Volume 1, Issue 1
Bifurcation of a Modified Leslie-Gower System with Discrete and Distributed Delays

Zhongkai Guo, Haifeng Huo, Qiuyan Ren & Hong Xiang

J. Nonl. Mod. Anal., 1 (2019), pp. 73-91.

Published online: 2021-04

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  • Abstract

A modified Leslie-Gower predator-prey system with discrete and distributed delays is introduced. By analyzing the associated characteristic equation, stability and local Hopf bifurcation of the model are studied. It is found that the positive equilibrium is asymptotically stable when $\tau$ is less than a critical value and unstable when $\tau$ is greater than this critical value and the system can also undergo Hopf bifurcation at the positive equilibrium when $\tau$ crosses this critical value. Furthermore, using the normal form theory and center manifold theorem, the formulae for determining the direction of periodic solutions bifurcating from positive equilibrium are derived. Some numerical simulations are also carried out to illustrate our results.

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@Article{JNMA-1-73, author = {Guo , ZhongkaiHuo , HaifengRen , Qiuyan and Xiang , Hong}, title = {Bifurcation of a Modified Leslie-Gower System with Discrete and Distributed Delays}, journal = {Journal of Nonlinear Modeling and Analysis}, year = {2021}, volume = {1}, number = {1}, pages = {73--91}, abstract = {

A modified Leslie-Gower predator-prey system with discrete and distributed delays is introduced. By analyzing the associated characteristic equation, stability and local Hopf bifurcation of the model are studied. It is found that the positive equilibrium is asymptotically stable when $\tau$ is less than a critical value and unstable when $\tau$ is greater than this critical value and the system can also undergo Hopf bifurcation at the positive equilibrium when $\tau$ crosses this critical value. Furthermore, using the normal form theory and center manifold theorem, the formulae for determining the direction of periodic solutions bifurcating from positive equilibrium are derived. Some numerical simulations are also carried out to illustrate our results.

}, issn = {2562-2862}, doi = {https://doi.org/10.12150/jnma.2019.73}, url = {http://global-sci.org/intro/article_detail/jnma/18869.html} }
TY - JOUR T1 - Bifurcation of a Modified Leslie-Gower System with Discrete and Distributed Delays AU - Guo , Zhongkai AU - Huo , Haifeng AU - Ren , Qiuyan AU - Xiang , Hong JO - Journal of Nonlinear Modeling and Analysis VL - 1 SP - 73 EP - 91 PY - 2021 DA - 2021/04 SN - 1 DO - http://doi.org/10.12150/jnma.2019.73 UR - https://global-sci.org/intro/article_detail/jnma/18869.html KW - Modified Leslie-Gower system, discrete and distributed delays, stability, Hopf bifurcation. AB -

A modified Leslie-Gower predator-prey system with discrete and distributed delays is introduced. By analyzing the associated characteristic equation, stability and local Hopf bifurcation of the model are studied. It is found that the positive equilibrium is asymptotically stable when $\tau$ is less than a critical value and unstable when $\tau$ is greater than this critical value and the system can also undergo Hopf bifurcation at the positive equilibrium when $\tau$ crosses this critical value. Furthermore, using the normal form theory and center manifold theorem, the formulae for determining the direction of periodic solutions bifurcating from positive equilibrium are derived. Some numerical simulations are also carried out to illustrate our results.

Guo , ZhongkaiHuo , HaifengRen , Qiuyan and Xiang , Hong. (2021). Bifurcation of a Modified Leslie-Gower System with Discrete and Distributed Delays. Journal of Nonlinear Modeling and Analysis. 1 (1). 73-91. doi:10.12150/jnma.2019.73
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