1. 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 (dW/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 !  2.

3. as the burning of a tree is not a given event but one that has only a chance of occurring, and many trees are given a similar probability you should end up with a Wfinal that can not be accurately predicted because any number of trees could have burned, the number that burn follows a bell curve.

additionally the tree should not keep growing at the same rate after a fire. the reduction in photosynthetic capability will reduce the tree's ability to grow after the fire.

to answer any further i would need to know more about how you are applying probability to the system. for instance is it a simple application where each tree has a set propability P of burning down to size 0 over every time interval T? in that case the total mass will not appear asymptotic on a graph but will exhibit the fluctuations over and under the population limit that natural populations exhibit in logistical growth ecology.  4. Hi !

Many thanks for your reply. I've made some progress on that but still dont have a straightforward solution. To answer your question, the growth function I use means that after a fire the tree (or burned forest patch) wont keep growing at the same rate, it will go back to t=To and thus start growing slowly again.

The idea is that I just want to know the final average biomass in my forest. That is, I take a forest that is big enough so that the probabilistic model of fire occurence wont make my average biomass varying up and down when t is high (i.e. when I've reached the equilibrium between forest growth and forest disturbance). I should say here if it wasn't clear that at each timestep, only a patch of my forest will burn (i.e. I dont burn my whole forest with probability p). I burn my trees inside the forest with probability p. So basically if my forest is big enough, almost exactly p% of it will burn every year.

Based on that, I can say that at t1, after I planted my forest, my forest biomass is:

W = (1-p) * Q.[1-exp(-K*t1)]^2 + p * Q.[1-exp(-K*t0]^2;
First element is the biomass of 90% of my forest that wasn't burned, second element are the 10% that burned down (thus having to, back to initial growth).

And at t2:

W = (1-p) - (1-p)*p * Q.[1-exp(-K*t2)]^2 + (p - p*p) * Q.[1-exp(-K*t1]^2 + p * Q.[1-exp(-K*t0]^2;
First element is the biomass of the 81% (90% - 10% of 90%) of my forest that still didnt burn since the beginning, second element is the biomass of my 9% of the forest that burned once at t1, and third element are the newly burned trees (10% : 9% from the first element at t1, and 1% of unlucky trees that burned twice). I dont consider any fuel threshold for a tree to burn, i.e. it can burn even if it burned the year before already.

I take it that in my case, at t400, my forest will be at equilibrium, so I could write a very long function by expanding the t1 and t2 steps up to t400... but that's not convenient. I did some simplification of that to come up with a way to compute W at t400 with a loop on t:

for t=1 to t=400
W = W + (p) * exp(-p * (400-t)) * (Q*(1-exp(-k*(400-t))^2;
end

But I'd really like to have a way to compute Wt400 (or Wfinal) without looping... it's part of a big model and I need to save running time as much as I can.

Hope that wasn't too confusing ! It's not anymore about derivatives but I guess the topic should stay here once it started...

Thank you !  Bookmarks
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