scholarly journals Effect of Time-Delay on a Ratio-Dependent Food Chain Model

2009 ◽  
Vol 14 (2) ◽  
pp. 199-216 ◽  
Author(s):  
B. Patra ◽  
A. Maiti ◽  
G. P. Samanta
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Food Chain ◽  
Chain Model ◽  
Functional Responses ◽  
Simple Criterion ◽  
Delayed System ◽  
Ratio Dependent ◽  
Food Chain Model ◽  
Practical Implications

This paper aims to study the effect of time-delay on a tritrophic food chainmodel with Michaelis-Menten type ratio-dependent functional responses. Boundednessof the time-delayed system is established. A simple criterion for deterministic extinctionis derived. It has been shown that the time-delay may introduce instability in the systemthrough Hopf bifurcation. Computer simulations are carried out to explain the analyticalfindings. It is discussed how these ideas illuminate some of the observed properties ofreal populations in the field, and explores practical implications.

RICH DYNAMICS OF A TIME-DELAYED FOOD CHAIN MODEL

2006 ◽  
Vol 14 (03) ◽  
pp. 387-412 ◽  
Author(s):  
ALAKES MAITI ◽  
G. P. SAMANTA
Keyword(s):  
Computer Simulation ◽  
Time Delay ◽  
Functional Response ◽  
Food Chain ◽  
Complex Dynamics ◽  
Chain Model ◽  
Dynamical Behaviors ◽  
Discrete Time Delay ◽  
Food Chain Model ◽  
Practical Implications

Complex dynamics of a tritrophic food chain model is discussed in this paper. The model is composed of a logistic prey, a classical Lotka-Volterra functional response for prey-predator and a ratio-dependent functional response for predator-superpredator. Dynamical behaviors such as boundedness, stability and bifurcation of the model are studied critically. The effect of discrete time-delay on the model is investigated. Computer simulation of various solutions is presented to illustrate our mathematical findings. How these ideas illuminate some of the observed properties of real populations in the field is discussed and practical implications are explored.

scholarly journals Dynamics of a food chain model with ratio-dependent and modified Leslie-Gower functional responses

2015 ◽  
Vol 20 (7) ◽  
pp. 2269-2290 ◽  
Author(s):  
Wen-Bin Yang ◽  
◽  
Yan-Ling Li ◽  
Jianhua Wu ◽  
Hai-Xia Li ◽  
...  
Keyword(s):  
Food Chain ◽  
Chain Model ◽  
Functional Responses ◽  
Ratio Dependent ◽  
Food Chain Model

Stability and Bifurcation Analysis of a Three-Species Food Chain Model with Delay

2015 ◽  
Vol 25 (09) ◽  
pp. 1550123 ◽  
Author(s):  
Nikhil Pal ◽  
Sudip Samanta ◽  
Santanu Biswas ◽  
Marwan Alquran ◽  
Kamel Al-Khaled ◽  
...  
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Bifurcation Analysis ◽  
Food Chain ◽  
Period Doubling ◽  
Equilibrium Analysis ◽  
Chain Model ◽  
Normal Form Theory ◽  
Center Manifold Theorem ◽  
Food Chain Model

In the present paper, we study the effect of gestation delay on a tri-trophic food chain model with Holling type-II functional response. The essential mathematical features of the proposed model are analyzed with the help of equilibrium analysis, stability analysis, and bifurcation theory. Considering time-delay as the bifurcation parameter, the Hopf-bifurcation analysis is carried out around the coexisting equilibrium. The direction of Hopf-bifurcation and the stability of the bifurcating periodic solutions are determined by applying the normal form theory and center manifold theorem. We observe that if the magnitude of the delay is increased, the system loses stability and shows limit cycle oscillations through Hopf-bifurcation. The system also shows the chaotic dynamics via period-doubling bifurcation for further enhancement of time-delay. Our analytical findings are illustrated through numerical simulations.

Limit cycles in a tritrophic food chain model with general functional responses

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Gamaliel Blé ◽  
Iván Loreto-Hernández
Keyword(s):  
Hopf Bifurcation ◽  
Food Chain ◽  
Stable Limit Cycle ◽  
Growth Function ◽  
Logistic Growth ◽  
Chain Model ◽  
Stable Limit ◽  
Functional Responses ◽  
Tritrophic Model ◽  
Food Chain Model

Abstract The conditions to have a stable limit cycle by an Andronov–Hopf bifurcation in a tritrophic model are given. A generalized logistic growth function for the prey is considered, and a general family of functional responses, including the Holling type, are taken for the predators. Some results obtained in previous works for tritrophic models, which consider logistic growth in the prey and Holling functional responses, are generalized.

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