The paper deals with a system of two nonlinear integro-differential equations of first order. Such a construction belongs to the so-called logistic equations, in this connection see the books by R. E. Gaines and J. L. Mawhin [Coincidence degree and nonlinear differential equations (1977; Zbl 0339.47031)] and by Y. Kuang [Delay differential equations with applications in population dynamics (1993; Zbl 0777.34002)]. Logistic equations describe the temporal evolution of a single species of population in a constant environment. Sufficient conditions are given for the existence of at least one positive periodic solution for the system under consideration and for the global asymptotic stability of such a solution.
The proofs of these results are based on the continuation theorem for the existence of at least one solution of the operator equation with Fredholm operator of index zero (in this connection see the above book by R. E. Gaines and J. L. Mawhin) and on the Lyapunov functional, respectively.