Uniform persistence for a two-species ratio-dependent predator–prey system with diffusion and variable time delays

2011 ◽  
Vol 217 (10) ◽  
pp. 4900-4910 ◽  
Author(s):  
Dongshu Wang ◽  
Quanyi Wang
Keyword(s):  
Time Delays ◽  
Uniform Persistence ◽  
Predator Prey ◽  
Variable Time ◽  
Prey System ◽  
Ratio Dependent

scholarly journals STABILITY AND BIFURCATION IN A DELAYED RATIO-DEPENDENT PREDATOR–PREY SYSTEM

2002 ◽  
Vol 46 (1) ◽  
pp. 205-220 ◽  
Author(s):  
Dongmei Xiao ◽  
Wenxia Li
Keyword(s):  
Periodic Solutions ◽  
Time Delays ◽  
Analytical Study ◽  
Real Life ◽  
Primary 34C25 ◽  
Predator Prey ◽  
Interior Equilibrium ◽  
Prey System ◽  
Ratio Dependent ◽  
The Stability

AbstractRecently, ratio-dependent predator–prey systems have been regarded by some researchers as being more appropriate for predator–prey interactions where predation involves serious searching processes. Due to the fact that every population goes through some distinct life stages in real-life, one often introduces time delays in the variables being modelled. The presence of time delay often greatly complicates the analytical study of such models. In this paper, the qualitative behaviour of a class of ratio-dependent predator–prey systems with delay at the equilibrium in the interior of the first quadrant is studied. It is shown that the interior equilibrium cannot be absolutely stable and there exist non-trivial periodic solutions for the model. Moreover, by choosing delay $\tau$ as the bifurcation parameter we study the Hopf bifurcation and the stability of the periodic solutions.AMS 2000 Mathematics subject classification: Primary 34C25; 92D25. Secondary 58F14

scholarly journals Permanence of a Semi-Ratio-Dependent Predator-Prey System with Nonmonotonic Functional Response and Time Delay

2009 ◽  
Vol 2009 ◽  
pp. 1-6 ◽  
Author(s):  
Xuepeng Li ◽  
Wensheng Yang
Keyword(s):  
Time Delay ◽  
Functional Response ◽  
Negative Feedback ◽  
Time Delays ◽  
Sufficient Conditions ◽  
Prey Population ◽  
Predator Prey ◽  
Prey System ◽  
Ratio Dependent ◽  
Nonmonotonic Functional Response

Sufficient conditions for permanence of a semi-ratio-dependent predator-prey system with nonmonotonic functional response and time delay    are obtained, where and stand for the density of the prey and the predator, respectively, and is a constant. stands for the time delays due to negative feedback of the prey population.

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