Nonlinear Physiologically Structured Population Models with Two Internal Variables
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Nonlinear Physiologically Structured Population Models with Two Internal Variables Hao Kang1 · Xi Huo1 · Shigui Ruan1 Received: 30 December 2019 / Accepted: 8 June 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract First-order hyperbolic partial differential equations with two internal variables have been used to model biological and epidemiological problems with two physiological structures, such as chronological age and infection age in epidemic models, age and another physiological character (maturation, size, stage) in population models, and cell-age and molecular content (cyclin content, maturity level, plasmid copies, telomere length) in cell population models. In this paper, we study nonlinear double physiologically structured population models with two internal variables by applying integrated semigroup theory and non-densely defined operators. We consider first a semilinear model and then a nonlinear model, use the method of characteristic lines to find the resolvent of the infinitesimal generator and the variation of constant formula, apply Krasnoselskii’s fixed point theorem to obtain the existence of a steady state, and study the stability of the steady state by estimating the essential growth bound of the semigroup. Finally, we generalize the techniques to investigate a nonlinear age-size structured model with size-dependent growth rate. Keywords Physiological structure · Cauchy problem with non-dense domain · Integrated semigroups · Infinitesimal generator · Spectrum theory · Stability Mathematics Subject Classification 35L04 · 92D25 · 47A10
Communicated by Anthony Bloch. Research was partially supported by National Science Foundation (DMS-1853622).
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Shigui Ruan [email protected] Department of Mathematics, University of Miami, Coral Gables, FL 33146, USA
123
Journal of Nonlinear Science
Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Integrated Semigroup Setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Semilinear Double Physiologically Structured Models . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Existence of Nontrivial Steady States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Nonlinear Double Physiologically Structured Models . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Existence of Nontrivial Steady States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Age-Size Structured Models with Size-Dependent Growth Rate . . . . . . . . . . . . . . . . . . . . 6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A Appendix: Positive Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . .
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