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Zbl 1035.34046
Ou, Liuman; Luo, Guilie; Jiang, Youlin; Li, Yiping
The asymptotic behaviors of a stage-structured autonomous predator-prey system with time delay. (English)
[J] J. Math. Anal. Appl. 283, No. 2, 534-548 (2003). ISSN 0022-247X

Here, the authors investigate the stage-structured autonomous predator-prey Lotka-Volterra system with time delay: $$\align & \dot x_1 =b_1e^{-d_1\tau_1}x_1(t-\tau_1) D_1{x_1}^2(t) + k\Theta x_1(t)y_1(t),\\ & \dot x_2 = b_1x_1(t)-d_1x_2(t)- b_1e^{-d_1\tau_1}x_1(t-\tau_1),\\ & \dot y_1 = b_2e^{-d_2\tau_2}y_1(t-\tau_2) D_2{y_1}^2(t) + k\Theta y_1(t)y_1(t), \\ & \dot y_2 = b_2y_1(t)-d_2y_2(t)- b_2e^{-d_2\tau_2}y_1(t-\tau_2),\ t\geq 0,\ -\tau_i \leq t \leq 0,\ i=1,2,\endalign$$ where $x_1(t)$ and $x_2(t)$ represent the densities of mature and immature of predator species, respectively, while $y_1(t)$ and $y_2(t)$ represent the densities of mature and immature of prey species, respectively. $\tau_i$ denotes the length of time from the birth to maturity of ith species. The basic properties of the model investigated are the boundedness of positive solutions to the system. Further, the authors obtain some conditions for the global asymptotic stability of the unique positive equilibrium point. Moreover, in the system the prey population get extinction and the predator population get permanence are investigated. Finally, the authors present a theorem extending corresponding conditions when there are no two stage structures.
[Chen Lan Sun (Beijing)]

MSC 2000:: *34C60 Applications of qualitative theory of ODE
34D05 Asymptotic stability of ODE
92D25 Population dynamics

Keywords: Predator-prey system; stage-structure; global asymptotically stable; ascendancy

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