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Dirichlet problem for a diffusive logistic population model with two delays

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  • In this paper, we investigate a diffusive logistic equation with non-zero Dirichlet boundary condition and two delays. We first exclude the existence of positive heterogeneous steady states, which implies the uniqueness of constant positive steady state. Then, we analyze the local stability and local Hopf bifurcation at the positive steady state. We show that multiple delays can induce multiple stability switches. Furthermore, we prove global stability of the positive steady state under certain conditions and obtain global Hopf bifurcation results. Our theoretical results are illustrated with numerical simulations.

    Mathematics Subject Classification: Primary: 92D25, 35K57; Secondary: 34K18.


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  • Figure 1.  The phase portrait of (9)

    Figure 2.  The root distribution of (14)

    Figure 3.  Left: $ u_* $ is stable at $ \tau = 5 $. Right: $ u_* $ is unstable at $ \tau = 12 $

    Figure 4.  Left: $ u_* $ is stable at $ \tau = 18 $. Right: $ u_* $ is unstable at $ \tau = 30 $

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