Stability and Hopf bifurcation of a delayed ratio-dependent predator-prey system
W.Y. Wang1,L.J. Pei2
1. School of Aerospace Engineering and Applied Mechanics, Tongji University, 200092 Shanghai, China 2. Department of Mathematics, Zhengzhou University, 450001 Zhengzhou, China
Abstract Since the ratio-dependent theory reflects the fact that predators must share
and compete for food, it is suitable for describing the
relationship between predators and their preys and has recently
become a very important theory put forward by biologists. In order
to investigate the dynamical relationship between predators and
their preys, a so-called Michaelis-Menten ratio-dependent
predator-prey model is studied in this paper with gestation time
delays of predators and preys taken into consideration. The
stability of the positive equilibrium is investigated by the
Nyquist criteria, and the existence of the local Hopf bifurcation
is analyzed by employing the theory of Hopf bifurcation. By means
of the center manifold and the normal form theories, explicit
formulae are derived to determine the stability, direction and
other properties of bifurcating periodic solutions. The above
theoretical results are validated by numerical simulations with
the help of dynamical software WinPP. The results show that if
both the gestation delays are small enough, their sizes will keep
stable in the long run, but if the gestation delays of predators
are big enough, their sizes will periodically fluc tuate in the long term. In order to reveal the effects of time
delays on the ratio-dependent predator-prey model, a
ratio-dependent predator-prey model without time delays is
considered. By Hurwitz criteria, the local stability of positive
equilibrium of this model is investigated. The conditions under
which the positive equilibrium is locally asymptotically stable
are obtained. By comparing the results with those of the model
with time delays, it shows that the dynamical behaviors of
ratio-dependent predator-prey model with time delays are more
complicated. Under the same conditions, namely, with the same
parameters, the stability of positive equilibrium of
ratio-dependent predator-prey model would change due to the
introduction of gestation time delays for predators and preys.
Moreover, with the variation of time delays, the positive
equilibrium of the ratio-dependent predator-prey model subjects to
Hopf bifurcation.
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