Convex-concave utility function: Optimal blood-consumption for vampires

Abstract

This paper considers an optimal control problem for the dynamics of a predator-prey model. The predator population has to choose the predation intensity over time in a way that maximizes the present value of the utility stream derived by consuming prey. The utility function is assumed to be convex for small levels of consumption and concave otherwise. The problem is solved using the maximum principle and different time patterns of the optimal solution are obtained in the cases of small, medium and high rates of time preference. The model has features of both, convex and concave optimal control problems and therefore phase plane analysis has to be combined with the problem of synthesis of bang-bang, singular and chattering solution pieces. © 1983.