Global stability of a model for competing predators: An extension of the Ardito \& Ricciardi Lyapunov function
It is considered the following model for competing predators \[ \dot{s}=h(s)-x_{0}f_{0}(s)- x_{1}f_{1}(s),\quad \dot{x}_{0}={x}_{0}\psi_{0}(s),\quad \dot{x}_{1}={x}_{1}\psi_{1}(s). \tag{1} \] Here, \(s\) denotes the prey density, \(x_{0}\) and \(x_{1}\) are the densities of the specialist predators feeding on the same prey \(s.\) The model (1) is analyzed under the following general conditions: (A1) The functions \(h,f_{0},f_{1},\psi_{0},\psi_{1}\) are \(C^{1}[0,\infty).\) (A2) There exists a constant \(K, K>0,\) such that \(h\) satisfies \( h(s)>0 \) if \( 0<s<K \) and \( h(s)<0 \) if \( s<0 \) or \( s>K.\) (A3) The functions \(f_{0}\) and \(f_{1}\) are increasing and have unique zeros at \(s=0.\) (A4) The functions \(\psi_{0}\) and \(\psi_{1}\) are increasing and there exist \(\lambda_{i}\) such that \(\psi_{i}\) satisfies \(\psi_{i} <0 \) if \( 0<s<\lambda_{i} \) and \(\psi_{i}(s) > 0 \) if \( s>\lambda_{i} \) for \(i=0,1.\) Moreover, it is assumed that \(\lambda_{0}<\lambda_{1}<K.\) Let \(F_{0}=h(s)/f_{0}(s)\), \(F_{1}=h(s)/f_{1}(s),\) \((\lambda_{0},F_{0}(\lambda_{0}),0)\) is one of the four equilibria of system (1), \(\overline{F}_{0}(s)=F_{0}(\lambda_{0}) \exp(-\gamma) \int_{\lambda_{0}} ^{s}\psi_{0}(t)/f_{0}(t) dt,\) and \[ P(s)= \left\{ \begin{aligned} \alpha\psi_1(s)f_0(s)-\beta\psi_0(s)f_1(s)+ \alpha\gamma h(s)\psi_0(s),\quad s>\lambda_0,\\ \alpha\psi_1(s)f_0(s)-\beta\psi_0(s)f_1(s),\quad s\leq\lambda_0. \end{aligned} \right. \] The following theorem is the main result of this paper: Let (A1)--(A4) hold and assume there exist \(\alpha>0, \beta>0, \gamma\geq 0 \) such that \(({F}_{0}(s)-\overline{F}_{0}(s))(s-\lambda_0)<0, s\neq\lambda_0, P(s)<0 \) holds for \( 0 < s <K. \) Then the equilibrium \((\lambda_{0},F_{0}(\lambda_{0}),0)\) is globally asymptotically stable, except for initial conditions in the \((s,x_1)\)-coordinate plane and the \((x_0,x_1)\)-coordinate plane.
scientific article; zbMATH DE number 1419548