Dynamics analysis of a delayed reaction-diffusion Filippov pest management model with double economic thresholds.

A delayed reaction-diffusion Filippov pest-natural enemy model with double thresholds is proposed to describe switching control in pest management. Compared with related Filippov pest models, the coupled effects of the double thresholds, diffusion, and delay are considered, allowing spatial pattern formation to be linked with nonsmooth dynamics. Sufficient conditions for the existence and stability of positive equilibria are established. Turing instability criteria and amplitude equations are derived, and Hopf bifurcation caused by reproductive delay is analyzed. Attracting sliding regions and pseudo-equilibria are determined, and boundary node and boundary focus bifurcations are investigated. Numerical simulations show that decreasing the pest diffusion coefficient changes the spatial pattern from spotted to mixed and then to striped. As the pest threshold approaches the equilibrium density, spatial heterogeneity is weakened and Turing patterns disappear through a boundary focus bifurcation. In addition, increasing the reproductive delay enlarges the sliding region and leads to touching, buckling, and crossing sliding bifurcations. These results reveal the interaction among spatial diffusion, delay, and threshold switching and provide useful insights for threshold selection and pest management.