Abstract
We continue studying systems whose state depends on time and whose resources are renewably based on functional operators with shift. In previous articles, we considered the term which described results of reproductive processes as a linear expression or as a shift summand. In this work, the reproductive term is represented using an integral with a degenerate kernel. A cyclic model of evolution of the system with a renewable resource is developed. We propose a method for solving the balance equation and we determine an equilibrium state of the system. Having applied this model, we can investigate problems of natural systems and their resource production.
Highlights
We continue studying systems whose state depends on time and whose resources are renewably based on functional operators with shift
We considered the term which described results of reproductive processes as a linear expression or as a shift summand
We propose a method for solving the balance equation and we determine an equilibrium state of the system
Summary
A great number of works are dedicated to systems with renewable resources, for example [1,2,3]. Separation of the individual parameter and the group parameter, and discretization of time lead us to functional equations with shift [4,5,6]. An essential difference between this work and [7] is a description of the reproductive process as an integral term with degenerate kernel. The balance equation of the cyclic model represents a lineal functional operator with shift and an integral term with degenerate kernel. For solving the functional balance equation with shift and the integral term with degenerate kernel in the weighted Hölder spaces the Fredholm method [8,9] is proposed. 2. Cyclic Model with the Reproduction Term Taking as Integrals with Degenerate Kernels.
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