Bifurcation and optimal harvesting of a diffusive predator-prey system with delays and interval biological parameters

- Consider the model with imprecise data as form of an interval in nature.
- Apply PDE theory to discuss the stability and bifurcation of the model.
- A Hopf bifurcation occurs as the delays increase through a certain threshold.
- Pay attention to the exploitation or harvesting of biological resources.

This paper deals with a delayed reaction-diffusion three-species Lotka-Volterra model with interval biological parameters and harvesting. Sufficient conditions for the local stability of the positive equilibrium and the existence of Hopf bifurcation are obtained by analyzing the associated characteristic equation. Furthermore, formulas for determining the direction of Hopf bifurcation and the stability of the bifurcating periodic solutions are derived by applying the normal form method and center manifold theorem. Then an optimal control problem has been considered. Finally, numerical simulation results are presented to validate the theoretical analysis. Numerical evidence shows that the presence of harvesting can impact the existence of species and over harvesting can result in the extinction of the prey or the predator which is in line with reality.