##
**Global dynamical properties of Lotka-Volterra systems.**
*(English)*
Zbl 0844.34006

Singapore: World Scientific. xii, 302 p. (1996).

This is an excellent and quite substantial book on global dynamical properties of Lotka-Volterra systems, such as persistence or permanence, global stability of nonnegative equilibrium points, periodic and chaotic motions – even if it does not claim for completeness: As the author states in the preface, “it is impossible to cover all of the topics in a book of this size and the selection of the material depends on my personal interests. Except well-known basic results on ordinary differential equations, the content of this book is my own and my collaborators’ recent work.” The book consists of four parts. Part I (15 pp.) is a quite short introduction to single-species and multiple-species growth models. Part II (74 pp.) is devoted to “standard” Lotka-Volterra systems \(\dot x_i = x_i (b_i + \sum^n_{j = 1} a_{ij} x_j)\), \(i = 1, \dots, n\). Lyapunov functions and complementarity theory are used to establish the existence of a globally stable nonnegative equilibrium point of the system. By using graph theory, several structures which ensure the existence of a globally stable equilibrium are given. The possibility of predator-mediated coexistence is analyzed. By the introduction of predators into the system, the species’ coexistence at a stable equilibrium point or in a periodic or chaotic motion is shown to be possible, even if all the species without the predators cannot coexist. Part III (116 pp.) discusses Lotka-Volterra systems with diffusion, i.e. systems of the type
\[
\begin{aligned} \dot x_i & = x_i \left( b_i + \sum^n_{j = 1} a_{ij} x_j \right) + D_i (y_i - x_i), \\ \dot y_i & = y_i \left( \overline b_i + \sum^n_{j = 1} \overline a_{ij} y_i \right) + \overline D_i (x_i - y_i), \quad i = 1, \dots, n. \end{aligned}
\]
Here the total system is partitioned into several subsystems (called patches), and the biological species are allowed to disperse among the patches. Interesting questions are as follows: (i) can the total system continue to be globally stable under species diffusion when each isolated patch is globally stable? (ii) conversely, can the system be made globally stable or persistent when each patch is unstable or not persistent? The final Part IV (58 pp.) deals with global dynamical properties of Lotka-Volterra systems with time delay, of the type
\[
\dot x_i = x_i \left( b_i + \sum^n_{j = 1} a_{ij} x_j + \sum^n_{j = 1} \gamma_{ij} \int^t_{- \infty} F _{ij} (t - \tau) x_j (r)dr \right) + d_i + D_i (y_i - x_i),
\]

\[ \dot y_i = y_i \left( \overline b_i + \sum^n_{j = 1} \overline a_{ij} y_j + \sum^n_{j = 1} \overline \gamma_{ij} \int^t_{- \infty} \overline F_{ij} (t - \tau) y_j (r)dr \right) + \overline d_i + \overline D_i (x_i - y_i). \] The effects of distributed and discrete delays on global stability and persistence are discussed. The existence of a nonnegative and globally stable equilibrium point is shown by applying homotopy techniques and Lyapunov functions. The book is completed by a bibliography of about 250 items and by four appendices which recall certain basic results on fundamental properties of solutions of ODE, stability of equilibria, the Hopf bifurcation theorem and on the linear complementarity problem in the theory of mathematical programming.

\[ \dot y_i = y_i \left( \overline b_i + \sum^n_{j = 1} \overline a_{ij} y_j + \sum^n_{j = 1} \overline \gamma_{ij} \int^t_{- \infty} \overline F_{ij} (t - \tau) y_j (r)dr \right) + \overline d_i + \overline D_i (x_i - y_i). \] The effects of distributed and discrete delays on global stability and persistence are discussed. The existence of a nonnegative and globally stable equilibrium point is shown by applying homotopy techniques and Lyapunov functions. The book is completed by a bibliography of about 250 items and by four appendices which recall certain basic results on fundamental properties of solutions of ODE, stability of equilibria, the Hopf bifurcation theorem and on the linear complementarity problem in the theory of mathematical programming.

Reviewer: W.Müller (Berlin)

### MSC:

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

92D25 | Population dynamics (general) |

34K18 | Bifurcation theory of functional-differential equations |

34K20 | Stability theory of functional-differential equations |

34D05 | Asymptotic properties of solutions to ordinary differential equations |

34D20 | Stability of solutions to ordinary differential equations |