J. Math. Biol. (1996) 34:857-877 Journalof Mathematical 6101o9y © Springer-Verlag 1996 Mechanisms for stabilisation and destabilisation of systems of reaction-diffusion equations S.A. Gourley 1, N.F. Britton 2, M.A.J. Chaplain 2, H.M. Byrne 2 1Department of Mathematical and Computing Sciences, University of Surrey, Guildford GU2 5XH, UK 2 School of Mathematical Sciences, University of Bath, Claverton Down, Bath BA2 7AY, UK Received 10 January 1995; received in revised form 5 June 1995 Abstract. Potential mechanisms for stabilising and destabilising the spatially uniform steady states of systems of reaction-diffusion equations are examined. In the first instance the effect of introducing small periodic perturbations of the diffusion coefficients in a general system of reaction-diffusion equations is studied. Analytical results are proved for the case where the uniform steady state is marginally stable and demonstrate that the effect on the original system of such perturbations is one of stabilisation. Numerical simulations carried out on an ecological model of Levin and Segel (1976) confirm the analysis as well as extending it to the case where the perturbations are no longer small. Spatio-temporal delay is then introduced into the model. Ana- lytical and numerical results are presented which show that the effect of the delay is to destabilise the original system. Key words: Reaction-diffusion equations - Time-periodic diffusion co- efficients - Spatio-temporal delay 1 Introduction The development of a heterogeneous spatial pattern from an underlying homogeneous steady-state via diffusion-driven instability is well known and so-called "Turing-systems" have been postulated to explain the occurrence of pattern formation arising in many biological situations (e.g. Turing, 1952; Gierer and Meinhardt, 1972). Indeed it has recently been realised that the question to be asked should not be "how does pattern arise?" but rather "how does pattern not arise?" (Dillon et al., 1994), since the crucial aspect of pattern formation is that it should occur in a robust way, time after time, generation after generation, largely unaffected by any noise in the system. It has been shown (Murray, 1982) that the difference between different reaction-diffusion systems is the size of the region of parameter space which gives rise to pattern.