- P K Maini, J D Murray and G F Oster, A mechanical model for biological pattern formation: a nonlinear bifurcation analysis, in
*Ordinary and partial differential equations, Dundee, 1984*(Lecture Notes in Math., 1151, Springer, Berlin, 1985), 252-269.We present a mechanical model for cell aggregation in embryonic development. The model is based on the large traction forces exerted by fibroblast cells which deform the extracellular matrix (ECM) on which they move. It is shown that the subsequent changes in the cell environment can combine to produce pattern. A linear analysis is carried out for this model. This reveals a wide spectrum of different types of dispersion relations. A non-linear bifurcation analysis is presented for a simple version of the field equations: a non-standard element is required. Biological applications are briefly discussed.

- P K Maini, J D Murray and G F Oster, An analysis of one- and two-dimensional patterns in a mechanical model for morphogenesis, in
*Nonlinear oscillations in biology and chemistry, Salt Lake City, Utah, 1985*(Lecture Notes in Biomath.,**66**, Springer, Berlin, 198), 46-656.In early embryonic development, fibroblast cells move through an extracellular matrix (ECM) exerting large traction forces which deform the ECM. We model these mechanical interactions mathematically and show that the various effects involved can combine to produce pattern in cell density. A linear analysis exhibits a wide selection of dispersion relations, suggesting a richness in pattern forming capability of the model. A nonlinear bifurcation analysis is presented for a simple version of the governing field equations. The one-dimensional analysis requires a non-standard element. The two-dimensional analysis shows the possibility of roll and hexagon pattern formation. A realistic biological application to the formation of feather germ primordia is briefly discussed.

- A S Perelson, P K Maini, J D Murray, J M Hyman and G F Oster, Nonlinear pattern selection in a mechanical model for morphogenesis,
*J. Math. Biol.***24**(5) (1986), 525-541.We present a numerical study of the nonlinear mechanical model for morphogenesis proposed by Oster et al. (1983) with the aim of establishing the pattern forming capability of the model. We present a technique for mode selection based on linear analysis and show that, in many cases, it is a reliable predictor for nonlinear mode selection. In order to determine the set of model parameters that can generate a particular pattern we develop a technique based on nonlinear least square fitting to a dispersion relation. As an application we present a scenario for sequential pattern formation of dermal aggregations in chick embryos which leads to the hexagonal array of cell aggregations observed in feather germ formation in vivo.

- P K Maini and J D Murray, A nonlinear analysis of a mechanical model for biological pattern formation,
*SIAM J. Appl. Math.***48**(5) (1988), 1064-1072.This paper studies a simplified but biologically relevant version of a mechanical model for morphogenesis proposed by Oster, Murray, and Harris [

*J. Embryol. Exp. Morph.***78**(1983), 83-125]. A nonlinear bifurcation analysis of the partial differential system is presented. In the one-dimensional version, the derivation of the amplitude equation involves a nonstandard element. The analysis of a caricature of the two-dimensional system predicts the formation of rolls and hexagons. The biological significance of these results to feather germ formation is briefly discussed. - C L Frenzen and P K Maini, Enzyme kinetics for a two-step enzymic reaction with comparable initial enzyme-substrate ratios,
*J. Math. Biol.***26**(6) (1988), 689-703.We extend the validity of the quasi-steady state assumption for a model double intermediate enzyme-substrate reaction to include the case where the ratio of initial enzyme to substrate concentration is not necessarily small. Simple analytical solutions are obtained when the reaction rates and the initial substrate concentration satisfy a certain condition. These analytical solutions compare favourably with numerical solutions of the full system of differential equations describing the reaction. Experimental methods are suggested which might permit the application of the quasi-steady state assumption to reactions where it may not have been obviously applicable before.

- J D Murray, P K Maini and R T Tranquillo, Mechanochemical models for generating biological pattern and form in development,
*Phys. Rep.***171**(2) (1988), 59-84.The central issue in development is the formation of spatial patterns of cells in the early embryo. The mechanisms which generate these patterns are unknown. Here we describe the new Oster-Murray mechanochemical approach to the problem, the elements of which are experimentally well documented. By way of illustration we derive one of the basic models from first principles and apply it to a variety of problems of current interest and research. We specifically discuss the formation of skin organ patterns, such as feather and scale germs, cartilage condensations in the developing vertebrate limb and finally wound healing.

- L Goldwasser, P K Maini and J D Murray, Splitting of cell clusters and bifurcation of bryozoan branches,
*J. Theoret. Biol.***137**(3) (1989), 271-279.We present a cell mechanical model that exhibits the pattern-forming behaviour of a cluster of cells at the growing tips of bryozoan branches. The crucial event in the production of the overall branching pattern is the splitting of the cluster into two clusters, rather than the formation of the cluster from a uniform distribution. In simulations, the uniform cell distribution initially evolved, as indicated by a linear analysis, either to a spatially patterned state, or to a temporally oscillating state. We suggest that the same processes responsible for the formation and behaviour of the cell cluster may also be responsible for tip branching.

