Introduction to the Modeling and Analysis of Complex Systems

272 CHAPTER 14. CONTINUOUS FIELD MODELS II: ANALYSISwhich will make all the derivatives (both temporal and spatial ones) become zero. For example,consider obtaining homogeneous equilibrium states of the following Turing patternformation model:∂u∂t = a(u − h) + b(v − k) + D u∇ 2 u (14.10)∂v∂t = c(u − h) + d(v − k) + D v∇ 2 v (14.11)The only thing you need to do is to replace the spatio-temporal functions u(x, t) and v(x, t)with the constants u eq and v eq , respectively:∂u eq= a(u eq − h) + b(v eq − k) + D u ∇ 2 u eq (14.12)∂t∂v eq= c(u eq − h) + d(v eq − k) + D v ∇ 2 v eq (14.13)∂tNote that, since u eq and v eq no longer depend on either time or space, the temporal derivativeson the left hand side and the Laplacians on the right hand side both go away. Thenwe obtain the following:0 = a(u eq − h) + b(v eq − k) (14.14)0 = c(u eq − h) + d(v eq − k) (14.15)By solving these equations, we get (u eq , v eq ) = (h, k), as we expected.Note that we can now represent this equilibrium state as a “point” in a two-dimensional(u, v) vector space. This is another reason why homogeneous equilibrium states areworth considering; they provide a simpler, low-dimensional reference point to help usunderstand the dynamics of otherwise complex spatial phenomena. Therefore, we willalso focus on the analysis of homogeneous equilibrium states for the remainder of thischapter.Obtain homogeneous equilibrium states of the following “Orego-Exercise 14.2nator” model:ɛ ∂uu − q= u(1 − u) −∂t u + q fv + D u∇ 2 u (14.16)∂v∂t = u − v + D v∇ 2 v (14.17)