Hopf bifurcations in a predator-prey system of population allelopathy with a discrete delay and a distributed delay
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A delayed Lotka–Volterra predator-prey system of population allelopathy with discrete delay and distributed maturation delay for the predator population described by an integral with a strong delay kernel is considered. By linearizing the system at the positive equilibrium and analyzing the associated characteristic equation, the asymptotic stability of the positive equilibrium is investigated and Hopf bifurcations are demonstrated. Furthermore, the direction of Hopf bifurcation and the stability of the bifurcating periodic solutions are determined by the normal form theory and the center manifold theorem for functional differential equations. Finally, some numerical simulations are carried out for illustrating the theoretical results.
KeywordsLotka–Volterra predator-prey system Discrete delay Distributed delay Stability Hopf bifurcation Periodic solution
The authors express their gratitude to the anonymous referees for their helpful suggestions and the partial support of Science Foundation of Yunnan Province (2011FZ086).
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