Analysis of Hopf bifurcations in a Colorado potato beetle invasion model with time delay

The Colorado potato beetle (CPB) is an agricultural pest known for its significant damage to potato crops. In this paper, we focused on the formulation of CPB invasion model with time delay. Using eigenvalue theory, we initially prove the asymptotically stabling of the positive equilibrium in the absence of time delay. On the other hand, in the presence of a time delay, we determine a critical threshold parameter beyond which the equilibrium becomes unstable. Further, we establish the sufficient conditions for the occurrence of a Hopf bifurcation. Furthermore, using the center manifold theorem and normal form theory, we investigate the characteristics of the bifurcating periodic solutions. Finally, the validity of the analysis is demonstrated by means of comprehensive numerical simulation.