Investigation of solutions to one family of mathematical models of living systems

We consider a family of integral equations used as models of some living systems. We prove that an integral equation is reducible to the equivalent Cauchy problem for a non-autonomous differential equation with point or distributed delay dependently on the choice of the survival function of system elements. We also study the issues of the existence, uniqueness, nonnegativity, and continuability of solutions. We describe all stationary solutions and obtain sufficient conditions for their asymptotic stability. We have found sufficient conditions for the existence of a limit of solutions on infinity and present an example of equations where the rate of generation of elements of living systems is described by a unimodal function (namely, the Hill function).