Necessary optimality conditions in the one discrete boundary problem of population dynamics control

One discrete problem of optimal population dynamics control with a controllable initial condition is considered. The process is described by a nonlinear system of Fredholm-type difference equations. An analogue of the L.S. Pontryagin’ maximum principle is proved by imposing a series of smoothness conditions on the right side the equation. The linearized maximum principle and the analogue of the Euler equation are proved for the case of convexity and openness of the control domain, which are first-order necessary conditions for optimality.