## 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.

### MSC:

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

 [1] DOI: 10.1016/0025-5564(90)90019-U · Zbl 0719.92017 [2] DOI: 10.1137/0152048 · Zbl 0760.92018 [3] DOI: 10.1137/S0036141000376086 · Zbl 1013.92034 [4] DOI: 10.1016/S0898-1221(99)00316-8 · Zbl 0968.92018 [5] Gourley S. A., J. Math. Biol. 49 pp 188– [6] Kuang Y., Delay Differential Equations with Applications in Population Dynamics (1993) · Zbl 0777.34002 [7] DOI: 10.1007/BF03167566 · Zbl 0758.34065 [8] DOI: 10.1016/S0092-8240(03)00008-9 · Zbl 1334.92349 [9] DOI: 10.1137/S0036139993252839 · Zbl 0847.34076 [10] DOI: 10.1016/S0895-7177(02)00279-0 · Zbl 1077.92516 [11] DOI: 10.1016/S0025-5564(00)00068-7 · Zbl 1028.34049 [12] Song X., Acta Math. Appl. Sinica 18 pp 307– [13] DOI: 10.1016/S0895-7177(02)00104-8 · Zbl 1024.92015 [14] DOI: 10.1016/j.amc.2003.11.008 · Zbl 1056.92063
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