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The asymptotic behaviors of a stage-structured autonomous predator-prey system with time delay. (English) Zbl 1035.34046

Here, the authors investigate the stage-structured autonomous predator-prey Lotka-Volterra system with time delay:

x ˙ 1 =b 1 e -d 1 τ 1 x 1 (t-τ 1 )D 1 x 1 2 (t)+kΘx 1 (t)y 1 (t),x ˙ 2 =b 1 x 1 (t)-d 1 x 2 (t)-b 1 e -d 1 τ 1 x 1 (t-τ 1 ),y ˙ 1 =b 2 e -d 2 τ 2 y 1 (t-τ 2 )D 2 y 1 2 (t)+kΘy 1 (t)y 1 (t),y ˙ 2 =b 2 y 1 (t)-d 2 y 2 (t)-b 2 e -d 2 τ 2 y 1 (t-τ 2 ),t0,-τ i t0,i=1,2,

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. τ 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.

MSC:
34C60Qualitative investigation and simulation of models (ODE)
34D05Asymptotic stability of ODE
92D25Population dynamics (general)