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Local dynamics of a predator–prey community in a moderate period of time

机译：局部动态的捕食者-猎物社区适度的时间

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In this work, we have introduced an ecological model of a prey–predator system. It is assumed that the prey species grows logistically, but the total number of predator is constant in the time interval. Positivity and boundedness of the solution ensure that the proposed model is well-posed. Local stability conditions of the equilibrium points have been analysed by the Routh–Hurwitz criterion. The persistence of the system has also been shown under a parametric restriction. Numerical analysis has indicated that both axial and interior steady states can exist only for moderate consumption rate (searching efficiency). But if this rate becomes high (or low), then only the prey-free equilibrium (or one of the interior equilibriums) exists as a steady state. Further, the equilibrium points can change their stability through transcritical and saddle-node bifurcations by varying the consumption rate of the predator. Analytical results provide an interesting phenomenon about this model: the system can never show any oscillating behaviour for any parametric values, i.e. no limit cycle can occur through Hopf bifurcation around an equilibrium point. The axial equilibrium becomes stable from an unstable situation when the consumption rate becomes high and the interior state which is stable remains stable as time goes by.
机译：在这项工作中,我们引入了生态捕食者-食饵系统的模型。猎物物种生长在逻辑上,但是捕食者的总数是不变的时间间隔。确保模型的解决方案适定的。平衡的点进行了分析Routh-Hurwitz标准。系统也被证明在一个参数限制。轴向和内部稳定的状态只存在适度消费(搜索效率)。高(或低),那么只有prey-free平衡(或内部均衡的)是一种稳定状态。平衡点可以改变他们的稳定性通过超临界和界定分岔消费率的变化捕食者。关于这个模型有趣的现象:系统无法显示任何振荡行为对于任何一个参数值,即无限循环可以通过霍普夫分岔发生在吗平衡点。稳定的从一个不稳定的情况率就高,内部消费状态是稳定的保持稳定,随着时间的推移通过。

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