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Dirichlet problem for a delayed diffusive hematopoiesis model. (English) Zbl 1447.35334

Summary: We study the dynamics of a delayed diffusive hematopoiesis model with two types of Dirichlet boundary conditions. For the model with a zero Dirichlet boundary condition, we establish global stability of the trivial equilibrium under certain conditions, and use the phase plane method to prove the existence and uniqueness of a positive spatially heterogeneous steady state. We further obtain delay-independent as well as delay-dependent conditions for the local stability of this steady state. For the model with a non-zero Dirichlet boundary condition, we show that the only positive steady state is a constant solution. Results for the local stability of the constant solution are also provided. By using the delay as a bifurcation parameter, we show that the model has infinite number of Hopf bifurcation values and the global Hopf branches bifurcated from these values are unbounded, which indicates the global existence of periodic solutions.

MSC:

35Q92 PDEs in connection with biology, chemistry and other natural sciences
92C37 Cell biology
35B35 Stability in context of PDEs
35B32 Bifurcations in context of PDEs
35B10 Periodic solutions to PDEs
35R07 PDEs on time scales
35A01 Existence problems for PDEs: global existence, local existence, non-existence
35A02 Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness
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