Dynamics of a synergistic biological control model with additional food for predators and delayed development of parasitoids

Lycium barbarum L., a key edible and medicinal crop, faces a persistent and serious threat to its production safety from Bactericera gobica Loginova. In this study, we develop a synergistic biological control model to investigate the combined effects of additional food supply to the predatory natural enemy Deraeocoris punctulatus Fallen and the developmental delay in the parasitoid Psyllaephagus arenarius Trjapitzin. With additional food supply, our equilibrium analysis gives sharp conditions for the occurrence and stability of extinction, psyllid-only, predator-absent, parasitoid-absent, and predator-prey coexistence. Theoretical analysis reveals that variations in additional food supply can cause instability in parasitoid-absent equilibrium and induce a Hopf bifurcation. Furthermore, a Hopf bifurcation also occurs when the developmental delay of the parasitoid falls below a critical threshold at the coexistence equilibrium. Numerical simulations confirm that changes in both additional food supply and developmental delay can trigger the Hopf bifurcations. Finally, PRCC sensitivity analysis identifies the critical parameters that influence the dynamic behavior of the system, and the stability of the predator directly influences the overall control efficacy.