具有恐惧诱导的群体防御和猎物庇护所的生态系统
Ecological system with fear induced group defence and prey refuge
AI总结：
本研究构建含恐惧效应、猎物群体防御及庇护所的猎物-捕食者模型，通过数学分析与数值模拟揭示恐惧水平等参数对系统稳定性、分岔及时空模式的调控作用，拓展了传统捕食模型的动力学认知。
摘要：
本研究探究了包含以下要素的空间与非空间猎物-捕食者相互作用模型的动力学特性：（i）纳入猎物出生率中的恐惧效应；（ii）猎物对捕食者的群体防御；（iii）猎物庇护所。我们对猎物和捕食者两个物种的灭绝与存续情形开展了全面的数学分析。为更深入探究系统的动力学特性，我们以恐惧水平、猎物出生率、种内竞争导致的猎物死亡率为分岔参数，开展了详尽的分岔分析。针对部分相关参数值集合，我们还探究了所有可能出现的双稳态动力学情形。\n我们的数值计算结果表明，高水平的恐惧可通过消除周期解的可能性，使猎物-捕食者系统趋于稳定。此外，与忽略恐惧成本、通常具有超临界Hopf分岔的传统猎物-捕食者模型不同，本模型的Hopf分岔为亚临界型。与普遍趋势相反，在猎物出生率较高时，捕食者物种会灭绝。我们还发现，随着猎物种内竞争水平显著提升，猎物和捕食者种群可在系统中共存，这与猎物物种通常会灭绝的典型趋势相悖。\n我们还通过解析方式研究了对应空间模型的稳定性与图灵不稳定性。同时开展了数值模拟，以观测不同参数对物种种群密度分布的影响。通过改变时间演化进程、捕食者种群扩散系数、恐惧因子水平及猎物庇护所参数，我们观测到了斑点、斑点-条纹混合等不同类型的时空模式。恐惧水平参数（k）对模型系统的空间动力学特性具有显著影响。
英文摘要：
In this study, we investigate the dynamics of a spatial and non spatial prey-predator interaction model that includes the following: (i) fear effect incorporated in prey birth rate; (ii) group defence of prey against predators; and (iii) prey refuge. We provide comprehensive mathematical analysis of extinction and persistence scenarios for both prey and predator species. To better explore the dynamics of the system, a thorough investigation of bifurcation analysis has been performed using fear level, prey birth rate, and prey death rate caused by intra-prey competition as bifurcation parameter. All potential occurrences of bi-stability dynamics have also been investigated for some relevant sets of parametric values. Our numerical evaluations show that high levels of fear can stabilize the prey-predator system by ruling out the possibility of periodic solutions. Also, our model Hopf bifurcation is subcritical in contrast to traditional prey-predator models, which ignore the cost of fear and have supercritical Hopf bifurcations in general. In contrast to the general trend, predator species go extinct at higher values of prey birth rates. We have also found that, contrary to the typical tendency for prey species to go extinct, both prey and predator populations may coexist in the system as intra-prey competition level grows noticeably. The stability and Turing instability of associated spatial model have also been investigated analytically. We also perform the numerical simulation to observe the effect of different parameters on the density distribution of species. Different types of spatiotemporal patterns like spot, mixture of spots and stripes have been observed via variation of time evolution, diffusion coefficient of predator population, level of fear factor and prey refuge. The fear level parameter (k) has a great impact on the spatial dynamics of model system.
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