Augmentation in competition interaction model: a discrete time optimal control analysis

Abstract

Many species are classified as threatened or endangered and are declining due to factors such as competition for limited resources. Competition affects the fitness of both populations as one organism’s use of a scarce resource decreases its availability to the other. In this study, a discrete-time competition interaction model with controls is formulated. We consider four populations represented by a discrete competition model with strong Allee effects. We define two objective functionals that account for linear and nonlinear representations of the translocation costs of the reserve populations at each time step. The objective functionals seek to maximize the populations and minimize the cost of augmentation. We employ the generalization of Pontryagin’s Maximum Principle for the optimal control of discrete-time state systems to obtain the necessary conditions. The discrete version of the forward-backward sweep method is employed to solve the optimality system numerically. The short-term and long-term qualitative dynamics of the model are discussed through the numerical simulations. Objective functional values indicating a percentage increase with optimal controls are calculated for each plot. The numerical simulation examines various scenarios, including the effects of cost constants, competition coefficients, and augmentation coefficients in the model.