Dynamics of One-Dimensional Maps and Gurtin–Maccamy’s Population Model. Part I. Asymptotically Constant Solutions

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] to the asymptotic behavior of solutions of the Gurtin–MacCamy’s system. According to 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.