Topics in mathematical biology.

*(English)*Zbl 1382.92003
Lecture Notes on Mathematical Modelling in the Life Sciences. Cham: Springer (ISBN 978-3-319-65620-5/pbk; 978-3-319-65621-2/ebook). xiv, 353 p. (2017).

From the introduction: “Chapter 1 is about diffusive coupling of two dynamical systems or compartments. I compare the Lie-Trotter approach of splitting in time (as used in advanced numerical analysis) to diffusive coupling and find similar limiting equations. I restrict mostly to two compartments because that case shows already most phenomena. Also, I do not want to get into oscillator chains, lattice differential equations, and other complicated structures. Then I take the view that a reaction diffusion equation can be obtained by coupling a reaction equation to a (linear) diffusion equation, thus having a first glance at modeling spread in space.

“Then I enter quiescent phases (no action at all in the second compartment) and I present results on stability and bifurcations of stationary points and periodic orbits. I explain the Turing bifurcation and I show various related results on matrix stability. I show a general principle, extending the little matrix example above, saying that heterogeneity stabilizes. Finally, similar results on quiescent phases are shown for discrete time systems.

“The link to the next chapter (Chap. 2), on delay equations, is the observation that other exit distributions than the exponential distribution lead to delay equations. In particular, if the exponential distribution is combined with the Dirac distribution then some classes of vector-valued delay equations arise. For these I get stability and bifurcation results similar to those for the ordinary differential equations case.

“Another approach to delay equations are one-dimensional convection equations and age structure models. To establish this connection, I have included a section on demography. The reduction of demographic models to delay equations is described in detail, also to state-dependent delay equations.

“The third chapter has some algebraic flavor. Lotka-Volterra systems have played a great role in the history of mathematical biology although their general mathematical properties are not well understood. Lotka-Volterra systems and quadratic replicator equations are equivalent. The latter are more symmetric, live on a compact state space, and have been used to describe social behavior. I present first various types of such equations that have been used as models in Mendelian genetics and then give a short account on game theory to support following results on Nash equilibria and evolutionary stable strategies.

“The next chapter (Chap. 4) covers ecological models, first one species models, then various types of predator-prey models where the results from the first chapter are used. In particular, there are strongly stable and also excitable predator-prey systems. The fifth chapter is about homogeneous systems preparing for epidemic models in the sixth chapter. Epidemic models allow ways of quiescence like quarantine as well as excitability.

“The seventh chapter is about spread in space. Starting from the diffusion equation, I make a transition to correlated random walks, Cattaneo systems, Langevin (Kramers) equations, and some very specific walks that are mimicking the behavior of certain bacteria and beetles. In each case the ultimate goal is to derive a diffusion equation where the diffusion coefficient contains information on the specific properties of the given walk. The eighth chapter deals with traveling front solutions of reaction diffusion equations and epidemic models.”

This reviewer has three comments: The death of K. P. Hadeler is a significant loss to mathematical biology specialists. The choice of topics is excellent. This book would benefit from a section on networks to complete Section 8.4.

“Then I enter quiescent phases (no action at all in the second compartment) and I present results on stability and bifurcations of stationary points and periodic orbits. I explain the Turing bifurcation and I show various related results on matrix stability. I show a general principle, extending the little matrix example above, saying that heterogeneity stabilizes. Finally, similar results on quiescent phases are shown for discrete time systems.

“The link to the next chapter (Chap. 2), on delay equations, is the observation that other exit distributions than the exponential distribution lead to delay equations. In particular, if the exponential distribution is combined with the Dirac distribution then some classes of vector-valued delay equations arise. For these I get stability and bifurcation results similar to those for the ordinary differential equations case.

“Another approach to delay equations are one-dimensional convection equations and age structure models. To establish this connection, I have included a section on demography. The reduction of demographic models to delay equations is described in detail, also to state-dependent delay equations.

“The third chapter has some algebraic flavor. Lotka-Volterra systems have played a great role in the history of mathematical biology although their general mathematical properties are not well understood. Lotka-Volterra systems and quadratic replicator equations are equivalent. The latter are more symmetric, live on a compact state space, and have been used to describe social behavior. I present first various types of such equations that have been used as models in Mendelian genetics and then give a short account on game theory to support following results on Nash equilibria and evolutionary stable strategies.

“The next chapter (Chap. 4) covers ecological models, first one species models, then various types of predator-prey models where the results from the first chapter are used. In particular, there are strongly stable and also excitable predator-prey systems. The fifth chapter is about homogeneous systems preparing for epidemic models in the sixth chapter. Epidemic models allow ways of quiescence like quarantine as well as excitability.

“The seventh chapter is about spread in space. Starting from the diffusion equation, I make a transition to correlated random walks, Cattaneo systems, Langevin (Kramers) equations, and some very specific walks that are mimicking the behavior of certain bacteria and beetles. In each case the ultimate goal is to derive a diffusion equation where the diffusion coefficient contains information on the specific properties of the given walk. The eighth chapter deals with traveling front solutions of reaction diffusion equations and epidemic models.”

This reviewer has three comments: The death of K. P. Hadeler is a significant loss to mathematical biology specialists. The choice of topics is excellent. This book would benefit from a section on networks to complete Section 8.4.

Reviewer: E. Ahmed (Mansoura)