Volterra-Hamilton models in the ecology and evolution of colonial organisms.

*(English)*Zbl 0930.92031
Series in Mathematical Biology and Medicine. 2. Singapore: World Scientific Publishing. xxiv, 201 p. (1996).

This book presents a modelling methodology developed by the authors for the study of colonial organisms. By a “colonial” organism the authors mean a colony (or a “superorganism”, such as the Portuguese Man-of-War) consisting of “polyp persons” with adaptations enabling them to perform specialized tasks for the colony as a whole. Their goal is “to model the various evolutionary constraints on morphological diversity and its proliferation in colonial animals for the purposes of quantitative analysis”.

The starting points (Chapter 1) are logistic dynamics for population numbers and Gompertz dynamics for biomass. A generalization of Gompertz growth (utilizing second order differential equations) is introduced for individuals making up populations. Volterra’s production and real growth potential are also introduced. In Chapter 2 intra- and inter-specific competition are studied by means of a modification of classical models that utilize quadratic nonlinearities, derived by the authors’ method of “homogenization”. This is done to account for “cooperative social” interactions of higher order, which are asserted to arise from changes of adaptations in the timing and sequencing of growth and development of individuals (“heterochrony”). Energy is considered in Chapter 3, where the “general Maupertuis energy functional” and metric are introduced. From the calculus of variations, the authors obtain minimizing extremal curves (growth or production curves). After an example (to the Crown-of-Thorns Starfish) is given in Chapter 4, the authors study the differential geometry determined by the metric and the stability of the geodesic curves.

The final Chapter 6 contains an introduction to their dynamical theory of heterochrony in the evolution of colonial invertebrates. The focus is on changes in timing and sequencing of growth and development that produce changes in the castes of a \(K\)-selected colony. Their results combine ecological theory with Wilson’s ergonomic theory and the “principle of division of labour”. Several appendices provide background material, as well as related material on the use of fuzzy differential inclusions to model stochastic effects.

The starting points (Chapter 1) are logistic dynamics for population numbers and Gompertz dynamics for biomass. A generalization of Gompertz growth (utilizing second order differential equations) is introduced for individuals making up populations. Volterra’s production and real growth potential are also introduced. In Chapter 2 intra- and inter-specific competition are studied by means of a modification of classical models that utilize quadratic nonlinearities, derived by the authors’ method of “homogenization”. This is done to account for “cooperative social” interactions of higher order, which are asserted to arise from changes of adaptations in the timing and sequencing of growth and development of individuals (“heterochrony”). Energy is considered in Chapter 3, where the “general Maupertuis energy functional” and metric are introduced. From the calculus of variations, the authors obtain minimizing extremal curves (growth or production curves). After an example (to the Crown-of-Thorns Starfish) is given in Chapter 4, the authors study the differential geometry determined by the metric and the stability of the geodesic curves.

The final Chapter 6 contains an introduction to their dynamical theory of heterochrony in the evolution of colonial invertebrates. The focus is on changes in timing and sequencing of growth and development that produce changes in the castes of a \(K\)-selected colony. Their results combine ecological theory with Wilson’s ergonomic theory and the “principle of division of labour”. Several appendices provide background material, as well as related material on the use of fuzzy differential inclusions to model stochastic effects.

Reviewer: J.M.Cushing (Tucson)