This model represents a spatio-temporal predator-prey model of population dynamics. It may or may not represent something deep and important. It makes pretty moving patterns. The ‘science bit’ is similar to that in shampoo adverts – the kind where a ‘scientist’ (someone in a lab-coat) talks about ‘nano-spheres’. But less so.

The model consists of a toroidal array of cells (easily generalisable to the continuum). Each cell has a value consisting of three components:

• The density of grass
• The density of rabbits
• The density of foxes

Naturally, rabbits eat grass and foxes eat rabbits. Without enough grass to eat, the rabbit population declines. Without enough rabbits to eat, the fox population declines. The density of both grass and rabbits increases if their respective 'predator' populations are not too large. However, the environment itself prevents population densities from growing too large.

Also, the three-tuple in each cell is drawn to the average value of its neighbourhood – this should be understood as a gradual diffusion of high density populations into neighbouring cells of lower density.

The expression for the growth of each component, e_xy, of population density in each cell x, y is:

δexy/δt

=

exy.(1 - exy).(N + I.qxy - D.rxy) + K.(ēxy - exy)

Where:

• e_xy is the population density of the component of the value in cell x, y for the population of interest
• ē_xy is the mean population density of the component in a neighbourhood around cell x, y
• q_xy is the prey population density of cell x, y
• r_xy is the predator population density of cell x, y
• N is the rate at which a small population density will grow, before it is inhibited by the environment
• I is the amount by which the prey density increases the rate of growth of the population density
• D is the amount by which the predator density decreases the rate of growth of the population density
• K is the spatial diffusion rate i.e. the rate at which cells of high density diffuse to neighbours of lower density

These equations, for all components across all cells, form a large set of simultaneous non-linear differential equations – Sound like some interesting behaviour might emerge? Let's see...