- P K Maini, Spatial and spatio-temporal patterns in a cell-haptotaxis model,
*J. Math. Biol.***27**(5) (1989), 507-522.We investigate a cell-haptotaxis model for the generation of spatial and spatio-temporal patterns in one dimension. We analyse the steady state problem for specific boundary conditions and show the existence of spatially hetero-geneous steady states. A linear analysis shows that stability is lost through a Hopf bifurcation. We carry out a nonlinear multi-time scale perturbation procedure to study the evolution of the resulting spatio-temporal patterns. We also analyse the model in a parameter domain wherein it exhibits a singular dispersion relation.

- P K Maini, Superposition of modes in a caricature of a model for morphogenesis,
*J. Math. Biol.***28**(3) (1990), 307-315.In a model proposed for cell pattern formation by Nagorcka et al. (

*J. Theor. Biol.*1987) linear analysis revealed the possibility of an initially spatially uniform cell density going unstable to perturbations of two distinct spatial modes. Here we examine a simple one-dimensional caricature of their model which exhibits similar linear behaviour and present a nonlinear analysis which shows the possibility of superposition of modes subject to appropriate parameter values and initial conditions. - M R Myerscough, P K Maini, J D Murray and K H Winters, Two-dimensional pattern formation in a chemotactic system. With comments by Y Cohen and Sean McElwain, in
*Math. Model., 6, Dynamics of complex interconnected biological systems, Albany, 1989*(Birkhäuser Boston, Boston, MA, 1990), 65-83.Chemotaxis is known to be important in ceil aggregation in a variety of contexts. We propose a simple partial differential equation model for a chemotactic system of two species, a population of cells and a chemo-attractant to which cells respond. Linear analysis shows that there exists the possibility of spatially inhomogeneous solutions to the model equations for suitable choices of parameters. We solve the full nonlinear steady state equations numerically on a two dimensional rectangular domain. By using mode selection from the linear analysis we produce simple pattern elements such as stripes and regular spots. More complex patterns evolve from these simple solutions as parameter values or domain shape change continuously. An example bifurcation diagram is calculated using the chemotactic response of the cells as the bifurcation parameter. These numerical solutions suggest that a chemotactic mechanism can produce a rich variety of complex patterns.

- P K Maini, D L Benson and J A Sherratt, Pattern formation in reaction-diffusion models with spatially inhomogeneous diffusion coefficients,
*IMA J. Math. Appl. Med. Biol.***9**(3) (1992), 197-213.Reaction-diffusion models for biological pattern formation have been studied extensively in a variety of embryonic and ecological contexts. However, despite experimental evidence pointing to the existence of spatial inhomogeneities in various biological systems, most models have only been considered in a spatially homogeneous environment. The authors consider a two-chemical reaction-diffusion mechanism in one space dimension in which one of the diffusion coefficients depends explicitly on the spatial variable. The model is analysed in the case of a step function diffusion coefficient and the insight gained for this special case is used to discuss pattern generation for smoothly varying diffusion coefficients. The results show that spatial inhomogeneity may be an important biological pattern regulator, and possible applications of the model to chondrogenesis in the vertebrate limb are suggested.

- D L Benson, P K Maini and J A Sherratt, Analysis of pattern formation in reaction-diffusion models with spatially inhomogeneous diffusion coefficients,
*Math. Comput. Modelling***17**(12) (1993), 29-34.We consider a reaction diffusion system in one spatial dimension in which the diffusion coefficients are spatially varying. We present a non-standard linear analysis for a certain class of spatially varying diffusion coefficients and show that it accurately predicts the behaviour of the full nonlinear system near bifurcation. We show that the steady state solutions exhibit qualitatively different behaviour to that observed in the usual case with constant diffusion coefficients. Specifically, the modified system can generate patterns with spatially varying amplitude and wavelength. Application to chondrogenesis in the limb is discussed.

- R Dillon, P K Maini and H G Othmer, Pattern formation in generalized Turing systems. I. Steady-state patterns in systems with mixed boundary conditions,
*J. Math. Biol.***32**(4) (1994), 345-393.Turing's model of pattern formation has been extensively studied analytically and numerically, and there is recent experimental evidence that it may apply in certain chemical systems. The model is based on the assumption that all reacting species obey the same type of boundary condition pointwise on the boundary. We call these scalar boundary conditions. Here we study mixed or nonscalar boundary conditions, under which different species satisfy different boundary conditions at any point on the boundary, and show that qualitatively new phenomena arise in this case. For example, we show that there may be multiple solutions at arbitrarily small lengths under mixed boundary conditions, whereas the solution is unique under homogeneous scalar boundary conditions. Moreover, even when the same solution exists under scalar and mixed boundary conditions, its stability may be different in the two cases. We also show that mixed boundary conditions can reduce the sensitivity of patterns to domain changes.

