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**Delay differential equations: with applications in population dynamics.**
*(English)*
Zbl 0777.34002

Mathematics in Science and Engineering. 191. Boston, MA: Academic Press, Inc.. xii, 398 p. (1993).

This is an excellent and clearly written book on the qualitative theory of delay differential equations (DDEs) and on applications of results of this theory in the investigation of global qualitative properties of population dynamics models.

The book is divided into two main parts encompassing nine chapters. The first three chapters constitute part I, devoted to the theory of DDEs. In the first chapter, various examples of DDEs arising in real life problems are presented, and the possible effects of time delays on some of these systems are shortly discussed. Examples are presented which show that small delays can have large influence on the global behavior of the system. Chapter 2 which, as the author states, “is a small subset of J. K. Hale’s book [Theory of functional differential equations (1977; Zbl 0352.34001)]”, contains the basic theory of delay DDEs. Since the local stability of a steady state in a DDE is determined by the locations of the roots of its characteristic equation, in chapter 3 methods developed by several researchers (H. I. Freedman and the author; K. L. Cooke and P. van den Driessche, revised by F. G. Boese; H. L. Smith; D. M. Fargue; H.-O. Walther; V. B. Kolmanovskij and V. R. Nosov; H. I. Freeman and K. Gopalsamy) in order to investigate the characteristic equations of various types of DDEs are reported.

Part II consists of the remaining six chapters. It deals with the analyses of various delayed population models. Chapter 4 focuses on the global stability analyses of positive steady states of single species models of the type \(\dot x=f(x_ t)-g(x(t))\). Chapter 5 mainly concerns the oscillatory aspects of the dynamics in these models (including transition to chaos). Other kinds of single species models are also discussed. Chapter 6 contains the global stability analyses of various multispecies models of the type \(\dot x_ i(t)=x_ i(t)F_ i(t,x_ t)\), \(i=1,\dots,n\) (competition, predator-prey or cooperation models). Chapter 7 covers the oscillatory aspects of the dynamics in these models. Global existence of periodic solutions in both autonomous and periodic systems are established. Chapter 8 presents some recent results on permanence theory (persistence, permanent coexistence, cooperativeness) of delayed systems. Chapter 9 documents some initial attempts on the study of several neutral delay models. Each chapter is complemented by a “remarks and open problems” section. The book ends with an extensive up-to-date bibliography of about 370 items representing the actual “state of the art” in this field and with an effective algorithm (presented as a VAX FORTRAN 77 code written by E. Lo and Z. Jackiewicz from Arizona State University) for performing numerical simulations of delayed models.

The book is divided into two main parts encompassing nine chapters. The first three chapters constitute part I, devoted to the theory of DDEs. In the first chapter, various examples of DDEs arising in real life problems are presented, and the possible effects of time delays on some of these systems are shortly discussed. Examples are presented which show that small delays can have large influence on the global behavior of the system. Chapter 2 which, as the author states, “is a small subset of J. K. Hale’s book [Theory of functional differential equations (1977; Zbl 0352.34001)]”, contains the basic theory of delay DDEs. Since the local stability of a steady state in a DDE is determined by the locations of the roots of its characteristic equation, in chapter 3 methods developed by several researchers (H. I. Freedman and the author; K. L. Cooke and P. van den Driessche, revised by F. G. Boese; H. L. Smith; D. M. Fargue; H.-O. Walther; V. B. Kolmanovskij and V. R. Nosov; H. I. Freeman and K. Gopalsamy) in order to investigate the characteristic equations of various types of DDEs are reported.

Part II consists of the remaining six chapters. It deals with the analyses of various delayed population models. Chapter 4 focuses on the global stability analyses of positive steady states of single species models of the type \(\dot x=f(x_ t)-g(x(t))\). Chapter 5 mainly concerns the oscillatory aspects of the dynamics in these models (including transition to chaos). Other kinds of single species models are also discussed. Chapter 6 contains the global stability analyses of various multispecies models of the type \(\dot x_ i(t)=x_ i(t)F_ i(t,x_ t)\), \(i=1,\dots,n\) (competition, predator-prey or cooperation models). Chapter 7 covers the oscillatory aspects of the dynamics in these models. Global existence of periodic solutions in both autonomous and periodic systems are established. Chapter 8 presents some recent results on permanence theory (persistence, permanent coexistence, cooperativeness) of delayed systems. Chapter 9 documents some initial attempts on the study of several neutral delay models. Each chapter is complemented by a “remarks and open problems” section. The book ends with an extensive up-to-date bibliography of about 370 items representing the actual “state of the art” in this field and with an effective algorithm (presented as a VAX FORTRAN 77 code written by E. Lo and Z. Jackiewicz from Arizona State University) for performing numerical simulations of delayed models.

Reviewer: W.Müller (Berlin)

### MSC:

34-02 | Research exposition (monographs, survey articles) pertaining to ordinary differential equations |

92D25 | Population dynamics (general) |

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

34K20 | Stability theory of functional-differential equations |

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

34K05 | General theory of functional-differential equations |

34-04 | Software, source code, etc. for problems pertaining to ordinary differential equations |