This special issue (divided into 2 partsthe second part appearing in Vol. 11(1)) focusses on travelling waves. The phenomenon of travelling waves is an ubiquitous example of spatio-temporal patterning. It is a crucial process in that it describes how a system goes from one state to another. The contributions to this special issue illustrate how travelling waves appear in diverse areas of science: the papers by Honoso, Lewis and Schmitz, and to some extent, Sherratt, consider travelling waves in ecology, while the paper by Cook considers how waves of organisation can lead to aggregations in animal populations. Chaplain and Orme, and Dale et al. consider examples in medicine, specifically, tumour growth and wound healing, respectively. The Metcalf et al. paper concerns travelling waves in chemistry. The papers by Cook, Kulesa and Murray, and to some extent, Sherratt, consider applications in embryology and development. The papers by Hadeler, and Sanchez-Garduno et al. address important technical aspects that underpin much of the analysis of travelling wave systems.
The paper by Chaplain and Orme (pp. 147-170) investigates tumour-induced angiogenesis and tumour invasion. A simplified model of the former, in the form of a single, nonlinear partial differential equation, is shown to exhibit travelling wave behaviour of capillary outgrowth. The latter is modelled by a coupled system of nonlinear partial differential equations, which is shown to exhibit an advancing front of invasive proliferating tumour cells with the collapse of the vasculature in its wake.
Cook's paper (pp. 171-203) focusses on how individuals in a population move and reorient and derives a partial differental equation model to describe the evolution of orientation tensors in space and time. Application to cell-cell interactions show how waves of cell alignment can propagate through populations of fibroblasts, consistent with experimental observation. A further application to animal herds and fish schools shows how aggregations can result from such a model.
Dale et al. (pp. 205-222) consider three problems in wound healing that can be addressed using travelling wave analysis, namely, factors which affect the speed of healing in corneal surface wounds, the quality of healing in adult dermal wounds, and abnormal healing in fibrocontractive diseases. The paper reviews recent results obtained by the authors.
Hadeler's paper (pp. 223-233) replaces the usual Brownian motion approximation in the scalar reaction diffusion equation by a correlated random walk. Exact conditions for the existence of travelling waves are found.
Hosono's paper (pp. 235-257) investigates travelling wave solutions for a diffusive Lotka-Volterra competition model in which the stronger competitor diffuses. Using geometric ideas it is shown that travelling wave solutions exist above a minimum wavespeed, which is determined as a function of the parameters of the model.
The paper by Kulesa and Murray (pp. 259-280) presents a model for the spatio-temporal wave initiation of tooth primordia in the alligator jaw. A key element in this model is the role that domain growth plays in determining the patterning sequence. A number of experimentally testable predictions are made from the model.
The paper by Lewis and Schmitz (Vol. 11(1)) investigates travelling waves in a population which has two separate states, namely, mobile and stationary. The travelling wave analysis of this system reveals that the minimum wavespeed is one half of that in the corresponding Fisher equation. This has obvious implications in measuring population parameters from field data.
Metcalf et al. (Vol. 11(1)) consider interaction of Belousov-Zhabotinsky waves across a semi-permeable membrane. By analysing the strength of coupling and the distance between the waves when coupling is switched on, it is shown that the system can exhibit self-initiation of waves or can cause waves to annihilate.
Sanchez-Garduno et al. (Vol. 11(1)) review some results on travelling wave solutions of one-dimensional reaction-diffusion equations with non-constant diffusion coefficients. It is shown that one can obtain sharp type solutions, oscillatory solutions and aggregative solutions.
Sherratt's paper (Vol. 11(1)) investigates travelling waves in Lambda-omega systems and illustrates that solutions can evolve into travelling waves that leave in their wake periodic waves which can move with the front or opposite to it. It is also shown that irregular spatio-temporal oscillations can occur behind such a transition front and that numerical evidence strongly suggests that these oscillations are chaotic. Application to intracellular calcium signalling and to predator-prey models are discussed.
Part of this work was carried out while I was on leave at the School of Mathematics and Statistics, University of Sydney, and at the Department of Mathematics, Williams College, Massachusetts. I thank both Departments for their hospitality and support.
Philip K. Maini
Centre for Mathematical Biology
24-29 St Giles'
Oxford OX1 3LB, U.K.