Traveling-Waves
This demo computes traveling wave solutions to an activator-
inhibitor enzyme model given by the PDE
ds d^2s
-- = beta1 ---- - lambda [ rho R(s,a) - (s0 - s) ]
dt dx^2
da d^2a
-- = beta2 ---- - lambda [ rho R(s,a) - alpha (a0 - a) ]
dt dx^2
where
R(s,a) = s a / [ (kappa1 + a) (1 + s + kappa2*s^2) ]
The diffusion constants are beta1=1 and beta2=5, as set in the
subroutine STPNT in the equations-file, which use PAR(15) and
PAR(16) this purpose. The wave speed is also initialized there,
namely, c = 0.05, as stored in PAR(10). The fixed parameters are
given the values lambda = 3, s0 = 145, rho = 210, kappa1 = 3.4,
kappa2 = 0.023, and alpha = 0.2. The parameter a0 is used as the
primary continuation parameter.
This demo has the following subdirectories:
- ODE:
Computes the bifurcation diagram of the 2D system of first
order ODEs that results when the diffusion constants beta1
and beta2 are set to zero. This diagram is instructive for
the actual traveling wave computations in subdirectories
"Basic-Waves" and "More-Waves".
- Basic-Waves:
This demo computes a bifurcation diagram for the "reduced
equations", namely the 4D system of first order ODEs whose
periodic solutions correspond to traveling waves of the
PDE of wavespeed c=0.05 and varying wave length L.
Subsequently a traveling wave of selected wave length is
integrated in time to test its stability. (It is stable.)
- More-Waves:
This demo is similar to "Traveling-Waves/Basic-Waves", in
that it re-computes the bifurcation diagram of the reduced
system. In addition, a traveling wave of selected wave length
is continued keeping its wave length L fixed, while the problem
parameter "a0" and the wave speed c are variable. Thereafter,
several of the fixed wave length waves are integrated in time
to test their stability. This leads to the detection of new
stationary solutions ("patterns"), which are subsequently
continued in the parameter "a0".
Reference:
E. J. Doedel, J. P. Kernevez, A numerical analysis of wave phenomena
in a reaction diffusion model, in: Nonlinear Oscillations in Biology
and Chemistry, H. G. Othmer, ed., Lecture Notes in Biomathematics 66,
Springer Verlag, 1986, 261-273.