EXISTENCE OF PERIODIC SOLUTIONS OF NONAUTONOMOUS PREDATOR-PREY MODEL WITH BEDDINGTON-DEANGELIS FUNCTIONAL RESPONSE AND TIME DELAY

2021 ◽  
Vol 10 (5) ◽  
pp. 2641-2652
Author(s):  
S. Mahalakshmi ◽  
V. Piramanantham
Keyword(s):  
Time Delay ◽  
Periodic Solutions ◽  
Functional Response ◽  
Sufficient Conditions ◽  
Coincidence Degree ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
Existence Of Periodic Solutions

In this paper we establish some easily verifiable sufficient conditions for the existence of periodic solutions of nonautonomous Predator-Prey Model with Beddington-DeAngelis Functional response and time delay using Mowhins Coincidence degree method.

scholarly journals Existence and Uniqueness of Positive Periodic Solutions for a Delayed Predator-Prey Model with Dispersion and Impulses

2014 ◽  
Vol 2014 ◽  
pp. 1-21
Author(s):  
Zhenguo Luo ◽  
Liping Luo ◽  
Liu Yang ◽  
Zhenghui Gao ◽  
Yunhui Zeng
Keyword(s):  
Periodic Solutions ◽  
Time Delays ◽  
Sufficient Conditions ◽  
Coincidence Degree ◽  
Degree Theory ◽  
Predator Population ◽  
Positive Periodic Solutions ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model

An impulsive Lotka-Volterra type predator-prey model with prey dispersal in two-patch environments and time delays is investigated, where we assume the model of patches with a barrier only as far as the prey population is concerned, whereas the predator population has no barriers between patches. By applying the continuation theorem of coincidence degree theory and by means of a suitable Lyapunov functional, a set of easily verifiable sufficient conditions are obtained to guarantee the existence, uniqueness, and global stability of positive periodic solutions of the system. Some known results subject to the underlying systems without impulses are improved and generalized. As an application, we also give two examples to illustrate the feasibility of our main results.

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.

COMPLEXITY OF A DELAYED PREDATOR–PREY MODEL WITH IMPULSIVE HARVEST AND HOLLING TYPE II FUNCTIONAL RESPONSE

2008 ◽  
Vol 11 (01) ◽  
pp. 77-97 ◽  
Author(s):  
GUANGZHAO ZENG ◽  
FENGYAN WANG ◽  
JUAN J. NIETO
Keyword(s):  
Periodic Solutions ◽  
Functional Response ◽  
Coincidence Degree ◽  
Degree Theory ◽  
Type Ii ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
Delay Differential ◽  
Type Ii Functional Response

We study an impulsive delay differential predator–prey model with Holling type II functional response. The stability of the trivial equilibrium is analyzed by means of impulsive Floquet theory providing a sufficient condition for extinction. Using coincidence degree theory we show the existence of positive periodic solutions. The system is then analyzed numerically, revealing that the presence of delays and impulses may lead to chaotic solutions, quasi-periodic solutions, or multiple periodic solutions. Several simulations and examples are presented.

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 Mathematical Analysis for a Discrete Predator-Prey Model with Time Delay and Holling II Functional Response

2015 ◽  
Vol 2015 ◽  
pp. 1-8 ◽  
Author(s):  
Dehong Ding ◽  
Kui Fang ◽  
Yang Zhao
Keyword(s):  
Functional Response ◽  
Sufficient Conditions ◽  
Coincidence Degree ◽  
Degree Theory ◽  
Positive Periodic Solutions ◽  
Mawhin’S Continuation Theorem ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
Holling Ii Functional Response

This paper is concerned with a discrete predator-prey model with Holling II functional response and delays. Applying Gaines and Mawhin’s continuation theorem of coincidence degree theory and the method of Lyapunov function, we obtain some sufficient conditions for the existence global asymptotic stability of positive periodic solutions of the model.

BIFURCATION AND CHAOS IN A DISCRETE PREDATOR–PREY MODEL WITH HOLLING TYPE-III FUNCTIONAL RESPONSE AND HARVESTING EFFECT

2021 ◽  
pp. 1-28
Author(s):  
ANURAJ SINGH ◽  
PREETI DEOLIA
Keyword(s):  
Functional Response ◽  
Sufficient Conditions ◽  
Flip Bifurcation ◽  
Bifurcation Structure ◽  
Bifurcation And Chaos ◽  
Predator Prey ◽  
Type Iii ◽  
Predator Prey Model ◽  
Prey Model ◽  
Wide Range

In this paper, we study a discrete-time predator–prey model with Holling type-III functional response and harvesting in both species. A detailed bifurcation analysis, depending on some parameter, reveals a rich bifurcation structure, including transcritical bifurcation, flip bifurcation and Neimark–Sacker bifurcation. However, some sufficient conditions to guarantee the global asymptotic stability of the trivial fixed point and unique positive fixed points are also given. The existence of chaos in the sense of Li–Yorke has been established for the discrete system. The extensive numerical simulations are given to support the analytical findings. The system exhibits flip bifurcation and Neimark–Sacker bifurcation followed by wide range of dense chaos. Further, the chaos occurred in the system can be controlled by choosing suitable value of prey harvesting.

Dynamical behaviors of a diffusive predator–prey model with Beddington–DeAngelis functional response and disease in the prey

2017 ◽  
Vol 10 (08) ◽  
pp. 1750119 ◽  
Author(s):  
Wensheng Yang
Keyword(s):  
Lyapunov Function ◽  
Iterative Method ◽  
Functional Response ◽  
Local Stability ◽  
Sufficient Conditions ◽  
Positive Equilibrium ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Dynamical Behaviors ◽  
Prey Model

The dynamical behaviors of a diffusive predator–prey model with Beddington–DeAngelis functional response and disease in the prey is considered in this work. By applying the comparison principle, linearized method, Lyapunov function and iterative method, we are able to achieve sufficient conditions of the permanence, the local stability and global stability of the boundary equilibria and the positive equilibrium, respectively. Our result complements and supplements some known ones.

scholarly journals Uniformly Strong Persistence for a Delayed Predator-Prey Model

2012 ◽  
Vol 2012 ◽  
pp. 1-7 ◽  
Author(s):  
Changjin Xu ◽  
Yuanfu Shao ◽  
Peiluan Li
Keyword(s):  
Time Delay ◽  
Sufficient Conditions ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
Asymptotically Periodic

An asymptotically periodic predator-prey model with time delay is investigated. Some sufficient conditions for the uniformly strong persistence of the system are obtained. Our result is an important complementarity to the earlier results.

scholarly journals Dynamics of a Diffusive Predator-Prey Model with General Nonlinear Functional Response

2014 ◽  
Vol 2014 ◽  
pp. 1-10
Author(s):  
Wensheng Yang
Keyword(s):  
Functional Response ◽  
Sufficient Conditions ◽  
Positive Equilibrium ◽  
Nonlinear Functional ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
Parameter Configuration ◽  
Sufficient And Necessary Conditions ◽  
Steady State Solutions

We study a diffusive predator-prey model with nonconstant death rate and general nonlinear functional response. Firstly, stability analysis of the equilibrium for reduced ODE system is discussed. Secondly, sufficient and necessary conditions which guarantee the predator and the prey species to be permanent are obtained. Furthermore, sufficient conditions for the global asymptotical stability of the unique positive equilibrium of the system are derived by using the method of Lyapunov function. Finally, we show that there are no nontrivial steady state solutions for certain parameter configuration.

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