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

Franco Herrera; Sergei Trofimchuk

doi:10.3842/umzh.v75i12.7678

Franco Herrera Instituto de Matemáticas, Universidad de Talca, Chile
Sergei Trofimchuk Instituto de Matemáticas, Universidad de Talca, Chile

DOI: https://doi.org/10.3842/umzh.v75i12.7678

Keywords: Gurtin-MacCamy's population model, Volterra unimodal integral equation, asymptotic convergence

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.

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