Chapter 3: Growth and Yield

Learning Objectives

After reading this chapter, you should be able to:

  • Explain how growth curves are fitted to forest inventory data

  • Use the ws3.core.Curve class to define and manipulate growth curves

  • Interpolate between curve points

  • Perform arithmetic operations on curves (add, subtract, multiply, divide)

  • Validate growth curves for biological plausibility

What Are Growth Curves?

Growth curves describe how forest attributes change over time. They are the engine of wood supply models — without them, the model would just track area without understanding how the forest grows.

Common attributes tracked by growth curves:

  • Volume: Total merchantable volume (m³/ha)

  • Basal area: Cross-sectional area of tree trunks (m²/ha)

  • Height: Dominant height (m)

  • Biomass: Total above-ground biomass (tonnes/ha)

  • Value: Dollar value per unit volume ($/m³)

Growth curves typically follow a sigmoidal (S-shaped) pattern:

        graph TD
  YOUNG["Young stands<br/>Slow growth"] --> MATURE["Mature stands<br/>Rapid growth"] --> OLD["Old stands<br/>Asymptoting"]

The curve starts slowly (young trees growing), accelerates (rapid middle-age growth), and then plateaus (trees reach physiological limits).

The Curve Class

ws3’s ws3.core.Curve class provides a flexible interface for defining and manipulating growth curves.

Defining a Curve

from ws3.core import Curve

# Define a volume curve for Douglas-fir on Site Index 50
volume_curve = Curve(
    label="DF-SI50_volume",
    is_volume=True,
    points=[(0, 0), (10, 5), (20, 25), (30, 65), (40, 120),
            (50, 200), (60, 300), (70, 400), (80, 470),
            (90, 500), (100, 510)]
)

The x values are ages (years), and the y values are attribute values (e.g., volume in m³/ha).

Interpolation

Curves support interpolation between defined points:

# Get volume at age 25 (interpolated between ages 20 and 30)
vol_25 = volume_curve(25)
print(f"Volume at age 25: {vol_25:.1f} m³/ha")

# Get volume at age 55
vol_55 = volume_curve(55)
print(f"Volume at age 55: {vol_55:.1f} m³/ha")

Arithmetic Operations

Curves support arithmetic operations, which is useful for combining curves or calculating differences:

# Create a value curve (volume * price)
price_curve = Curve(
    label="DF-SI50_value",
    points=[(0, 0), (10, 250), (20, 1250), (30, 3250), (40, 6000),
            (50, 10000), (60, 15000), (70, 20000), (80, 23500),
            (90, 25000), (100, 25500)]
)

# Net value = value curve - harvesting cost curve
cost_curve = Curve(
    label="harvest_cost",
    points=[(0, 0), (10, 100), (20, 400), (30, 900), (40, 1600),
            (50, 2500), (60, 3600), (70, 4900), (80, 6400),
            (90, 8100), (100, 10000)]
)

net_value = price_curve - cost_curve
print(f"Net value at age 50: ${net_value(50):,.0f}")

Curve Algebra

The Curve class supports all basic arithmetic operations:

Operator

Description

curve1 + curve2

Element-wise addition

curve1 - curve2

Element-wise subtraction

curve1 * curve2

Element-wise multiplication

curve1 / curve2

Element-wise division

curve * scalar

Scale all values by a constant

curve / scalar

Divide all values by a constant

Fitting Growth Curves

Growth curves are typically fitted to field data using statistical methods. Common approaches include:

  1. Polynomial regression: Simple but can produce unrealistic curves

  2. Logistic growth: Captures sigmoidal shape well

  3. Richards function: Flexible sigmoidal model

  4. Provincial yield tables: Government-published curves for common species

Using Provincial Yield Tables

In British Columbia, the Ministry of Forests publishes growth-and-yield tables for common species. These tables provide volume estimates for different species, site indices, and ages.

# Example: Load a provincial yield table
# (This is a simplified example — actual tables are more complex)

yield_data = {
    "species": "Pseudotsuga menziesii",
    "site_index": 50,
    "ages": [0, 10, 20, 30, 40, 50, 60, 70, 80, 90, 100],
    "volumes": [0, 5, 25, 65, 120, 200, 300, 400, 470, 500, 510]
}

curve = Curve(
    label=f"{yield_data['species']}-SI{yield_data['site_index']}_volume",
    is_volume=True,
    points=list(zip(yield_data["ages"], yield_data["volumes"]))
)

Validating Growth Curves

Before using a growth curve in a model, validate it:

def validate_curve(curve, min_age=0, max_age=500):
    """Validate a growth curve for biological plausibility."""

    # Check that x values are increasing
    x = curve.x
    if not all(x[i] < x[i+1] for i in range(len(x)-1)):
        raise ValueError("x values must be strictly increasing")

    # Check that y values are non-negative
    if any(y < 0 for y in curve.y):
        raise ValueError("y values must be non-negative")

    # Check that curve starts at zero (or near zero)
    if curve.y[0] > 10:
        raise ValueError("Curve should start near zero volume")

    # Check for reasonable maximum values
    max_vol = max(curve.y)
    if max_vol > 10000:  # 10,000 m³/ha is unrealistically high
        raise ValueError(f"Maximum volume {max_vol} m³/ha is unrealistic")

    # Check that curve eventually plateaus (optional)
    if len(curve.y) > 10:
        recent_growth = curve.y[-1] - curve.y[-2]
        if recent_growth > 100:  # More than 100 m³/ha growth in one period
            print("Warning: Curve may not be plateauing")

validate_curve(volume_curve)

Common Growth Curve Shapes

Different species have characteristic growth curve shapes:

Species

Growth Pattern

Douglas-fir

Fast initial growth, peaks around age 80-100, then declines

Western red cedar

Slow initial growth, continues increasing slowly past age 200

Sitka spruce

Very rapid growth, peaks early (age 40-60), then declines sharply

Lodgepole pine

Moderate growth, peaks around age 60-80, relatively flat plateau

Exercises

Exercise 1 (Easy): Create a volume curve for Sitka spruce on Site Index 45 and interpolate to find the volume at age 35.

Exercise 2 (Medium): Write a function that takes two curves (volume and price) and returns a net value curve (price * volume - cost).

Exercise 3 (Hard): Fit a logistic growth curve to the following data points using scipy.optimize.curve_fit:

ages = [10, 20, 30, 40, 50, 60, 70, 80]
volumes = [10, 40, 90, 150, 210, 260, 300, 330]

Further Reading