Abstract
In this paper, we investigate a delay reaction–diffusion–advection model with ideal free dispersal. The stability of positive steady-state solutions and the existence of the associated Hopf bifurcation are obtained by analyzing the principal eigenvalue of an elliptic operator. By the normal form theory and the center manifold reduction, the stability and bifurcation direction of Hopf bifurcating periodic solutions are obtained. Moreover, numerical simulations and a brief discussion are presented to illustrate our theoretical results.
Highlights
In recent years, biological mathematics has developed to be one of the most active research directions in the field of applied mathematics
Their results imply that the increase of time delay can make the spatially nonhomogeneous positive steady state unstable for (1.6), and the model can exhibit an oscillatory pattern through Hopf bifurcation
We introduce the notion of the ideal free distribution (IFD) into the Hopf bifurcation problems to understand the Hopf bifurcation and bifurcation direction of the spatially nonhomogeneous positive steady state, and consider the following system:
Summary
Biological mathematics has developed to be one of the most active research directions in the field of applied mathematics. Their results imply that the increase of time delay can make the spatially nonhomogeneous positive steady state unstable for (1.6), and the model can exhibit an oscillatory pattern through Hopf bifurcation. They considered the effect of advection on Hopf bifurcation values, and the Hopf bifurcation is more likely to occur with the increase of advection rate. By virtue of [22], we have the local Hopf bifurcation theorem for partial functional differential equations as follows.
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