Periodic control of biocommunity and circle homeomorphisms

To solve a problem of preserving a predator-prey biocommunity species structure, a mathematical model of periodic control process is proposed and investigated. A model is based on reducing of continuous dynamics to a discrete one generated by circle homeomorphisms.
The biocommunity dynamics is described by a three dimensional system of ordinary differential equations. Two equations present the Lotka-Volterra system, and the third one describes the dynamics of food attractivity the notion of which was introduced in [1]. The specifics of the system is such that its trajectories belong to cylindrical surfaces. The latter permits to conduct a qualitative research of the system.
The following process is modeled. At some point of time, in order to diminish the growth of prey population, which is considered as a harmful one, a predator population is transferred to a patch. The latter procedure is widely spread in practice while controlling the growth of harmful, invasive, species. If, after a while, the value of food attractivity becomes less than some threshold then the predator population leaves the patch. Thus, there arises a control problem consisting in removal of some part of predator population in such a way that for the remaining part the patch attractivity value becomes more than a threshold.
A notion of admissible piecewise constant control is proposed. The latter takes into account a possibility of its realization with the less anthropogenic load on a patch. To solve the formulated problem, a method of tangent control is proposed. On the basis of this method, a periodic control process, as the most natural if one takes into account the periodicity of the free Lotka-Volterra system, is constructed. In this case, a continuous dynamical system is reduced to a discrete one which generates circle homeomorphisms. The conditions under which a dynamical system is periodic are obtained. The explicit expressions for periods are found. The set of attainability is constructed. Also, there is considered a generalization of the control problem consisting in generating of a discrete dynamics which induces double circle rotations. In this case, the problem of finding periodic trajectories is posed.