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

  • Authors:

  • Lin-Lin Wang;Wan-Tong Li

  • Affiliations:

  • Department of Mathematics, Lanzhou University, Gansu, Lanzhou 730000, People's Republic of China;Department of Mathematics, Lanzhou University, Gansu, Lanzhou 730000, People's Republic of China

  • Venue:
  • Journal of Computational and Applied Mathematics
  • Year:
  • 2004

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Abstract

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.