Pontificia Universidad Católica de Valparaíso, Chile.
Departamento de Matemáticas, Física y Estadística, Facultad de Ciencias Básicas, Universidad Católica del Maule, Talca, Chile.
Math Biosci Eng. 2020 Nov 5;17(6):7708-7731. doi: 10.3934/mbe.2020392.
In the ecological literature, mutual interference (self-interference) or competition among predators (CAP) to effect the harvesting of their prey has been modeled through different mathematical formulations. In this work, the dynamical properties of a Leslie-Gower type predation model is analyzed, incorporating one of these forms, which is described by the function $g\left(y\right) =y^{\beta }$, with $0<\beta <1$. This function $g$ is not differentiable for $y=0$, and neither the Jacobian matrix of the system is not defined in the equilibrium points over the horizontal axis ($x-axis$). To determine the nature of these points, we had to use a non-standard methodology. Previously, we have shown the fundamental properties of the Leslie-Gower type model with generalist predators, to carry out an adequate comparative analysis with the model where the competition among predators (CAP) is incorporated. The main obtained outcomes in both systems are: (i) The unique positive equilibrium point, when exists, is globally asymptotically stable (g.a.s), which is proven using a suitable Lyapunov function. (ii) There not exist periodic orbits, which was proved constructing an adequate Dulac function.
在生态文献中，通过不同的数学公式来模拟捕食者之间的相互干扰（自干扰）或竞争（CAP）对其猎物的捕食效果。在这项工作中，分析了包含其中一种形式的 Leslie-Gower 型捕食模型的动态特性，该形式由函数$g\left(y\right) =y^{\beta }$描述，其中$0<\beta <1$。对于$y=0$，这个函数$g$不可微，并且系统的雅可比矩阵在水平轴（$x$轴）上的平衡点处也没有定义。为了确定这些点的性质，我们必须使用非标准方法。之前，我们已经展示了具有广义捕食者的 Leslie-Gower 型模型的基本性质，以便与包含捕食者之间竞争（CAP）的模型进行适当的比较分析。在这两个系统中主要得到的结果是：（i）当存在唯一的正平衡点时，它是全局渐近稳定的（g.a.s），这是通过使用合适的李雅普诺夫函数证明的。（ii）不存在周期轨道，这是通过构建适当的 Dulac 函数证明的。
