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.

We investigate the impact of cyclic competition on pattern formation in the rock-paper-scissors game. By separately considering random and prepared initial conditions, we observe a critical influence of the competition rate p on the stability of spiral waves and on the emergence of biodiversity. In particular, while increasing values of p promote biodiversity, they may act detrimentally on spat...

The topic of Pattern Formation in Growing Sandpiles was first looked at in the 1980’s through a model of sand being added to a two dimensional system at a constant rate.[1] How this sand creates patterns of inactivity and unpredictability is an interesting natural phenomena. The Abelian Sandpile model is one model created to model these patterns, showing that the ending stable state of the mode...

The balance between ice and water controls the habitability of an important fraction of the globe and influences the majority of the world’s population. The freezing of water to form ice is one of the most common phase transformations in the natural environment. However, a complete understanding of its microscopics and their influence on macroscopic phenomena still eludes us. As this millennium...

We present a simple model based on a reaction-diffusion equation to explain pattern formation in a multicellular bacterium (Streptomyces). We assume competition for resources as the basic mechanism that leads to pattern formation; in particular we are able to reproduce the spatial pattern formed by bacterial aerial mycelium in the case of growth in minimal (low resources) and maximal (large res...

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Topics covered: 1. What is disturbance? 2. Characteristics of disturbances and disturbance regimes 3. Role of disturbance in ecosystems and landscapes 4. Factors affecting disturbances 5. Feedback mechanisms affecting disturbance regimes 6. Interactions among the physical template, biotic processes and disturbance regimes 7. Human effects on disturbance regimes 8. Resource management implications.

We address the slow generation of crack networks as a problem of pattern formation. Issues of pattern selection and the associated statistical properties were considered by means of a detailed theoretical analysis and simulations of a discrete spring-block model. Developed after observations in desiccation experiments, the model describes the nucleation and propagation of cracks in a layer in c...

During morphogenesis regular patterns often develop behind a frontier of pattern formation which travels across the prospective tissue. Here the authors consider the propagating patterns exhibited in a two-dimensional domain by a tissue interaction mechanochemical model for skin pattern formation. It is shown that the model can exhibit travelling waves of complex spatial pattern formation. Two ...

In the present article, evolution of microstructure during solidification, as a function of various parameters, is discussed. Macrosegregation is described as being due to insufficient diffusivity of solute in the solid. Pattern formation is discussed in the light of instabilities at the solidification growth front. An overview of the scaling relations for various microstructures is given. Meta...