# Logistic ODE as a source term

Hello Fipy,


I've run into a problem in designing/implementing my model and was wondering
what the best practice would be.

Currently, I am trying to model a 1D PDE that is a cell species, H, along a
linear axis, x:

$$\label{FiPyFinal} \underbrace{\frac{\delta H}{\delta t}}_{Transient} + \underbrace{(v_A + C_\Phi \Phi \frac{\delta S}{\delta x})\frac{\delta H}{\delta x}}_{Convection} = \underbrace{D\frac{\delta^2 H}{\delta x^2}}_{Diffusion} + \underbrace{r_H C_A A H\Big(1 - \frac{\bar{H}}{K_H}\Big) - \delta_H H - \delta_D S H}_{Source}$$

H is the cell species I am interested in modeling, but S and A are also
CellVariables. Currently they are invariant, but this will probably change as
the model develops.

Importantly, \bar{H} is the total population of H. \Phi is also a globally
calculated feedback function.

My questions pertain to uncertainty in how to implement the Convection and
Source terms.

For the source term:

I see in the FiPy docs that for source terms, "The dependence can only be
included in a linear manner".

So does this imply that the correct way to implement this source term would be:

ImplicitSourceTerm(coeff = r_H * C_A * A * (1 - H_bar()/K_H) - delta_H -
delta_D * S)

With H_bar() being a function:

# Set up the summing of HSCs
def H_bar():
return H.cellVolumeAverage * mesh.cellVolumes.sum()

Am I incorrect in manually computing H_bar() at every timestep and then using
it as the source term coefficient?

Also, will there be any problems with A and S also being CellVariables in this
coupled PDE system?

For the Convection term:

And as another question, for the convection term, is it appropriate to have it
as:

ConvectionTerm(coeff = v_A + C_phi + Phi() + S.faceGrad)

As in the source term, will there be issues with having Phi() as a function
which changes at every timestep?

Thank you,

Derek Park

---------------------------
Derek Park | D. Phil student, University of Oxford


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