I am modelling demand and consumption (i.e. purchases) of a food system using coupled ordinary differential equations. I have modelled demand, $D$ as:
$\frac{dD}{dt} = rD(1 - \frac{D}{\kappa})$
and the amount of the food resource, $R$, will be:
$\frac{dR}{dt} = -g(D)$
where $g(D)$ is a non-linear function of demand that ideally should rise almost linearly providing there is a sufficient amount of resource $R$ (i.e. if there is enough food, demand is satisfied via consumption), and then asymptote at $R$ (i.e. consumption cannot exceed demand).
I have considered the function:
$R(1 - e^{-\alpha D})$
which asymptotes at $R$, but this means that $g(D)$ can become $>> D$ if $R$ increases much higher than $D$ and for certain values of $\alpha$. For instance, if $R=1000$ and $D = 100$, unless $\alpha$ is very small (e.g. $\alpha=0.001$) then $1000(1 - e^{-\alpha 100}) >> D$, which doesn't really make sense. If there is a sudden influx of resources, $R(t) >> R(t-1)$, this function implies that consumption will also increase without any change in demand.
So, I need a smooth function that returns $g(D) \approx D$ when $D < R$ and $g(D) = R$ when $D \geq R$.
Does anyone have any ideas?
Assuming the upper bound can also be approximate, one choice is $g=DR/(D+R)$. This encourages a change of variables say $x=1/D,\,y=1/R$. Failing that, we'd have to resort to something non-smooth, such as $\min\{D,\,R\}=\frac12\left(|D+R|-|D-R|\right)$. Another approximation is $D+(R-D)/(1+e^{-(R-D)})$.