I'm modelling some real-world gene expression data with various growth models including linear, exponential, and Verhulst growth but not all of the genes are showing these forms of time-dependence. Thus I am exploring other models, and that has led me to the hyperbolastic functions.
According to wikipedia, the type II hyperbolastic rate equation is given by
$$\frac{dP(x)}{dx} = \frac{\alpha \delta P^2(x)x^{\gamma-1}}{M} \tanh \left( \frac{M - P(x)}{\alpha P(x)} \right)$$
which has the solution
$$P(x) = \frac{M}{1 + \alpha \sinh^{-1} \left(e^{-\delta x \gamma} \right)}$$
that doesn't explicitly have the initial conditions in it. Then wikipedia says I can substitute
$$\alpha := \frac{M - P_0}{P_0 \sinh^{-1}\left(e^{-\delta x_0 \gamma} \right)}$$
which has the initial conditions $x_0$ and $P_0$ in it. Since $\alpha$ appears in both the rate equation and its solution, it seems that the initial condition doesn't fall out of the derivation of the solution as I often find in solving differential equations. I don't see why one can't have arbitrary constants in the rate equation, but I would like to confirm that I've understood correctly that the above assignment for $\alpha$ applies to both the rate equation and its solution. If so, I should be able to rewrite the rate equation as
$$\frac{dP(x)}{dx} = \frac{M - P_0}{P_0 \sinh^{-1}\left(e^{-\delta x_0 \gamma} \right)} \frac{\delta P^2(x)x^{\gamma-1}}{M} \tanh \left(\frac{P_0 \sinh^{-1}\left(e^{-\delta x_0 \gamma} \right)}{M - P_0} \frac{M - P(x)}{P(x)} \right)$$
The wikipedia page states "If one desires to use initial condition $P(x_0) = P_0$, then $\alpha$ can be expressed as [...]". While the hyperbolastic rate equation of type II doesn't have an initial condition in it in the general case, one can be produced by the above choice in assignment through substitution and rearrangement for $\alpha$. The constant $\alpha$ was likely chosen rather than $\delta$ or $\gamma$ because of some combination of (1) being harder to solve for, (2) resulting in a more 'complicated' expression in the solution, or (3) solutions that are numerically less well-behaved.