Dynamics of one-dimensional maps and Gurtin–Maccamy's population model. Part I. Asymptotically constant solutions
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 convolutio...
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Institute of Mathematics, NAS of Ukraine
2024
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| author | Herrera, Franco Trofimchuk, Sergei Herrera, Franco Trofimchuk, Sergei Трофимчук, Сергей |
| author_facet | Herrera, Franco Trofimchuk, Sergei Herrera, Franco Trofimchuk, Sergei Трофимчук, Сергей |
| author_sort | Herrera, Franco |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2024-06-19T00:34:58Z |
| description | 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. |
| doi_str_mv | 10.3842/umzh.v75i12.7678 |
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| institution | Ukrains’kyi Matematychnyi Zhurnal |
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| language | English |
| last_indexed | 2026-03-24T03:32:58Z |
| publishDate | 2024 |
| publisher | Institute of Mathematics, NAS of Ukraine |
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| spelling | umjimathkievua-article-76782024-06-19T00:34:58Z Dynamics of one-dimensional maps and Gurtin–Maccamy's population model. Part I. Asymptotically constant solutions Dynamics of one-dimensional maps and Gurtin–Maccamy's population model. Part I. Asymptotically constant solutions Herrera, Franco Trofimchuk, Sergei Herrera, Franco Trofimchuk, Sergei Трофимчук, Сергей Gurtin-MacCamy's population model, Volterra unimodal integral equation, asymptotic convergence 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. УДК 517.9 Динаміка одновимірних відображень та популяційна модель Гуртіна–Маккемі. Частина І. Асимптотично сталі розв’язки На основі нещодавньої роботи Ма та Магала [Proc. Amer. Math. Soc. (2021); https://doi.org/10.1090/proc/15629] щодо властивості глобальної стабільності популяційної моделі Гуртіна–Маккемі розглянуто сім’ю скалярних нелінійних рівнянь згортки з унімодальними нелінійностями. Зокрема, аналіз сингулярно збурениx диференціальних рівнянь із запізненням, запропонований  Івановим та Шарковським в [https://doi.org/10.1007/978-3-642-61243-5_5], пов’язано з асимптотикою розв’язків системи Гуртіна–Маккемі. За класифікацією, запропонованою в [https://doi.org/10.1007/978-3-642-61243-5_5],  можна виділити три основних типи неперервних розв’язків наших рівнянь, а саме: розв’язки асимптотично сталого типу,  релаксаційного та турбулентного типів. Наведено різні умови, які гарантують, що всі розв'язки належать до першого з трьох згаданих класів. У постановці унімодальних рівнянь згортки ці умови пропонують узагальнену версію відомої гіпотези Райта. Institute of Mathematics, NAS of Ukraine 2024-01-02 Article Article https://umj.imath.kiev.ua/index.php/umj/article/view/7678 10.3842/umzh.v75i12.7678 Ukrains’kyi Matematychnyi Zhurnal; Vol. 75 No. 12 (2023); 1635 - 1651 Український математичний журнал; Том 75 № 12 (2023); 1635 - 1651 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/7678/9932 Copyright (c) 2024 Сергій Іванович Трофімчук |
| spellingShingle | Herrera, Franco Trofimchuk, Sergei Herrera, Franco Trofimchuk, Sergei Трофимчук, Сергей Dynamics of one-dimensional maps and Gurtin–Maccamy's population model. Part I. Asymptotically constant solutions |
| title | Dynamics of one-dimensional maps and Gurtin–Maccamy's population model. Part I. Asymptotically constant solutions |
| title_alt | Dynamics of one-dimensional maps and Gurtin–Maccamy's population model. Part I. Asymptotically constant solutions |
| title_full | Dynamics of one-dimensional maps and Gurtin–Maccamy's population model. Part I. Asymptotically constant solutions |
| title_fullStr | Dynamics of one-dimensional maps and Gurtin–Maccamy's population model. Part I. Asymptotically constant solutions |
| title_full_unstemmed | Dynamics of one-dimensional maps and Gurtin–Maccamy's population model. Part I. Asymptotically constant solutions |
| title_short | Dynamics of one-dimensional maps and Gurtin–Maccamy's population model. Part I. Asymptotically constant solutions |
| title_sort | dynamics of one-dimensional maps and gurtin–maccamy's population model. part i. asymptotically constant solutions |
| topic_facet | Gurtin-MacCamy's population model Volterra unimodal integral equation asymptotic convergence |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/7678 |
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