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

• Published in: Applied mathematics and computation Vol. 217; no. 22; pp. 9198 - 9208
• Main Authors: Cheng, Huidong, Zhang, Tongqian
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier Inc 15.07.2011; Elsevier
• Subjects: Delay; Delay of digestion; Differential equations; Digestion; Dynamical systems; Exact sciences and technology; Global analysis, analysis on manifolds; Global attractivity; Harvesting; Impulsive effect; Mathematical analysis; Mathematical models; Mathematics; Nonlinear algebraic and transcendental equations; Numerical analysis; Numerical analysis. Scientific computation; Ordinary differential equations; Permanence; Perturbation methods; Predator–prey model; Sciences and techniques of general use; Time delay; Topology. Manifolds and cell complexes. Global analysis and analysis on manifolds; Global attractivity; Permanence; Impulsive effect; Predator–prey model; Delay of digestion; Transcendental equation; Differential equation; Differential system; Impulsive equation; Mathematical method; Discrete system; Dynamical system; Periodic solution; Delay equation; Equation system; Non linear equation; Numerical analysis; Applied mathematics; Algebraic equation; Numerical simulation; Predator-prey model
• ISSN: 0096-3003; 1873-5649
• DOI: 10.1016/j.amc.2011.03.159

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.