Hopf bifurcation of a turbidostat model with nutrient recycling and multiple delay effects

In the ecosystem, species need time to transform resources into energy for growth. In addition, the degradation period after species' mortality will lead to the reuse of resources. The two phenomena above contribute to the delay effects during the species' evolution. We here develop a turbidostat model incorporating delay effects to investigate the impact of resource consumption and reusing period on population evolution. We first analyze the existence and the stability of the positive equilibrium. Our results reveal the perturbation effect of delay parameters on its dynamics. When the time lag periods cross the critical values, the positive equilibrium becomes unstable, and periodic solutions are produced. The populations of the species will oscillate along with the resource consumption and recycling period. In addition, the multiple delay effect will decrease the negative effect of delay phenomenon on positive solutions. By controlling the delay period of decomposition, we can moderate the oscillations caused by the delay effect in resource consumption. To further understand the dynamics of periodic solutions, the direction of the Hopf bifurcation and the periodic solutions' stability are also discussed here. Several numerical examples are tested to validate the theoretical results in this work.