scholarly journals Double Periodic Solutions for a Ratio-Dependent Predator-Prey System with Harvesting Terms on Time Scales

2009 ◽  
Vol 2009 ◽  
pp. 1-12 ◽  
Author(s):  
Zhenjie Liu
Keyword(s):  
Periodic Solutions ◽  
Time Scales ◽  
Difference Equations ◽  
Coincidence Degree ◽  
Degree Theory ◽  
Discrete Models ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey System ◽  
Ratio Dependent

We investigate a nonautonomous ratio-dependent predator-prey model with Beddington-DeAngelis functional response and multiple harvesting (or exploited) terms on time scales. By means of using a continuation theorem based on coincidence degree theory, we obtain sufficient criteria for the existence of at least two periodic solutions for the system. Moreover, when the time scale𝕋is chosen asℝorℤ, the existence of the periodic solutions of the corresponding continuous and discrete models follows. Therefore, the methods are unified to provide the existence of the desired solutions for the continuous differential equations and discrete difference equations.

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 Periodic Solutions for Gause-Type Ratio-Dependent Predator-Prey Systems with Delays on Time Scales

2013 ◽  
Vol 2013 ◽  
pp. 1-8
Author(s):  
Xiaoquan Ding ◽  
Hongyuan Liu ◽  
Fengye Wang
Keyword(s):  
Periodic Solutions ◽  
Time Scales ◽  
Coincidence Degree ◽  
Degree Theory ◽  
Continuation Theorem ◽  
Coincidence Degree Theory ◽  
Predator Prey ◽  
Existence Of Periodic Solutions ◽  
Ratio Dependent

This paper is devoted to periodic Gause-type ratio-dependent predator-prey systems with monotonic or nonmonotonic numerical responses on time scales. By using a continuation theorem based on coincidence degree theory, we establish easily verifiable criteria for the existence of periodic solutions. In particular, our results improve and generalize some known ones.

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 Positive Periodic Solutions for Neutral Delay Ratio-Dependent Predator-Prey Model with Holling-Tanner Functional Response

2011 ◽  
Vol 2011 ◽  
pp. 1-14
Author(s):  
Guirong Liu ◽  
Sanhu Wang ◽  
Jurang Yan
Keyword(s):  
Periodic Solutions ◽  
Functional Response ◽  
Coincidence Degree ◽  
Degree Theory ◽  
Positive Periodic Solutions ◽  
Neutral Delay ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
Ratio Dependent

By using a continuation theorem based on coincidence degree theory, we establish some easily verifiable criteria for the existence of positive periodic solutions for neutral delay ratio-dependent predator-prey model with Holling-Tanner functional responsex'(t)=x(t)[r(t)-a(t)x(t-σ(t))-b(t)x'(t-σ(t))]-c(t)x(t)y(t)/h(t)y(t)+x(t),y'(t)=y(t)d(t)-f(t)y(t-τ(t))/x(t-τ(t)).

scholarly journals Multiple Positive Periodic Solutions for a Gilpin-Ayala Competition Predator-Prey System with Harvesting Terms

2012 ◽  
Vol 2012 ◽  
pp. 1-13
Author(s):  
Yongzhi Liao ◽  
Yongkun Li ◽  
Xiaoyan Dou
Keyword(s):  
Periodic Solutions ◽  
Sufficient Conditions ◽  
Coincidence Degree ◽  
Degree Theory ◽  
Continuation Theorem ◽  
Coincidence Degree Theory ◽  
Positive Periodic Solutions ◽  
Mawhin’S Continuation Theorem ◽  
Predator Prey ◽  
Prey System

By applying Mawhin’s continuation theorem of coincidence degree theory, we study the existence of multiple positive periodic solutions for a Gilpin-Ayala competition predator-prey system with harvesting terms and obtain some sufficient conditions for the existence of multiple positive periodic solutions for the system under consideration. The result of this paper is completely new. An example is employed to illustrate our result.

scholarly journals Existence of Positive Periodic Solutions for a Nonautonomous Generalized Predator-Prey System with Time Delay in Two-Patch Environment

2012 ◽  
Vol 2012 ◽  
pp. 1-12
Author(s):  
Huilan Wang ◽  
Zhengqiu Zhang ◽  
Weiping Zhou
Keyword(s):  
Periodic Solutions ◽  
Topological Degree ◽  
Sufficient Conditions ◽  
Coincidence Degree ◽  
Degree Theory ◽  
Positive Periodic Solutions ◽  
Predator Prey ◽  
Prey System ◽  
System With Time Delay ◽  
Function Matrix

By using continuation theorem of coincidence degree theory, sufficient conditions of the existence of positive periodic solutions are obtained for a generalized predator-prey system with diffusion and delays. In this paper, we construct a V-function to make the prior estimation for periodic solutions, which makes the discussion more concise. Moreover, to compute the mapping's topological degree, a polynomial function matrix is constructed straightforwardly as a homotopic mapping for the generalized one, which improves the methods of computation on topological degree for a generalized mapping.

scholarly journals Multiple Periodic Solutions of Generalized Gause-Type Predator-Prey Systems with Nonmonotonic Numerical Responses and Impulse

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
Zhenguo Luo ◽  
Liping Luo ◽  
Yunhui Zeng
Keyword(s):  
Periodic Solutions ◽  
Coincidence Degree ◽  
Degree Theory ◽  
Continuation Theorem ◽  
Coincidence Degree Theory ◽  
Sufficient Condition ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
Multiple Periodic Solutions

We consider an impulsive periodic generalized Gause-type predator-prey model with nonmonotonic numerical responses. Using the continuation theorem of coincidence degree theory, we present an easily verifiable sufficient condition on the existence of multiple periodic solutions. As corollaries, some applications are listed. In particular, our results extend and improve some known criteria.

POSITIVE PERIODIC SOLUTION FOR A GAUSE-TYPE RATIO-DEPENDENT PREDATOR-PREY SYSTEM WITH DIFFUSION AND TIME DELAY

2008 ◽  
Vol 01 (03) ◽  
pp. 339-354 ◽  
Author(s):  
XIAOQUAN DING ◽  
YUANYUAN WANG
Keyword(s):  
Periodic Solution ◽  
Time Delay ◽  
Coincidence Degree ◽  
Degree Theory ◽  
Positive Periodic Solution ◽  
Continuation Theorem ◽  
Predator Prey ◽  
Prey System ◽  
Ratio Dependent ◽  
System With Time Delay

A two-species Gause-type ratio-dependent predator-prey system with time delay in a two-patch environment is investigated. By using a continuation theorem based on coincidence degree theory, we establish easily verifiable criteria for the existence of periodic solution for the system. As corollaries, some applications are listed. In particular, our results extend and improve some known results.

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