Stability and Hopf bifurcation of a modified predator-prey model with a time delay and square root response function

Stability and Hopf bifurcation of a modified predator-prey model with a time delay and square... In this paper, we consider a two-dimensional predator-prey model with a time delay and square root response function. We analyze the stability of equilibria with the delay τ increasing and the critical value of τ when Hopf bifurcation occurs. Because the model has the term of square root, the zero point is a singularity. In order to clearly study the stability of the zero point, we rescale the variable x ( t ) $x(t)$ , say x ( t ) = X 2 ( t ) $x(t)=X^{2}(t)$ . The conclusion is that the zero point is not stable and the instability is not affected by the delay τ. We apply the normal form method and center manifold theorem to obtain the direction and stability of the Hopf bifurcation. Finally, we make several numerical simulations which is consistent with the conclusion of theoretical analysis. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Advances in Difference Equations Springer Journals

Stability and Hopf bifurcation of a modified predator-prey model with a time delay and square root response function

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Publisher
Springer International Publishing
Copyright
Copyright © 2017 by The Author(s)
Subject
Mathematics; Difference and Functional Equations; Mathematics, general; Analysis; Functional Analysis; Ordinary Differential Equations; Partial Differential Equations
eISSN
1687-1847
D.O.I.
10.1186/s13662-017-1292-1
Publisher site
See Article on Publisher Site

Abstract

In this paper, we consider a two-dimensional predator-prey model with a time delay and square root response function. We analyze the stability of equilibria with the delay τ increasing and the critical value of τ when Hopf bifurcation occurs. Because the model has the term of square root, the zero point is a singularity. In order to clearly study the stability of the zero point, we rescale the variable x ( t ) $x(t)$ , say x ( t ) = X 2 ( t ) $x(t)=X^{2}(t)$ . The conclusion is that the zero point is not stable and the instability is not affected by the delay τ. We apply the normal form method and center manifold theorem to obtain the direction and stability of the Hopf bifurcation. Finally, we make several numerical simulations which is consistent with the conclusion of theoretical analysis.

Journal

Advances in Difference EquationsSpringer Journals

Published: Aug 14, 2017

References

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