Good morning !
I have an equation for the growth of a tree: let B be the amount of wood in my tree, and t the time since I planted it.
The growth is as follow:
dW/dt = a.(W^0.5) - b.W
This is the normal growth of my tree, when nothing special happens. Parameters a and b are such that my tree starts growing slowly, then reaches a maximum growth, and progressively goes down asymptotically towards zero growth.
The accumulation of vegetation in my forest, after integration of that function, is:
W(t) = Q.[1-exp(-Kt)]^2
(Q = (a/b)^0.5 and K=b/2)
Up to now, everything is fine. The tree nicely accumulates vegetation following a sigmoid curve towards its maximum, which is equal to Q. Now I plant a whole forest, and I use the same equations to represent the average amount of wood per tree in my forest.
Now I want to have some disturbances in my forest. Every day, there is a probability p for a tree to burn. When a tree burns, at t+1, it starts regrowing, but may burn again anytime with the same probability p.
And now I want to know the new asymptotic value of wood in my forest after it reaches equilibrium under the new fire probability (let's call it Wfinal).
I've built a model of my forest to do that, which is fine, but I need to be able to predict Wfinal mathematically. Basically I need the new function W(t) as a function of a, b and p.
I modified the growth function to have the fires going on (dB/dt = a.(W^0.5) - b.W - p.W), but that's wrong, because the growth of my trees is not linear through time.
I gave it long thoughts and many many scribbles on paper but couldn't make it...
Many thanks for your help !