Stochastic Differential Equation Model for Tree Growth

1. The Deterministic Growth (Logistic Model)

First, we model the ideal, predictable part of the growth. A tree's growth is fastest when it's young and slows down as it approaches its maximum possible height ($H_{max}$). This is an asymptotic behavior well-described by the logistic growth model. The rate of change of height ($H$) is:

$dH(t)=r\cdot H(t)\cdot (1-H(t)/H_{max})\cdot dt$

Here, $r$ is the intrinsic growth rate. This equation ensures that growth tapers off as $H(t)$ gets closer to $H_{max}$.

2. Introducing Randomness

To account for random environmental "shocks" (like a sudden drought or a particularly sunny week), we add a stochastic term. The effect of this randomness is often proportional to the size of the tree. We model this using a Brownian motion term, $dW(t)$.

$\sigma \cdot H(t)\cdot dW(t)$

The parameter $\sigma$ (sigma) represents the volatility or the magnitude of the random effects. $dW(t)$ is the noise process.

3. The Complete SDE

Combining both parts gives us the full SDE for the tree's height $H(t)$:

This model beautifully captures both the determined, goal-seeking nature of biological growth and the unpredictable reality of its environment.

See this simulation: https://ajpancollantes.github.io/cajondes/sdetree.html.