Abstract
In this paper, the effect of the two different delays on the dynamics of a three-species ratio-dependent predator-prey food-chain model is considered. By regarding the delay as the bifurcation parameter, the local stability of the positive equilibrium and the existence of Hopf bifurcation are investigated. Explicit formulas determining the properties of a Hopf bifurcation are obtained by using the normal form method and the center manifold theorem. Special attention is paid to the global continuation of local Hopf bifurcation when the delay $\tau _{1}\neq\tau_{2}$ . Finally, several numerical simulations supporting the theoretical analysis are also given.
Highlights
There has been great interest in dynamical characteristics of population models during the last few decades, among these models, predator-prey systems play an important role in population dynamics
It is an important subject to investigate if these nonconstant periodic solutions which are obtained through local Hopf bifurcations exist globally due to theoretical and practical significance
Let n = and make use of sufficient conditions of the global stability of the positive equilibrium of system ( . ) in [ ], we investigate the Hopf bifurcation and global periodic solutions of three-species ratio-dependent predator-prey model with two delays:
Summary
There has been great interest in dynamical characteristics of population models during the last few decades, among these models, predator-prey systems play an important role in population dynamics. Arditi and Ginzburg [ ] proposed the following ratio-dependent predator-prey model with a Michaelis-Menten-type or Holling-type II functional response: x ax(. The ratio-dependent predator-prey models with time delays have been studied by many researchers recently and rich dynamics has been observed (see, for example, [ – ] and references cited therein). It is an important subject to investigate if these nonconstant periodic solutions which are obtained through local Hopf bifurcations exist globally due to theoretical and practical significance. ) in [ ], we investigate the Hopf bifurcation and global periodic solutions of three-species ratio-dependent predator-prey model with two delays:. ), and it is a mathematical subject to investigate whether the nontrivial periodic solutions which are obtained through local Hopf bifurcations exist globally.
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