A nonautonomous epidemic prey-predator system with impulsive effects and multiple delays is considered; further, there is an epidemic disease in the predator. By the mean-value theorem of multiple variables, integral inequalities, differential inequalities, and other mathematical analysis skills, sufficient conditions which guarantee the permanence of the system are obtained. Furthermore, by constructing a series of Lyapunov functionals it is proved that there exists a unique uniformly asymptotically stable almost periodic solution of the system.
We investigate the dynamics of a food-chain system with digest delay and periodic harvesting for the prey. By using the comparison theorem, small amplitude skills in the impulsive differential equation, and a special qualitative analysis method in the delay differential equation, we prove that there exists a predators-eradication periodic solution which is globally attractive and show that the pest population can be controlled under the economic threshold level and the system can be uniformly permanent when the harvest periodTis long enough or the harvesting rateδis not too large. Furthermore, we perform a series of numerical simulations to display the effects of the digest delay and periodic harvesting on the dynamic behavior of the food-chain system.
A nonautonomous food-chain system with Holling II functional response is studied, in which multiple delays of digestion are also considered. By applying techniques in differential inequalities, comparison theorem in ordinary differential equations, impulsive differential equations, and functional differential equations, some effective control strategies are obtained for the permanence of the system. Furthermore, effects of some important coefficients and delays on the permanence of the system are intuitively and clearly shown by series of numerical examples. − 3 ( ) ( ) ( ) 3 ( ) + ( ) , = ( ) ( 2 ( ) 2 ( ) ( − 5 ) 2 ( ) + ( − 5 ) + 3 ( ) 3 ( ) ( − 6 ) 3 ( ) + ( − 6 ) − ( ) ( − 3 )) , ̸ = , ( + ) = (1 + 1 ) ( ) , Hindawi Publishing Corporation
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