In this article, we consider a bioeconomic model for minimax control problems which are governed by periodic competing parabolic Lotka-Volterra equations. Firstly the existence and uniqueness results to the state equations are proved as well as stability under mild assumptions. Afterwards, we formulate the minimax control problem. The optimality criteria are to minimize damage and trapping costs of the first species (nuisance), and to maximize the difference between economic revenue and cost of the second species. The existence of an optimal solution is obtained. Also, necessary conditions for a saddle point optimality are derived.