Pontificia Universidad Católica de Valparaíso, Chile.
Facultad de Educación, Universidad de Las Américas, Chile.
Math Biosci Eng. 2019 Sep 3;16(6):7995-8024. doi: 10.3934/mbe.2019403.
In the ecological literature,many models for the predator-prey interactions have been well formulated but partially analyzed.Assuming this analysis to be true and complete,some authors use that results to study a more complex relationship among species (food webs).Others employ more sophisticated mathematical tools for the analysis,without further questioning.The aim of this paper is to extend,complement and enhance the results established in an earlier article referred to a modified Leslie-Gower model.In that work,the authors proved only the boundedness of solutions,the existence of an attracting set,and the global stability of a single equilibrium point at the interior of the first quadrant.In this paper,new results for the same model are proven,establishing conditions in the parameter space for which up two positive equilibria exist.Assuming there exists a unique positive equilibrium point,we have proved,the existence of:i) a separatrix curve Σ,dividing the trajectories in the phase plane,which can have different ,ii) a subset of the parameter space in which two concentric limit cycles exist,the innermost unstable and the outermost stable.Then,there exists the phenomenon of tri-stability,because simultaneously,it has:a local stable positive equilibrium point, a stable limit cycle,and an attractor equilibrium point over the vertical axis.Therefore,we warn the model studied have more rich and interesting properties that those shown that earlier papers.Numerical simulations and a bifurcation diagram are given to endorse the analytical results.
在生态文献中，已经很好地制定了许多用于捕食者-猎物相互作用的模型，但部分分析。假设这种分析是正确和完整的，一些作者使用这些结果来研究物种之间更复杂的关系（食物网）。其他人则使用更复杂的数学工具进行分析，而不进一步质疑。本文的目的是扩展、补充和增强在一篇早期文章中建立的关于改进的 Leslie-Gower 模型的结果。在该工作中，作者仅证明了解的有界性、吸引集的存在性以及第一象限内部单个平衡点的全局稳定性。在本文中，为同一模型证明了新的结果，确定了在参数空间中存在两个正平衡点的条件。假设存在唯一的正平衡点，我们已经证明了存在：i）一条分隔曲线 Σ，将相平面中的轨迹分开，Σ 可能具有不同的形状，ii）参数空间的一个子集，其中存在两个同心极限环，最内不稳定，最外稳定。然后，存在三稳定性现象，因为同时存在：局部稳定的正平衡点、稳定的极限环和垂直轴上的吸引平衡点。因此，我们警告所研究的模型具有比早期论文中显示的更丰富和有趣的性质。数值模拟和分岔图给出了对分析结果的支持。
