A new predator-prey model with a profitless delay of digestion and impulsive perturbation on the prey

Abstract In this paper, we formulate a robust prey-dependent consumption predator–prey model with a delay of digestion and impulsive perturbation on the prey. Using the discrete dynamical system determined by the stroboscopic map, we obtain a ‘predator-eradication’ periodic solution and show that the ‘predator-eradication’ periodic solution is globally attractive when harvesting for the prey is over certain value. Using a new qualitative analysis method for impulsive and delay differential equations, we prove the system is uniformly persistent when harvesting for the prey is under certain value. Further, we show the delay of digestion is a “profitless” time delay. Moreover, we show our theoretical results by numerical simulation. In this paper, the main feature is that we introduce a delay of digestion and impulsive effects into the predator–prey model and exhibit a new mathematical method which is applied to investigate the system which is governed by both impulsive and delay differential equations.

[1]  Jianjun Jiao,et al.  A stage-structured Holling mass defence predator-prey model with impulsive perturbations on predators , 2007, Appl. Math. Comput..

[2]  R R Kao,et al.  The dynamics of an infectious disease in a population with birth pulses. , 1998, Mathematical biosciences.

[3]  Y. Kuang.  Delay Differential Equations: With Applications in Population Dynamics , 2012 .

[4]  Shujing Gao,et al.  Analysis of a delayed epidemic model with pulse vaccination and saturation incidence. , 2006, Vaccine.

[5]  Jianjun Jiao,et al.  A pest management SI model with periodic biological and chemical control concern , 2006, Appl. Math. Comput..

[6]  Lansun Chen,et al.  The periodic competing Lotka-Volterra model with impulsive effect. , 2004, Mathematical medicine and biology : a journal of the IMA.

[7]  Jurang Yan,et al.  Stability for impulsive delay differential equations , 2005 .

[8]  Elena Braverman,et al.  Linearized oscillation theory for a nonlinear delay impulsive equation , 2003 .

[9]  Lansun Chen,et al.  A NEW MATHEMATICAL MODEL FOR OPTIMAL CONTROL STRATEGIES OF INTEGRATED PEST MANAGEMENT , 2007 .

[10]  Lansun Chen,et al.  Two profitless delays for the SEIRS epidemic disease model with nonlinear incidence and pulse vaccination , 2007, Appl. Math. Comput..

[11]  Bing Liu,et al.  The dynamical behaviors of a Lotka–Volterra predator–prey model concerning integrated pest management ☆ , 2005 .

[12]  Lansun Chen,et al.  The dynamics of an impulsive delay predator-prey model with variable coefficients , 2008, Appl. Math. Comput..

[13]  D. Baĭnov,et al.  Systems with impulse effect : stability, theory, and applications , 1989 .

[14]  V. Lakshmikantham,et al.  Theory of Impulsive Differential Equations , 1989, Series in Modern Applied Mathematics.

[15]  Xinzhi Liu,et al.  Boundedness for impulsive delay differential equations and applications to population growth models , 2003 .

[16]  H. I. Freedman,et al.  A time-delay model of single-species growth with stage structure. , 1990, Mathematical biosciences.

[17]  Xinyu Song,et al.  The Stage-structured Predator-Prey System with Delay and Harvesting , 2002 .

[18]  Xinzhu Meng,et al.  Almost periodic solution of non-autonomous Lotka-Volterra predator-prey dispersal system with delays. , 2006, Journal of theoretical biology.

[19]  Alberto d’Onofrio,et al.  Pulse vaccination strategy in the sir epidemic model: Global asymptotic stable eradication in presence of vaccine failures , 2002 .

[20]  Deming Zhu,et al.  Dynamic complexities for prey-dependent consumption integrated pest management models with impulsive effects , 2006 .

[21]  Y. Kuang,et al.  A Stage Structured Predator-Prey model and its Dependence on Through-Stage Delay and Death Rate , 2003 .

[22]  Wan-Tong Li,et al.  Existence of positive periodic solution of a neutral impulsive delay predator-prey system , 2007, Appl. Math. Comput..

[23]  Guilie Luo,et al.  Extinction and permanence in competitive stage structured system with time-delays , 2002 .

[24]  Yanni Xiao,et al.  Dynamical behavior for a stage-structured SIR infectious disease model , 2002 .