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This paper considers the optimal harvesting control of a biological species, whose growth is governed by the parabolic diffusive Volterra-Lotka equation. We prove that such equation with L" periodic coefficients has an unique positive periodic solution. We show the existence and uniqueness of an optimal control, and under certain conditions, we characterize the optimal control in terms of a parabolic optimality system. A monotone sequence which converges to the optimal control is constructed.[$ 0 on S = aR x(0, T ) and u(x, T ) = u(x, 0) for X E R . (1.1) (def) Remark 1.1. All results of this paper can be formulated for general elliptic operator instead of -A with appropriate assumptions.Next, let K , M > 0 be given constants; we define the payoff functional byUi(X, 0) = Ui(X, T ) for x E Q,

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