Stability and bifurcation in a ratio-dependent Holling-III system with diffusion and delay
Wenjie Zuo
Harbin Institute of Technology; China University of Petroleum
Junjie Wei
China University of Petroleum
Published 2014-01-20


Hopf bifurcation
Holling-type III
global stability
uniformly permanent

How to Cite

Zuo W. and Wei J. (2014) “Stability and bifurcation in a ratio-dependent Holling-III system with diffusion and delay”, Nonlinear Analysis: Modelling and Control, 19(1), pp. 132-153. doi: 10.15388/NA.2014.1.9.


A diffusive ratio-dependent predator-prey system with Holling-III functional response and delay effects is considered. Global stability of the boundary equilibrium and the stability of the unique positive steady state and the existence of spatially homogeneous and inhomogeneous periodic solutions are investigated in detail, by the maximum principle and the characteristic equations. Ratio-dependent functional response exhibits rich spatiotemporal patterns. It is found that, the system without delay is dissipative and uniformly permanent under certain conditions, the delay can destabilize the positive constant equilibrium and spatial Hopf bifurcations occur as the delay crosses through some critical values. Then, the direction and the stability of Hopf bifurcations are determined by applying the center manifold reduction and the normal form theory for partial functional differential equations. Some numerical simulations are carried out to illustrate the theoretical results.

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