We consider a renewable resource distributed in a periodic environment, which is taken as n-dimensional torus. The dynamics of the resource is described by Kolmogorov-Piskunov-Petrovsky-Fisher equation [1], [7], [8] in divergent form

p_t = (α(x)p_x)_x + a(x)p − b(x)p^2.

Here p = p(x, t) is the density of the resource at the point x of its distribution area at the time t, and functions α, a and b characterize the diffusion of the resource, the rates of its renewal and saturation of the environment with it, respectively. It is assumed that these functions continuously depend on a point of this area, but do not depend on time. In addition, it is assumed that the function b is positive and separated from zero by some constant b0 > 0, the matrix α is positive definite, and its elements have derivatives satisfying the Holder condition with some positive exponent.

The resource is exploited by either the permanent harvesting, or periodic
impulse harvesting, or else in the case of the circle (when n = 1) by harvesting
control machine, which periodically moves along the circle and at each moment
collect a part of resource density that depends of the current machine position
and difficulties to search or extract the resource from this position.
Under natural assumptions we prove that for all considered harvesting modes
there exists admissible strategy which provide the maximum time averaged in-
come in kind, and the respective state of the resource is periodic in time or stationary [2], [3], [4], [5], [6].
 

References
 

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Springer, Cham. https://doi.org/10.1007/978-3-031-17558-9−5

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biological problem, Bull. Moscow Univ., 1937, Math. Mech. 1, pp. 1–25.