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In this paper, we consider systems with one resource, which can be in several states. The states differ significantly in their processes of mortality, reproduction and mutual influence. For instance, infected elements can have a higher mortality rate than healthy and recovered ones. For cyclic models, in which the initial state of the system coincides with the final state, balance relations are found. They represent a system with functional operators with shift and integrals with degenerate kernels. Modified Fredholm method, proposed in previous works to solve the integral equations of the second type with degenerate kernels and shifts, is applied. Equilibrium position of a system with a three-state resource is found.

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