## Forest growth equation -- continue

Hi all,

I posted a month ago a question on a forest growth model, to know the amount of biomass in my forest under a new disturbance (fire) rate, which has been solved in a very helpful way ! I have a new question about it, so here is quickly the summary of the problem:

In the forest growth model, the growth of any tree or patch of forest is:
dW/dt = a.(W^0.5) - b.W

And, after some time t, the total amount of biomass in my forest is:
W(t) = Q.[1-exp(-Kt)]^2
with Q = (a/b)^0.5 and K=b/2

Now I include fires in my forest (with the assumption of trees burning independently, even if unlikely), so that my equation is:
dW/dt = a.(W^0.5) - b.W - ${\lambda}$.W
$\lambda$ being the probability of a fire in one timestep.

I wanted to know the final asymptotic biomass value (Wfinal) that my forest would reach. SpringFan25 solved it:
$\lim_{t \to \infty} E(W_t) = -Q \lambda \left( \frac{2}{K + \lambda} - \frac{1}{2K + \lambda} -\frac{1}{\lambda} \right)$

Thanks again for that ! I would need some more insights on a similar question... I tried to get it done on my own but I'm quite limited on maths.

So, let's say I plant a brand new forest with that probability p of having a fire (with the assumption that it can burn the first year, no need for fuel accumulation). I would need to know at what time t_mature my forest will reach say 95% of the new asymptotic value (t_mature will depend on K and $\lambda$ I guess).

Many thanks !
Yannick

For more details on the previous post: http://www.mathhelpforum.com/math-he...tml#post663921