具有避难效应和非局部被捕食效应的捕食系统斑图形态研究

Spatial Patterns of a Predator-prey Model with Prey Refuge and Nonlocal Predation Effect

具有避难效应的捕食–食饵反应扩散模型是一类重要的种群动力学模型,其斑图形态为研究种群数量时空分布提供关键信息。建立了一个具有避难效应和非局部被捕食效应的Leslie-Gower捕食–食饵反应扩散模型。首先,通过线性稳定性分析和Turing失稳分析分别推导出正平衡点局部稳定及Turing失稳的条件。其次,通过数值模拟分别展示了食饵斑图形态随非局部被捕食效应和避难效应变化而演化的情况。最后,通过分析捕食者和食饵种群空间平均密度与非局部被捕食效应和避难效应参数的变化关系,研究表明:食饵种群数量随着非局部效应的增强而增大,捕食者种群数量随着非局部效应的增强而减少,说明非局部被捕食效应在促进食饵密度增长的同时,抑制捕食者种群的增长;捕食者和食饵种群的数量均随避难效应的增强而增大,说明较强避难效应对食饵种群具有庇护作用,并能促进捕食者和食饵种群持续共存。结论有助于人们更好地从数学层面理解非局部效应和避难效应对捕食者和食饵种群斑图形态影响的规律。

Predator-prey reaction-diffusion model with refuge effect is an important type of population dynamics model,and its pattern formation provides key information for studying the spatiotemporal distribution of population.A Leslie-Gower predator-prey reaction-diffusion model with both refuge and nonlocal predation effects is proposed.Firstly,the conditions for local stability and Turing instability of the positive equilibrium point are derived through linear stability analysis and Turing instability analysis,respectively.Secondly,the evolution of the patterns of the prey with nonlocal predation and refuge effects was demonstrated through numerical simulations.Finally,by analyzing the relationship between the spatial mean density of predator and prey populations and the changes in nonlocal prey and refuge effect parameters,it is shown that the spatial mean density of prey increases,while the number of predator decreases with the enhancement of nonlocal predation effects,indicating that nonlocal effects promote the growth of prey density while inhibiting the growth of predator populations;The spatial mean density of predator and prey increases simultaneously with the enhancement of refuge effect,indicating that strong refuge effect has a protective effect on prey and can promote the persistence and coexistence of predator and prey.The result helps people better understand the law of nonlocal effects and refuge effects on the pattern formation of predator and prey from a mathematical perspective.