Discrete mathematics and its application to ecology

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Editorial

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Applications of mathematical modelling to biological pattern formation,

The formation of spatiotemporal patterning in biology has intrigued experimentalists and theoreticians for many generations. Here we present a brief review of some mathematical models for pattern formation and then focus on three models which use the phenomenon of chemotaxis to generate pattern

Modelling aspects of solid cancer growth

The modelling of cancer provides an enormous mathematical challenge because of its inherent multiscale nature. For example, in vascular tumours, nutrient is transported by the vascular system, which operates on a tissue level. However, it affects processes occurring on a molecular level. Molecular and intra-cellular events in turn effect the vascular network and therefore the nutrient dynamics. Our modelling approach is to model, using partial differential equations, processes on the tissue level and couple these to the intercellular events (modelled by ordinary differential equations) via cells modelled as automaton units. Thusfar, within this framework we have modelled structural adaptation at the vessel level and the effects of growth factor production in response to hypoxia. We have also investigated the effects of acid production, mutation and phenotypic evolution driven by tissue environment. These results will be presented

A comparison of reaction diffusion and mechanochemical models for limb development

Several theoretical models have been proposed to attempt to elucidate the underlying mechanisms involved in the spatial patterning of skeletal elements in the limb. Here, I briefly compare two such models - reaction diffusion (RD) and mechanochemical (MC) - and highlight their properties and predictions

Superposition of modes in a caricature of a model for morphogenesis

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

Using mathematical models to help understand biological pattern formation

One of the characteristics of biological systems is their ability to produce and sustain spatial and spatio-temporal pattern. Elucidating the underlying mechanisms responsible for this phenomenon has been the goal of much experimental and theoretical research. This paper illustrates this area of research by presenting some of the mathematical models that have been proposed to account for pattern formation in biology and considering their implications.To cite this article: P.K. Maini, C. R. Biologies 327 (2004)

A Lyapunov function and global properties for SIR and SEIR epidemiological models with nonlinear incidence

Explicit Lyapunov functions for SIR and SEIR compartmental epidemic models with nonlinear incidence of the form $\beta I_p S_q$ for the case p<1 are constructed. Global stability of the models is thereby established