- F Sánchez-Garduño and P K Maini, Existence and uniqueness of a sharp travelling wave in degenerate non-linear diffusion Fisher-KPP equations,
*J. Math. Biol.***33**(2) (1994), 163-192.In spite of its biological importance, the introduction of dispersal into mathematical models to describe the space distribution patterns of species is relatively new. Continuous models to describe the spread of biological populations in their habitat can be derived in two ways: 1. Random walks followed by the so-called diffusion approximation and 2. Continuous media dynamics, based on conservation laws. ... there are biological (mating, attracting and repelling substances, overcrowding, spatial distribution of food, social behaviour, etc.) and physical (light, temperature, humidity, etc.) factors which influence population dispersal. The introduction of some of these factors into the derivation of the corresponding models implies that the probability p is no longer a space-symmetric function. It could depend on local biological and physical conditions such as population density or concentration of a chemoattractant or chemorepellant, for instance. In this paper we restrict ourselves to the case in which the probability is a density-dependent function.

- J D Murray, G C Cruywagen and P K Maini, Pattern formation in tissue interaction models, in
*Frontiers in mathematical biology*(Lecture Notes in Biomath.,**100**, Springer, Berlin, 1994), 104-116.Tissue interaction plays a major role in many morphogenetic processes, particularly those associated with skin organ primordia. We examine travelling wave solutions in a tissue interaction model for skin pattern formation which is firmly based on the known biology. From a phase space analysis we conjecture the existence of travelling waves with specific wave speeds. Subsequently, analytical approximations to the wave profiles are derived using perturbation methods. We then show numerically that such travelling wave solutions do exist and that they are in good agreement with our analytical results. Finally, the biological implications of our analysis are discussed.

- G A Ngwa and P K Maini, Spatio-temporal patterns in a mechanical model for mesenchymal morphogenesis,
*J. Math. Biol.***33**(5) (1995), 489-520.We present an in-depth study of spatio-temporal patterns in a simplified version of a mechanical model for pattern formation in mesenchymal morphogenesis. We briefly motivate the derivation of the model and show how to choose realistic boundary conditions to make the system well-posed. We firstly consider one-dimensional patterns and carry out a nonlinear perturbation analysis for the case where the uniform steady state is linearly unstable to a single mode. In two-dimensions, we show that if the displacement field in the model is represented as a sum of orthogonal parts, then the model can be decomposed into two sub-models, only one of which is capable of generating pattern. We thus focus on this particular sub-model. We present a nonlinear analysis of spatio-temporal patterns exhibited by the sub-model on a square domain and discuss mode interaction. Our analysis shows that when a two-dimensional mode number admits two or more degenerate mode pairs, the solution of the full nonlinear system of partial differential equations is a mixed mode solution in which all the degenerate mode pairs are represented in a frequency locked oscillation.

- A J Perumpanani, J A Sherratt and P K Maini, Phase differences in reaction-diffusion-advection systems and applications to morphogenesis,
*IMA J. Appl. Math.***55**(1) (1995), 19-33.The authors study the effect of advection on reaction-diffusion patterns. It is shown that the addition of advection to a two-variable reaction-diffusion system with periodic boundary conditions results in the appearance of a phase difference between the patterns of the two variables which depends on the difference between the advection coefficients. The spatial patterns move like a travelling wave with a fixed velocity which depends on the sum of the advection coefficients. By a suitable choice of advection coefficients, the solution can be made stationary in time. In the presence of advection a continuous change in the diffusion coefficients can result in two Turing-type bifurcations as the diffusion ratio is varied, and such a bifurcation can occur even when the inhibitor species does not diffuse. It is also shown that the initial mode of bifurcation for a given domain size depends on both the advection and diffusion coefficients. These phenomena are demonstrated in the numerical solution of a particular reaction-diffusion system, and finally a possible application of the results to pattern formation in Drosophila larvae is discussed.

- P D Dale, L Olsen, P K Maini and J A Sherratt, Travelling waves in wound healing,
*Forma***10**(3) (1995), 205-222.We illustrate the role of travelling waves in wound healing by considering three different cases. Firstly, we review a model for surface wound healing in the cornea and focus on the speed of healing as a function of the application of growth factors. Secondly, we present a model for scar tissue formation in deep wounds and focus on the role of key chemicals in determining the quality of healing. Thirdly, we propose a model for excessive healing disorders and investigate how abnormal healing may be controlled.