Xin Jiang, Zhikun She, Zhaosheng Feng
Abstract:
In this article, we study a density-dependent predator-prey
system with the Beddington-DeAngelis functional response for stability
and Hopf bifurcation under certain parametric conditions.
We start with the condition of the existence of the unique
positive equilibrium, and provide two sufficient conditions
for its local stability by the Lyapunov function method and the Routh-Hurwitz
criterion, respectively. Then, we establish sufficient conditions for
the global stability of the positive equilibrium by proving the
non-existence of closed orbits in the first quadrant
.
Afterwards, we analyze the Hopf bifurcation geometrically by
exploring the monotonic property of the trace of the Jacobean matrix with
respect to
and analytically verifying that there is a unique
such that the trace is equal to 0.
We also introduce an auxiliary map by restricting all the five parameters
to a special one-dimensional geometrical structure and analyze the Hopf
bifurcation with respect to all these five parameters. Finally, some
numerical simulations are illustrated which are in agreement with our
analytical results.
Submitted March 6, 2016. Published September 21, 2016.
Math Subject Classifications: 34D20, 34E05, 37G15.
Key Words: Density-dependent; local and global stability; Hopf bifurcation;
monotonicity; geometrical restriction.
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Xin Jiang School of Mathematics and Systems Science Beihang University Beijing 100191, China email: jiangxin1991@126.com | |
Zhikun She School of Mathematics and Systems Science Beihang University Beijing 100191, China email: zhikun.she@buaa.edu.cn | |
Zhaosheng Feng School of Mathematical and Statistical Sciences University of Texas-Rio Grande Valley Edinburg, TX 78539, USA email: zhaosheng.feng@utrgv.edu |
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