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 establish a reduction of the integral equation to the equivalent Cauchy problem for a non-autonomous differential equation with a point or distributed delay dependent on a choice of a elements survival function. We also investigate questions of existence, uniqueness, nonnegativity and extendibility of solutions. We describe all stationary solutions and obtain sufficient conditions of their asymptotic stability. We have found sufficient conditions of existence of a limit of solutions on infinity and present an example of a research of equations in which the production speed of elements of living systems is described by means of unimodal function (Hill function).