soft question - What does Peter Parkers formula represent?

Okay, so the trailer for the new Spider Man movie is out and appearently our friendly physicist from the neightborhood came up with something. However I can't find out what this is.

Transcription:

$$\begin{align}\frac{\text d \log\Phi}{\text d t} &= \alpha\Biggl({1 - g{{\biggl( {\frac{\Phi }{K}} \biggr)}^\beta }} \Biggr) \\ \Phi &= K\sum_i\prod_j\exp \Biggl(\frac{g^j(1 - E_a)^{j - 1}}{(j - 1)!\bigl(1 - ( {1 - E_a)^j\bigr)}}\Biggr) \log\bigl( \cdots\end{align}$$

(the last part is hidden)

I don't think that it's just jubberish and while my first guess was statistical physics or network theory, it could as well go in the biology direction.

Answer

The two equations are unrelated.

First equation

The first equation is a simple modification of the logistic differential equation, although it is somewhat disguised. The usual logistic equation is

$$ {dx\over dt} = x(1-x) $$

or in terms of the derivative of $\log(x)$, it's the equation Peter Parker writes, with $\beta=1$. The factor of K in conjunction with g sets the scale for $\Phi$, and it is irrelevant, the qualitative behavior for different $\beta$ at long times is not modified, since the equilibrium position is at

$$\Phi_\mathrm{eq}={K\over g^{1\over\beta}}$$

Values above this go down and values below this go up. Further there is a nonzero first derivative of $\Phi^\beta$ for all reasonable idea of what $\beta$ is supposed to be, so this is describing a quantity $\Phi$ which wants to go up exponentially, but is suppressed by competitive effects.

The exponent $\beta$ describes the competitive effects. The logistic equation describes (say) bacteria (or white blood cells) replicating where two bacteria compete for the same limited resources. In this case, the competition is $\beta$-fold, the bacteria crowd each other out worse than quadraticaly (or less worse, depending on whether $\beta<1$ or $\beta>1$).

This equation is consistent with a biological interpretation that $\Phi$ is the concentration of some replicating crowding out agent, like a disease model.

Second equation

The second equation is writing down $\Phi$ in a way that depends on g but not on t. It has a K in it, but there is an unrelated expansion of $\Phi$ in terms of the $E_n$'s, so it isn't the expression for the equilibrium value or the relaxation to this equilibrium value.

Further, you can massage the form by exponentiating, expanding the denominator in a power-series, and performing the sum on j, to produce a second infinite series, but only if you assume the hidden log part does not depend on j, but only on the variable "i" which has so far not been used.

${\Phi\over K} = \exp(g (\sum_{k=1}^{\infty} \Delta^k e^{g\Delta^k})) \sum_i log(...)$$

Where $\Delta=(1-E_k)$, and from the form, I will assume $0<\Delta<1$, so that $0