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**Reaction-diffusion equations and their applications to biology.**
*(English)*
Zbl 0602.92001

London etc.: Academic Press (Harcourt Brace Jovanovich, Publishers). IX, 277 p. $ 65.00 (1986).

Application of mathematics to modelling of deterministic aspects of biological phenomena means to some part application of differential equations. The partial reaction-diffusion (or production-diffusion) differential equations with their different versions play an important role in biological problems where physical and chemical processes constitute the essential properties of the dynamics of the biological systems in question, and where it is justified to neglect stochastic effects, as it is reasoned in chapter 1 of this book.

The general reaction-diffusion system, proves of existence and uniqueness of solutions, bounds for solutions of intractable systems, and numerical techniques for the computation of solutions are dealt with in the central chapter 5.

Steady and periodic solutions of intractable systems and applications to pattern formation are discussed in chapter 6 by using bifurcation theory. The equations for oscillatory systems, for systems with fast and slow variables, and for systems with small diffusion coefficients with their biological applications are considered in the last 3 chapters.

In chapter 2, an introduction to techniques of ordinary differential equations is given, and in the following one these equations as a particular class of reaction-diffusion equations without diffusion term are applied to systems without diffusional effects which are governed by conservative kinetic equations. Such systems play a role in epidemiology.

In chapter 4 the scalar reaction-diffusion equation is treated together with some applications as models for biological systems (wave of advance of an advantageous gene in a population; control of the spruce budworm of North America).

The general reaction-diffusion system, proves of existence and uniqueness of solutions, bounds for solutions of intractable systems, and numerical techniques for the computation of solutions are dealt with in the central chapter 5.

Steady and periodic solutions of intractable systems and applications to pattern formation are discussed in chapter 6 by using bifurcation theory. The equations for oscillatory systems, for systems with fast and slow variables, and for systems with small diffusion coefficients with their biological applications are considered in the last 3 chapters.

In chapter 2, an introduction to techniques of ordinary differential equations is given, and in the following one these equations as a particular class of reaction-diffusion equations without diffusion term are applied to systems without diffusional effects which are governed by conservative kinetic equations. Such systems play a role in epidemiology.

In chapter 4 the scalar reaction-diffusion equation is treated together with some applications as models for biological systems (wave of advance of an advantageous gene in a population; control of the spruce budworm of North America).

Reviewer: J.Peil

### MSC:

92-02 | Research exposition (monographs, survey articles) pertaining to biology |

93C20 | Control/observation systems governed by partial differential equations |

35K99 | Parabolic equations and parabolic systems |

35B10 | Periodic solutions to PDEs |

35B32 | Bifurcations in context of PDEs |

93C15 | Control/observation systems governed by ordinary differential equations |

92D40 | Ecology |

92D25 | Population dynamics (general) |

92Exx | Chemistry |