ИСТИНА

We consider an exploited population with dynamic described by the equation \begin{displaymath} x_t(t,l){+} [g(l, E(t, l)) x(t,l)]_l= -[ \mu (l,E(t,l)){+}u(l)]x(t,l). \end{displaymath} Here $x(t,l) $ is the density of individuals of size $l$ at the moment $t;$ $g$ and $\mu$ are growth and death rates, respectively, and $u$ stands for exploitation intensity. The competition level $E$ could have symmetric or asymmetric form. In the asymmetric case, for example, it could be defined as $$E(t,l)=\int_l^{L} \chi(l)x(t,l)dl$$ with some non-negative function $\chi$ being integrable on interval $[0,L], L>0, $ where we manage and exploit the population. It is also assumed that the inflow of new generation (of individuals of size $0$) is defined by equation $$ x(t,0)=\int\limits_{0}^{L} r(l,E(t,l))x^{\beta}(t,l)dl+p_0. $$ with some $\beta \in (0,1),$ birth rate $r\ge 0$ and industrial population renewal $p_0\ge 0.$ Under the natural assumptions on the model parameters we prove the existence of a positive stationary solution and optimal among them, providing the maximum benefit from the exploitation for different objective functionals.