##
**Stability and oscillations in delay differential equations of population dynamics.**
*(English)*
Zbl 0752.34039

Mathematics and its Applications (Dordrecht). 74. Dordrecht etc.: Kluwer Academic Publishers. xii, 501 p. (1992).

Although the study of functional differential equations has received a great deal of attention for the last thirty years, as yet there have been yet few expository books on the topic, and still fewer books not necessarily restricted to specialists of this field. The monograph under review fulfils this aim, and in our opinion represents a successful and motivating introduction to the subject, whilst still being of interest to specialists.

The book is divided into five chapters: all of them except chapter 2, which is a short one, devoted to the Hopf bifurcation theorem, are of comparable size. Each chapter is almost self-contained, with very few cross-references, which facilitates reading. Non classical mathematical notions (e.g. a fixed point theorem for sums of operators by Nashed and Wong) are stated wherever necessary.

The book considers three main aspects: stability, oscillations and population dynamics. Stability is the primary topic, while “oscillations [are studied] in order to make use of the knowledge of oscillatory solutions in stability investigations”; and population dynamics are treated as a ground for applying methods and results presented elsewhere in the book. This latter choice is well-justified both by the importance that FDE’s have gained over the last few years in population dynamics and the personal production of the author, who is a recognized specialist on the subject. Throughout the book, the author points out ideas and methods, voluntarily avoiding excessive generalizations, even restricting such or such a proof to a particular case. Thus, only discrete delays or distributed delays with an integrable kernel are considered. In the same way, when nonlinear systems are studied, only autonomous ones are considered, and only systems with one positive equilibrium. Most statements are formulated in a way accessible even to non specialists. As a result, the book should be a useful tool for applied research.

Amongst the methods provided, the most complete treatment is for the Lyapunov method. The author presents many examples, which cover known models, delay and neutral equations, predator-prey type equations, etc., where Lyapunov functions are built and used. Other methods are presented too, including Routh-Hurwitz criteria for stability, the Hopf bifurcation theorem. Special features such as cooperative systems, which lead to monotonically increasing dynamical systems, are dealt with in detail.

Oscillation theory for FDE’s is the second important issue of the book. The latest developments on this topic, some of which are contributions by the author, are included. A comparison method which has been extended recently to various classes of FDE’s (vectorial, neutral, etc.) is presented in detail. The book gives an account of the developments on the subject over the last few years: FDE’s with piecewise constant arguments, impulses, stability switches, neutral equations, etc. Lastly, each chapter is followed by numerous exercises which either apply the notions treated in the chapter, or aim at widening the scope of the text.

The book is divided into five chapters: all of them except chapter 2, which is a short one, devoted to the Hopf bifurcation theorem, are of comparable size. Each chapter is almost self-contained, with very few cross-references, which facilitates reading. Non classical mathematical notions (e.g. a fixed point theorem for sums of operators by Nashed and Wong) are stated wherever necessary.

The book considers three main aspects: stability, oscillations and population dynamics. Stability is the primary topic, while “oscillations [are studied] in order to make use of the knowledge of oscillatory solutions in stability investigations”; and population dynamics are treated as a ground for applying methods and results presented elsewhere in the book. This latter choice is well-justified both by the importance that FDE’s have gained over the last few years in population dynamics and the personal production of the author, who is a recognized specialist on the subject. Throughout the book, the author points out ideas and methods, voluntarily avoiding excessive generalizations, even restricting such or such a proof to a particular case. Thus, only discrete delays or distributed delays with an integrable kernel are considered. In the same way, when nonlinear systems are studied, only autonomous ones are considered, and only systems with one positive equilibrium. Most statements are formulated in a way accessible even to non specialists. As a result, the book should be a useful tool for applied research.

Amongst the methods provided, the most complete treatment is for the Lyapunov method. The author presents many examples, which cover known models, delay and neutral equations, predator-prey type equations, etc., where Lyapunov functions are built and used. Other methods are presented too, including Routh-Hurwitz criteria for stability, the Hopf bifurcation theorem. Special features such as cooperative systems, which lead to monotonically increasing dynamical systems, are dealt with in detail.

Oscillation theory for FDE’s is the second important issue of the book. The latest developments on this topic, some of which are contributions by the author, are included. A comparison method which has been extended recently to various classes of FDE’s (vectorial, neutral, etc.) is presented in detail. The book gives an account of the developments on the subject over the last few years: FDE’s with piecewise constant arguments, impulses, stability switches, neutral equations, etc. Lastly, each chapter is followed by numerous exercises which either apply the notions treated in the chapter, or aim at widening the scope of the text.

Reviewer: O.Arino (Pau)

### MSC:

34K20 | Stability theory of functional-differential equations |

92D40 | Ecology |

34K99 | Functional-differential equations (including equations with delayed, advanced or state-dependent argument) |

34K40 | Neutral functional-differential equations |

34-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to ordinary differential equations |

34C10 | Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations |