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In this paper, we investigate the role of cell death in promoting pattern formation within bacterial biofilms. To do this we utilise an extension of the model proposed by Dockery and Klapper [13], and consider the effects of two distinct death rates. Equations describing the evolution of a moving biofilm interface are derived, and properties of steady state solutions are examined. In particular, a comparison of the planar behaviour of the biofilm interface in the different cases of cell death is investigated. Linear stability analysis is carried out at steady state solutions of the interface, and it is shown that, under certain conditions, instabilities may arise. Analysis determines that, while the emergence of patterns is a possibility in 'deep' biofilms, it is unlikely that pattern formation will arise in 'shallow' biofilms. [ABSTRACT FROM AUTHOR]

We discuss travelling-wave solutions of a system of coupled reaction—diffusion equations used by the authors to describe the macroscopic behaviour of fungal mycelia. Such systems have been used in a multitude of applications; and, in particular, Merkin and Needham (1990, Proc. R. Soc. Lond. A 430, 315–45) have studied a certain formulation as a model for generic isothermal chemical reactions. We show that an alternative analysis provides more complete results in ascertaining the conditions under which travelling-wave solutions exist and that it allows a wider range of parameter values to be considered; this is essential to the application considered in the present case. Numerical investigations of travelling-wave solutions and the related initial-value problem are included to motivate and extend the analysis. [ABSTRACT FROM PUBLISHER]

In this paper we discuss steady-state solutions of the system of reaction-diffusion equations known as the Sel'kov model. This model has been the subject of much discussion; in particular, analytical and numerical results have been discussed by Lopez-Gomez et al. (1992, IMA J. Num. Anal. 12, 405–28). We show that a simple analysis of the bifurcation function associated with the system can explain many of the numerical observations, such as the formation and development of loops of nontrivial solutions, in a simpler and more complete manner than the analysis of Lopez-Gomez et al. This allows for a clearer understanding of the qualitative behaviour of the set of nontrivial solutions and hence of the bifurcation diagram. [ABSTRACT FROM PUBLISHER]