Growth bound and threshold dynamic for nonautonomous nondensely defined evolution problems

• Published in: Journal of mathematical biology Vol. 87; no. 2; p. 32
• Main Authors: Djidjou-Demasse, Ramsès, Goudiaby, Ibou, Seydi, Ousmane
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.08.2023; Springer Nature B.V; Springer
• Subjects: Analysis of PDEs; Applications of Mathematics; Banach spaces; Dynamical Systems; Epidemiology; Evolution; Linear operators; Malaria; Mathematical and Computational Biology; Mathematics; Mathematics and Statistics; Operators (mathematics); Pathogens; Population genetics; Population growth; Topology; Threshold dynamics; Evolutionary systems; 47N60; 92D25; 37B55; 47D62; Reproduction number; Growth bound; 34K20; Reproduction number Growth bound Threshold dynamics evolutionary systems Mathematics Subject Classification 34K20 37B55 47D62 47N60 92D25; evolutionary systems
• ISSN: 0303-6812; 1432-1416; 1432-1416
• DOI: 10.1007/s00285-023-01966-w

We propose a general framework for simultaneously calculating the threshold value for population growth and determining the sign of the growth bound of the evolution family generated by the problem below d v ( t ) d t = A v ( t ) + F ( t ) v ( t ) - V ( t ) v ( t ) , where A : D ( A ) ⊂ X → X is a Hille–Yosida linear operator (possibly unbounded, non-densely defined) on a Banach space ( X , ‖ · ‖ ) , and the maps t ∈ R ↦ V ( t ) ∈ L ( X 0 , X ) , t ∈ R ↦ F ( t ) ∈ L ( X 0 , X ) are p -periodic in time and continuous in the operator norm topology. We give applications of our approach for two general examples of an age-structured model, and a delay differential system. Other examples concern the dynamics of a nonlocal problem arising in population genetics and the dynamics of a structured human-vector malaria model.