Here is my simplified model. 
Let P be the bitcoin price or more preciously, the profitability at this moment.
Let X be the total machine power in the network.
v = dX/dt = the change in new machine power in the network, or the velocity.
a = dX^2/dt^2 = the change in velocity -- the acceleration, which I believe is a function of P.
Let f be the friction, which is a function of both P and difficulty level.
If we solve this 2nd order differential equation and set the boundary condition, we should be able to get a equation for X.
It should note that the a is a leading indicator, while X is a lagging indicator, meaning the surge in bitcoin price a few weeks ago increases a instantly, but the change in X reflects a couple weeks later. It is logical to think that when a surge in price, people will start investing in new machines, but the new machines take time to procure, ship, and assemble.
The change in difficulty level affect f, which is a feedback factor. It should adjust v and drive the profitability to zero in the long run when the system is in equilibrium.
Let P be the bitcoin price or more preciously, the profitability at this moment.
Let X be the total machine power in the network.
v = dX/dt = the change in new machine power in the network, or the velocity.
a = dX^2/dt^2 = the change in velocity -- the acceleration, which I believe is a function of P.
Let f be the friction, which is a function of both P and difficulty level.
If we solve this 2nd order differential equation and set the boundary condition, we should be able to get a equation for X.
It should note that the a is a leading indicator, while X is a lagging indicator, meaning the surge in bitcoin price a few weeks ago increases a instantly, but the change in X reflects a couple weeks later. It is logical to think that when a surge in price, people will start investing in new machines, but the new machines take time to procure, ship, and assemble.
The change in difficulty level affect f, which is a feedback factor. It should adjust v and drive the profitability to zero in the long run when the system is in equilibrium.