Abstract Several field data and experiments on a terrestrial vertebrates exhibited that the fear of predators would cause a substantial variability of prey demography. Fear for predator population enhances the survival probability of prey population, and it can greatly reduce the reproduction of prey population. Based on the experimental evidence, we proposed and analyzed a prey-predator system introducing the cost of fear into prey reproduction with Holling type-II functional response. We investigate all the biologically feasible equilibrium points, and their stability is analyzed in terms of the model parameters. Our mathematical analysis exhibits that for strong anti-predator responses can stabilize the prey-predator interactions by ignoring the existence of periodic behaviors. Our model system undergoes Hopf bifurcation by considering the birth rate r0 as a bifurcation parameter. For larger prey birth rate, we investigate the transition to a stable coexisting equilibrium state, with oscillatory approach to this equilibrium state, indicating that the greatest characteristic eigenvalues are actually a pair of imaginary eigenvalues with real part negative, which is increasing for r0. We obtained the conditions for the occurrence of Hopf bifurcation and conditions governing the direction of Hopf bifurcation, which imply that the prey birth rate will not only influence the occurrence of Hopf bifurcation but also alter the direction of Hopf bifurcation. We identify the parameter regions associated with the extinct equilibria, predator-free equilibria and coexisting equilibria with respect to prey birth rate, predator mortality rates. Fear can stabilize the predator-prey system at an interior steady state, where all the species can exists together, or it can create the oscillatory coexistence of all the populations. We performed some numerical simulations to investigate the relationship between the effects of fear and other biologically related parameters (including growth/decay rate of prey/predator), which exhibit the impact that fear can have in prey-predator system. Our numerical illustrations also demonstrate that the prey become less sensitive to perceive the risk of predation with increasing prey growth rate or increasing predators decay rate.

中文翻译：

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捕食者-猎物相互作用模型中恐惧效应对猎物生长的影响

摘要 陆生脊椎动物的一些野外数据和实验表明，对捕食者的恐惧会导致猎物人口统计学的显着变异。对捕食者种群的恐惧提高了猎物种群的生存概率，可以大大减少猎物种群的繁殖。基于实验证据，我们提出并分析了一种将恐惧成本引入具有 Holling II 型功能反应的猎物繁殖的猎物-捕食者系统。我们研究了所有生物学上可行的平衡点，并根据模型参数分析了它们的稳定性。我们的数学分析表明，对于强烈的反捕食者反应，可以通过忽略周期性行为的存在来稳定猎物-捕食者的相互作用。我们的模型系统通过将出生率 r0 作为分叉参数进行 Hopf 分叉。对于较大的猎物出生率，我们研究了向稳定共存平衡状态的过渡，对该平衡状态采用振荡方法，表明最大特征值实际上是一对实部为负的虚特征值，随着 r0 增加。我们得到了发生Hopf分岔的条件和控制Hopf分岔方向的条件，这意味着猎物出生率不仅会影响Hopf分岔的发生，还会改变Hopf分岔的方向。我们确定了与灭绝均衡、无捕食者均衡和与猎物出生率相关的共存均衡相关的参数区域，捕食者死亡率。恐惧可以将捕食者-猎物系统稳定在内部稳定状态，所有物种可以一起存在，或者它可以创建所有种群的振荡共存。我们进行了一些数值模拟，以研究恐惧的影响与其他生物相关参数（包括猎物/捕食者的生长/衰减率）之间的关系，这些参数展示了恐惧对猎物-捕食者系统的影响。我们的数值插图还表明，随着猎物增长率的增加或捕食者衰减率的增加，猎物对感知捕食风险的敏感度降低。我们进行了一些数值模拟，以研究恐惧的影响与其他生物相关参数（包括猎物/捕食者的生长/衰减率）之间的关系，这些参数展示了恐惧对猎物-捕食者系统的影响。我们的数值插图还表明，随着猎物增长率的增加或捕食者衰减率的增加，猎物对感知捕食风险的敏感度降低。我们进行了一些数值模拟，以研究恐惧的影响与其他生物相关参数（包括猎物/捕食者的生长/衰减率）之间的关系，这些参数展示了恐惧对猎物-捕食者系统的影响。我们的数值插图还表明，随着猎物增长率的增加或捕食者衰减率的增加，猎物对感知捕食风险的敏感度降低。