The dynamics of an impulsive delay predator–prey model with variable coefficients

• Published in: Applied mathematics and computation 2008-04, Vol.198 (1), p.361-374
• Main Authors: Meng, Xin-zhu, Chen, Lan-sun, Li, Qing-xue
• Format: Article
• Language: English
• Subjects: Algebra; Algebraic geometry; Exact sciences and technology; Global analysis, analysis on manifolds; Global attractivity; Impulsive effects; Mathematical analysis; Mathematics; Numerical analysis; Numerical analysis. Scientific computation; Numerical linear algebra; Ordinary differential equations; Periodic solution; Permanence; Predator–prey model; Sciences and techniques of general use; Time delays; Topology. Manifolds and cell complexes. Global analysis and analysis on manifolds
• ISSN: 0096-3003; 1873-5649
• DOI: 10.1016/j.amc.2007.08.075

In this paper, we formulate a new robust two-species nonautonomous predator–prey model with multi-delays and impulsive effects and perform a systematic mathematical and ecological study. Our results in this paper indicate that under the appropriate linear periodic impulsive perturbations, the system is permanent and has a unique positive globally attractive semi-trivial periodic solution. By using the Brouwer fixed point theorem, we prove that if the periodic system is permanent, then there is at least one positive periodic solution of the system. We show that the conditions for global attractivity of the positive semi-trivial periodic solution and permanence of the population of the model depend on time delay, so, we call it “profitless”. In this paper, the main feature is that we introduce multi-delays and impulses into the predator–prey model, exhibit a new modeling method which is applied to investigate multi-species impulsive multi-delays differential equations.