Bifurcation dynamics of a delayed chemostat system with spatial diffusion

The resource’s diffusion and the populations’ migration may alter the ecosystem’s structure such that the species’ dynamics are changed. Moreover, the delay phenomenon in population behaviors, such as digestion or maturation, will inevitably affect the species’ dynamics. We, in this work, investigate a chemostat system with delay and spatial diffusion. The existence conditions of the Hopf bifurcation from the time lag and diffusive terms are determined. The concentration of population in the chemostat approaches a positive value when the bifurcation parameter’s value does not cross the critical point. The microorganisms’ concentration will fluctuate periodically as the value of the bifurcation parameter passes through the critical point. By the theory of norm form and center manifold, we further talked about the direction of the Hopf bifurcation and the stability of the periodic solutions. Several numerical examples are provided to support the theoretical results in this work.