scholarly journals Periodic solutions for a semi-ratio-dependent predator–prey system with functional responses

2005 ◽  
Vol 18 (3) ◽  
pp. 313-320 ◽  
Author(s):  
Hai-Feng Huo
Keyword(s):  
Periodic Solutions ◽  
Functional Responses ◽  
Predator Prey ◽  
Prey System ◽  
Ratio Dependent

scholarly journals Periodic Solutions for a Semi-Ratio-Dependent Predator-Prey System with Delays on Time Scales

2012 ◽  
Vol 2012 ◽  
pp. 1-15 ◽  
Author(s):  
Xiaoquan Ding ◽  
Gaifang Zhao
Keyword(s):  
Periodic Solutions ◽  
Time Scales ◽  
Sufficient Conditions ◽  
Coincidence Degree ◽  
Functional Responses ◽  
Predator Prey ◽  
Prey System ◽  
Special Cases ◽  
Existence Of Periodic Solutions ◽  
Ratio Dependent

This paper is devoted to the existence of periodic solutions for a semi-ratio-dependent predator-prey system with time delays on time scales. With the help of a continuation theorem based on coincidence degree theory, we establish necessary and sufficient conditions for the existence of periodic solutions. Our results show that for the most monotonic prey growth such as the logistic, the Gilpin, and the Smith growth, and the most celebrated functional responses such as the Holling type, the sigmoidal type, the Ivlev type, the Monod-Haldane type, and the Beddington-DeAngelis type, the system always has at least one periodic solution. Some known results are shown to be special cases of the present paper.

scholarly journals Periodic solutions and stability for a delayed discrete ratio-dependent predator-prey system with Holling-type functional response

2004 ◽  
Vol 2004 (2) ◽  
pp. 325-343 ◽  
Author(s):  
Lin-Lin Wang ◽  
Wan-Tong Li
Keyword(s):  
Periodic Solutions ◽  
Functional Response ◽  
Sufficient Conditions ◽  
Coincidence Degree ◽  
Degree Theory ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey System ◽  
Globally Asymptotical Stability ◽  
Ratio Dependent

The existence of positive periodic solutions for a delayed discrete predator-prey model with Holling-type-III functional responseN1(k+1)=N1(k)exp{b1(k)−a1(k)N1(k−[τ1])−α1(k)N1(k)N2(k)/(N12(k)+m2N22(k))},N2(k+1)=N2(k)exp{−b2(k)+α2(k)N12(k−[τ2])/(N12(k−[τ2])+m2N22(k−[τ2]))}is established by using the coincidence degree theory. We also present sufficient conditions for the globally asymptotical stability of this system when all the delays are zero. Our investigation gives an affirmative exemplum for the claim that the ratio-dependent predator-prey theory is more reasonable than the traditional prey-dependent predator-prey theory.

EXISTENCE FOR PERIODIC SOLUTIONS OF A RATIO-DEPENDENT PREDATOR-PREY SYSTEM WITH TIME-VARYING DELAYS ON TIME SCALES

2010 ◽  
Vol 08 (03) ◽  
pp. 227-233
Author(s):  
HUIJUAN LI ◽  
ANPING LIU ◽  
ZUTAO HAO
Keyword(s):  
Periodic Solution ◽  
Periodic Solutions ◽  
Functional Response ◽  
Time Scales ◽  
Coincidence Degree ◽  
Degree Theory ◽  
Time Varying ◽  
Predator Prey ◽  
Prey System ◽  
Ratio Dependent

In this paper, by using the continuation theorem of coincidence degree theory we study the existence of periodic solution for a two-species ratio-dependent predator-prey system with time-varying delays and Machaelis–Menten type functional response on time scales. Some new results are obtained.

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

Sign in / Sign up

Export Citation Format

Share Document