resources, their asymptotic profiles were described explicitly. Moreover, numerical simulations were presented to illustrate the theoretical results. The obtained results provided a solution to Problem 16 in [<CitationRef CitationID="CR3">3</CitationRef>].</p>

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Asymptotic Profiles of Steady States in Diffusive Models with Perception and Learning Mechanism

  • 期刊论文

  • 发表日期：

    2025年10月2日

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  • Qigang Deng,
  • Ranchao Wu,
  • Biao Liu

摘要

In this paper, to describe the cognitive process, local perception and the simple learning mechanism were introduced into the resource-consumer models. Their effects on the spatial concentration distribution of the resources and consumers were investigated, and non-constant positive steady states were found to exist. Due to the presence of the convective effect, by using the theory of fixed point index in positive cones, the existence results of non-constant positive steady states were established. Further, their asymptotic profiles, as cognition process was extremely calmed or chaotic, were presented. When the diffusion rate of the cognition was tending to zero or infinity, with the fixed diffusion rates of consumers and resources, their asymptotic profiles were described explicitly. Moreover, numerical simulations were presented to illustrate the theoretical results. The obtained results provided a solution to Problem 16 in [3].