This textbook provides an introduction to the mathematical methods used to analyse deterministic models in life sciences, including population dynamics, epidemiology and ecology. The book covers both discrete and continuous models.

The presentation emphasises the solvability of the equations appearing in the mathematical modelling of natural phenomena and, in the absence of solutions, the analysis of their relevant properties. Of particular interest are methods that allow for determining the long-term behaviour of solutions. Thus, the book covers a range of techniques, from the classical Lyapunov theorems and positivity methods based on the PerronFrobenius theorem, to the more modern monotone dynamical system approach. The book offers a comprehensive presentation of the Lyapunov theory, including the inverse Lyapunov theorems with applications to perturbed equations and Vidyasagar theorem. Furthermore, it provides a coherent presentation of the foundations of the theory of monotone dynamical systems with its applications to epidemiological models. Another feature of the book is the derivation of the McKendrickvon Foerster equation from the discrete Leslie model and the analysis of the long-term behaviour of its solutions.

Designed for upper undergraduate courses and beyond, this textbook is written for students and researchers looking to master the mathematics of the tools commonly used to analyse life science models. It therefore goes somewhat deeper into mathematics than typical books at this level but should be accessible to anyone with a good command of calculus with elements of real and complex analysis and linear algebra; the necessary concepts are collected in the appendices.

An introduction to the mathematical models commonly used in life sciences Provides foundations of the theory of monotone dynamical systems and its applications to epidemiology Covers both discrete and continuous models

Autorentext

Jacek Banasiak was born on 15th March 1959 in ód, Poland. The current holder of the DSI/NRF SARChI Chair in Mathematical Models and Methods in Biosciences and Bioengineering at the University of Pretoria, he is also a research professor at the ód University of Technology in Poland and a visiting professor at the University of Strathclyde in Glasgow, Scotland. His research interests are nonlocal integro-differential models in kinetic theory, mathematical biology and fragmentation-coagulation theory, asymptotic analysis of multiple scale problems and epidemiological modelling. Up to this point in his career, he has authored/co-authored seven research monographs and over 145 refereed research papers. He is a recipient of the South African Mathematical Society Award for Research Distinction (2012), Cross of Merit (Silver) of the Republic of Poland (2013), 1st prize for the best paper in applied mathematics from the Centre for Applications of Mathematics of Gdask (2014), as well as the Minister of Science and Education of the Republic of Poland Award for Scientific Achievements in 2022. He is the Editor-in-Chief of Afrika Matematika (Springer journal) and a member of editorial boards of several other journals.

Inhalt

1 Mathematical modelling.- Part I Unstructured Models.- 2 Models with discrete time.- 3 Models with continuous time.- 4 Qualitative theory for a single equation.- Part II Models with discrete structure.- 5 Linear models with discrete structure.- 6 Continuous time non-linear models for interacting species and age-structured populations.- 7 Discrete time non-linear models for interacting species and structured populations.- 8 Positivity in natural science models.- Part III Models with continuous age structure.- 9 McKendrickvon Foerster model.- 10 Basic nonlinear models. Part IV Appendices.- A Spaces, norms and order.- B Differential equations and systems.- C Linear algebra tools.- D First order partial differential equations and the method of characteristics.