Bifurcation Analysis of a One-Prey and Two-Predators Model with Additional Food and Harvesting Subject to Toxicity

In this paper, a three-dimensional one-prey and two-predators model, with additional food and harvesting in the presence of toxicity is proposed. Additional food is being provided to one predator. The dynamics and bifurcations of the system are investigated using center manifold theorem, normal form theory and Sotomayor's theorem. It is proved that the system undergoes transcritical bifurcation, saddle-node bifurcation, Hopf bifurcation, generalized Hopf bifurcation, Bogdanov-Takens bifurcation and cusp bifurcation with respect to different parameters. Bifurcation diagrams of the system with respect to toxic effect and harvesting effect are illustrated. The phase portraits and solution curves are also presented to verify the dynamic behavior. The results show that the combined effect of the factors has the power of transforming simple ecosystems into complex ecosystems.