A mathematical model for the spatial spread and biocontrol of the asian longhorned beetle

We propose a mathematical model, of four coupled delay differential equations, for control of the Asian longhorned beetle Anoplophora glabripennis by one of its natural predators, the cylindrical bark beetle Dastarcus longulus or another predator with similar characteristics. It is a predator prey interaction at the larval rather than adult level which creates interesting modeling challenges. We specify the birth rate only for A. glabripennis and calculate the birth rate of the control agent D. longulus by keeping track of its consumption of larval A. glabripennis biomass and using the idea of conversion of biomass. We prove rigorous results on the stability of equilibria and on persistence of D. longulus, and we make an assessment of the kinds of characteristics that enable D. longulus, or any similar control agent, to effectively control A. glabripennis. A pest such as A. glabripennis will destroy its habitat and must continually find new host trees. Even though our model does not have explicit spatial dependence, we may use it to make some inferences about the likely spatial spread of an infestation.