Dynamics in a Plankton Model with Toxic Substances and Phytoplankton Harvesting

In this paper, a phytoplankton–zooplankton model incorporating toxic substances and nonlinear phytoplankton harvesting is established. The existence and stability of the equilibrium of this model are first investigated. The occurrence of transcritical, saddle-node, Hopf and Bautin bifurcations at different equilibria is then verified. In addition, the properties of Hopf bifurcation and Bautin bifurcation are discussed by using normal form method. These results demonstrate that phytoplankton and zooplankton populations will oscillate periodically when the harvesting level is high. More interestingly, it is found that the oscillations are always unstable for small phytoplankton carrying capacity, while the dynamics have close relations with the initial population densities for a large environmental capacity. The existence of Bautin bifurcation theoretically indicates that toxic phytoplankton can cause extinction once there exist harmful algal blooms for some time. These results are numerically illustrated for the model with spatial diffusion, which shows that local phytoplankton blooms will lead to global populations extinction.

Hopf and Bautin Bifurcation in a Tritrophic Food Chain Model with Holling Functional Response Types III and IV

In this paper, we analyze the Hopf and Bautin bifurcation of a given system of differential equations, corresponding to a tritrophic food chain model with Holling functional response types III and IV for the predator and superpredator, respectively. We distinguish two cases, when the prey has linear or logistic growth. In both cases we guarantee the existence of a limit cycle bifurcating from an equilibrium point in the positive octant of [Formula: see text]. In order to do so, for the Hopf bifurcation we compute explicitly the first Lyapunov coefficient, the transversality Hopf condition, and for the Bautin bifurcation we also compute the second Lyapunov coefficient and verify the regularity conditions.

Global Bifurcation Map of the Homogeneous States in the Gray–Scott Model

We study the spatially homogeneous time-dependent solutions and their bifurcations in the Gray–Scott model. We find the global map of bifurcations by a combination of rigorous verification of the existence of Takens–Bogdanov and a Bautin bifurcation, in the space of two parameters [Formula: see text]–[Formula: see text]. With the aid of numerical continuation of local bifurcation curves we give a global description of all the possible bifurcations.