Introduction to the Modeling and Analysis of Complex Systems

14.2. VARIABLE RESCALING 273Exercise 14.3Segel model:Obtain homogeneous equilibrium states of the following Keller-∂a∂t = µ∇2 a − χ∇ · (a∇c) (14.18)∂c∂t = D∇2 c + fa − kc (14.19)14.2 Variable RescalingVariable rescaling of continuous field models comes with yet another bonus variable, i.e.,space, which you can rescale to potentially eliminate more parameters from the model.In a 2-D or higher dimensional space, you can, theoretically, have two or more spatialvariables to rescale independently. But the space is usually assumed to be isotropic (i.e.,there is no difference among the directions of space) in most spatial models, so it may notbe practically meaningful to use different rescaling factors for different spatial variables.Anyway, here is an example. Let’s try to simplify the following spatial predator-preymodel, formulated as a reaction-diffusion system in a two-dimensional space:( )∂r∂∂t = ar − brf + D r∇ 2 2 rr = ar − brf + D r∂x + ∂2 r2 ∂y( 2 )∂f∂∂t = −cf + drf + D f∇ 2 2 ff = −cf + drf + D f∂x + ∂2 f2 ∂y 2(14.20)(14.21)Here we use r for prey (rabbits) and f for predators (foxes), since x and y are alreadytaken as the spatial coordinates. We can apply the following three rescaling rules to statevariables r and f, as well as time t and space x/y:r → αr ′ (14.22)f → βf ′ (14.23)t → γt ′ (14.24)x, y → δx, δy (14.25)