Principle of mass balance
Rate of change in volume = Net flux across boundary of V + net production in V
dM/dt = -I +Q
Mass balance, V is arbitrary
dc/dt = - nabla . J + q
where J is total flux and q is local reactions (birth and death)
What is diffusion
random movement of particles
Diffusion flux term
Jdiff = -D nabla c
proportional to concentration gradient
D is diffusion coefficient
What is chemotaxis
flux proportional to the gradient of attrcators/repellors
Chemotaxis flux term
Jchem = chi c nabla b
c is concentration of focus organism
chi is chemotactic parameter
b is chemo attractor/repellor
What is advection
chemical carried in a flow
Advection flux term
Jadv = vc
where v is the speed of the thing carrying c
What is logistic growth
rc(1-c/d)
where r and d are positive constants
co moving frame basic coord transformation
E = z - vt
c(z,t) —> c(E,t)
in a comoving frame a travelling wave is…
a stationary solution
diff transforms in comoving frame
dc/dz
d2c/dz2
dc/dt
dc/dz –> dc/dE
d2c/dz2 –> d2c/dE2
dc/dt –> -v dc/dE + dc/dt
