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A delayed predator-prey model with strong Allee effect in prey population growth. (English) Zbl 1242.92060
Summary: We consider a delayed predator-prey system with intraspecific competition among the predator and a strong Allee effect [{\it W.C. Allee, Animal aggregation. Univ. Chicago Press (1931)] in prey population growth. Using the delay as bifurcation parameter, we investigate the stability of coexisting equilibrium points and show that Hopf-bifurcations can occur when the discrete delay crosses some critical magnitude. The direction of the Hopf bifurcating periodic solution and its stability are determined by applying the normal form method and the centre manifold theory. In addition, special attention is paid to the global continuation of local Hopf bifurcations. Using the global Hopf-bifurcation result of {\it J. Wu} [Trans. Amer. Math. Soc. 350, No. 12, 4799--4838 (1998; Zbl 0905.34034)] for functional differential equations, we establish the global existence of periodic solutions. Numerical simulations are carried out to validate the analytical findings.

MSC:
92D40Ecology
34K18Bifurcation theory of functional differential equations
34K20Stability theory of functional-differential equations
34K13Periodic solutions of functional differential equations
65C20Models (numerical methods)
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References:
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