Periodic solutions and permanence for a delayed nonautonomous ratio-dependent predator-prey model with Holling type functional response

By Linlin Wang, Wan-Tong Li. 2004

76 citations recorded by the source corpus.

By using the continuation theorem of coincidence degree theory, the existence of positive periodic solutions for a delayed ratio-dependent predator-prey model with Holling type III functional response x'(t) = x(t)[a(t) - b(t) ∫t-∞ k(t - s)x(s) ds] - c(t)x2(t)y(t)/m2y2(t) + x2(t), y'(t) = y(t) [e(t)x2(t - τ)/m2y2(t - τ) + x2(t - τ) - d(t)], is established, where a(t), b(t), c(t), e(t) and d(t) are all positive periodic continuous functions with period ω > 0, m > 0 and k(s) is a measurable function with period ω, τ is a nonnegative constant. The permanence of the system is also considered. In particular, if k(s) : δ0(s), where δ0(s) is the Dirac delta function at s = 0, our results show that the permanence of the above system is equivalent to the existence of positive periodic solution.

Subfields: Applied Mathematics, Geometry and Topology, Public Health, Environmental and Occupational Health

View Paper: https://www.semanticscholar.org/paper/94b5299c0485efeea91ad5278ff0b99cb3babca4

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Citations in the corpus

Permanence for a delayed discrete ratio-dependent predator–prey system with Holling type functional response☆ (2004, methodology): “We refer to the monographs [10,16,21,24] for general delayed biological systems and to [18,22,23,25–27,30,31] for investigation on predator– prey systems.”
Permanence and Periodic Solution of Predator-Prey System with Holling Type Functional Response and Impulses (2007, background): “In [14], Wang and Li investigated the global existence of positive periodic solutions and permanent property of the ratio-dependent predator-prey system with Holling type III functional response and delay, which takes the form”
Existence and global attractivity of positive periodic solutions for impulsive predator–prey model with dispersion and time delays (2009, background): “In recent years, applications of theory of differential equations in mathematical ecology have developed rapidly, various mathematical models have been proposed in the study of population dynamics [2–8].”
Global asymptotic stability of a ratio-dependent predator-prey system with diffusion (2006, methodology): “We refer to the monographs [7,14,18,23] for general delayed biological systems and to [16,21,22,24,25,27,28] for investigations on predator–prey systems.”