The Mathematical Model
Stochastic Differential Equations (SDEs) are powerful tools for modeling systems that evolve over time while being influenced by random, unpredictable factors. A perfect example from nature is the growth of a tree, which is subject to variable conditions like weather, resource availability, and pests.
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 (Hmax). This is an asymptotic behavior well-described by the logistic growth model. The rate of change of height (H) is:
Here, r is the intrinsic growth rate. This equation ensures that growth tapers off as H(t) gets closer to Hmax.
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).
The parameter σ (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.
Live Simulation
Height: 0.10 m
Time: 0.00 years