Hopf Bifurcation Analysis in a Diffusive Food-Chain Model with Two Time Delays

2017 ◽  
Author(s):  
Fanghui Guo ◽  
Yan Meng ◽  
Tianjiao Zhang
Keyword(s):  
Hopf Bifurcation ◽  
Bifurcation Analysis ◽  
Food Chain ◽  
Time Delays ◽  
Chain Model ◽  
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.

Dynamics of an autonomous food chain model and existence of global attractor of the associated non-autonomous system

2019 ◽  
Vol 12 (08) ◽  
pp. 1950082 ◽  
Author(s):  
Jyotirmoy Roy ◽  
Shariful Alam
Keyword(s):  
Hopf Bifurcation ◽  
Food Chain ◽  
Migration Rate ◽  
Equilibrium Points ◽  
Model System ◽  
Seasonal Effect ◽  
Chain Model ◽  
Upward Migration ◽  
Downward Migration ◽  
Food Chain Model

In this paper, we have analyzed a tri-trophic food chain model consisting of phytoplankton, zooplankton and fish population in an aquatic environment. Here, the pelagic water column is divided into two layers namely, the upper layer and the lower layer. The zooplankton population makes a diel vertical migration (DVM) from lower portion to upper portion and vice-versa to trade-off between food source and fear from predator (Fish). Here, mathematical model has been developed and analyzed in a rigorous way. Apart from routine calculations like boundedness and positivity of the solution, local stability of the equilibrium points, we performed Hopf bifurcation analysis of the interior equilibrium point of our model system in a systematic way. It is observed that the migratory behavior of zooplankton plays a crucial role in the dynamics of the model system. Both the upward and downward migration rates of DVM leads the system into Hopf bifurcation. The upward migration rate of zooplankton deteriorates the stable coexistence of all the species in the system, whereas the downward migration rate enhance the stability of the system. Further, we analyze the non-autonomous version of the system to capture seasonal effect of environmental variations. We have shown that under certain parametric restrictions periodic coexistence of all the species of our system is possible. Finally, extensive numerical simulation has been performed to support our analytical findings.

Stability and Bifurcation Analysis in a Food Chain Model with Delay

2011 ◽  
Vol 110-116 ◽  
pp. 3382-3388
Author(s):  
Zhang Li
Keyword(s):  
Hopf Bifurcation ◽  
Food Chain ◽  
Center Manifold ◽  
Chain Model ◽  
Center Manifold Theory ◽  
Stability And Bifurcation Analysis ◽  
Existence And Stability ◽  
The Stability ◽  
Manifold Theory ◽  
Food Chain Model

In this paper, we investigate a delayed three-species food chain model. The existence and stability of equilibria are obtained. A explicit algorithm for determining the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions are derived by using the normal form and the center manifold theory.

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