A model for biological control in agriculture

Persistence and global stability of the coexistence equilibrium of a recently published model in biocontrol of crops are here shown both in absence and presence of delays, introduced to simulate the handling time of the prey. In the latter case, the system can behave in two different ways, in dependence of whether a suitably defined key parameter exceeds a certain threshold. Namely, below the threshold the delay is shown not to be able to influence the stability of the coexistence equilibrium; above it, existence of Hopf bifurcations is analytically proven. Further, in this range, numerical simulations reveal a route to chaotic behavior as function of the size of the delay. Some operative conclusions for agroecosystem management are drawn, although they ultimately depend on each particular situation.