Dynamics of one-dimensional maps and Gurtin–Maccamy's population model. Part I. Asymptotically constant solutions

Abstract

UDC 517.9 Motivated by the recent work by  Ma and Magal [Proc. Amer. Math. Soc. (2021); https://doi.org/10.1090/proc/15629] on  the global stability property of  the Gurtin–MacCamy's population model, we consider a family of scalar nonlinear convolution equations with unimodal nonlinearities.  In particular, we  relate the Ivanov and Sharkovsky analysis  of singularly perturbed delay differential equations in [https://doi.org/10.1007/978-3-642-61243-5_5] with the asymptotic behavior of solutions of the  Gurtin–MacCamy's system. According the classification proposed in  [https://doi.org/10.1007/978-3-642-61243-5_5], we can distinguish three fundamental  kinds of continuous solutions of our equations, namely, solutions of the asymptotically constant type, relaxation type  and turbulent type. We present various conditions assuring that all solutions belong to the first of these three classes. In the setting of unimodal convolution  equations, these conditions suggest a generalized version of the famous Wright's conjecture.