Global stability of a stage-structured predator-prey system. (English) Zbl 1173.34043

The authors consider the following two prey-predator systems with stage structure: \[ \begin{cases} \dot{x}(t)=rx(t)[1-x(t)/K]-ax(t)y(t),\\ \dot{y}(t)=be^{-\gamma \tau}x(t-\tau)y(t-\tau)-dy(t)-cy^2(t),\\ \dot{y_j}(t)=bx(t)y(t)-be^{-\gamma \tau}x(t-\tau)y(t-\tau)-\gamma y_j(t),\\ x(\theta),y(\theta),y_j(\theta)\geq 0\quad \text{is continuous on }-\tau\leq \theta <0,\\ \text{and }x(0),y(0),y_j(0)>0;\end{cases}\tag{1} \]
\[ \begin{cases}\dot{x}_j(t)=b_1x(t)-\gamma_1x_j(t)-b_1e^{-\gamma_1\tau_1}x(t-\tau_1),\\ \dot{x}(t)=b_1e^{-\gamma_1\tau_1}x(t-\tau_1)-c_1x^2(t)-ax(t)y(t),\\ \dot{y}(t)=be^{-\gamma \tau}x(t-\tau)y(t-\tau)-dy(t)-cy^2(t),\\ \dot{y}_j(t)=bx(t)y(t)-be^{-\gamma \tau}x(t-\tau)y(t-\tau)-\gamma y_j(t),\\ x(\theta),x_j(\theta),y(\theta),y_j(\theta)\geq 0\;\text{is continuous on }-\tau_2\leq \theta <0,\\ \tau_2=\max\{\tau_1,\tau\}, \text{ and }x(0),x_j(0),y(0),y_j(0)>0.\end{cases}\tag{2} \] Here \(y(t), y_j(t)\) represent the densities of immature and mature individual predators at time \(t\), and \(x(t), x_j(t)\) represent the densities of immature and mature individual preys at time \(t\). They discuss the existence of equilibria. By using the eigenvalue method, the local stability of each equilibrium is discussed. Furthermore, the global stability of each nonnegative equilibrium is also investigated. Numerical simulation suggests that time delay has both oscillatory dynamics and stabilizing effects.


34K20 Stability theory of functional-differential equations
92D25 Population dynamics (general)
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