Introduction to the Modeling and Analysis of Complex Systems

274 CHAPTER 14. CONTINUOUS FIELD MODELS II: ANALYSISWith these replacements, the model equation can be rewritten as follows:( )∂(αr ′ )∂∂(γt ′ ) = a(αr′ ) − b(αr ′ )(βf ′ 2 (αr ′ )) + D r∂(δx ′ ) + ∂2 (αr ′ )2 ∂(δy ′ ) 2( )∂(βf ′ )∂∂(γt ′ ) = −c(βf ′ ) + d(αr ′ )(βf ′ 2 (βf ′ )) + D f∂(δx ′ ) + ∂2 (βf ′ )2 ∂(δy ′ ) 2We can collect parameters and rescaling factors together, as follows:∂r ′= aγr ′ − bβγr ′ f ′ + D ( )rγ ∂ 2 r ′∂t ′ δ 2 ∂x + ∂2 r ′′2 ∂y ′2∂f ′∂t ′= −cγf ′ + dαγr ′ f ′ + D ( )fγ ∂ 2 f ′δ 2 ∂x + ∂2 f ′′2 ∂y ′2Then we can apply, e.g., the following rescaling choices(a(α, β, γ, δ) =d , a b , 1 √ )a , Dra(14.26)(14.27)(14.28)(14.29)(14.30)to simplify the model equations into∂r ′= r ′ − r ′ f ′ + ∇ 2 r ′ ,∂t ′ (14.31)∂f ′= −eγf ′ + r ′ f ′ + D∂t ′ ratio ∇ 2 f ′ , (14.32)with e = c/a and D ratio = D f /D r . The original model had six parameters (a, b, c, d, D r ,and D f ), but we were able to reduce them into just two parameters (e and D ratio ) witha little additional help from spatial rescaling factor δ. This rescaling result also tells ussome important information about what matters in this system: It is the ratio between thegrowth rate of the prey (a) and the decay rate of the predators (c) (i.e., e = c/a), as well asthe ratio between their diffusion speeds (D ratio = D f /D r ), which essentially determinesthe dynamics of this system. Both of these new parameters make a lot of sense from anecological viewpoint too.Exercise 14.4Simplify the following Keller-Segel model by variable rescaling:∂a∂t = µ∇2 a − χ∇ · (a∇c) (14.33)∂c∂t = D∇2 c + fa − kc (14.34)