modelling Level 3

Phenology and Growing Season Dynamics

When do leaves emerge in spring? When do crops mature? Phenology—the timing of biological events—is controlled by temperature accumulation (growing degree days) and day length (photoperiod). This model models bud break, leaf expansion, flowering, and senescence, and shows how phenology affects annual productivity.

Prerequisites: thermal time, accumulation, threshold response, seasonal dynamics

26 Feb 2026 Updated 12 Mar 2026 13 min read

1. The Question

Why do trees in Minnesota leaf out in May, while trees in Florida leaf out in February?

Phenology is the study of seasonal timing in organisms:

Bud break: When buds open and leaves emerge
Flowering: When plants produce flowers
Senescence: When leaves die and fall
Migration: When animals move (birds, butterflies)

These events are triggered by environmental cues:

Temperature: Accumulated warmth (thermal time)
Photoperiod: Day length
Chilling: Cold exposure needed for dormancy break

The mathematical question: How do we model phenological timing from weather data, and how does phenology affect ecosystem productivity?

2. The Conceptual Model

Growing Degree Days (GDD)

Thermal time: Plants accumulate heat above a base temperature.

\[\text{GDD} = \sum_{t=1}^{n} \max(0, T_{\text{avg}}(t) - T_{\text{base}})\]

Where:

$T_{\text{avg}}$ = daily average temperature (°C)
$T_{\text{base}}$ = base temperature (typically 5–10°C)
Sum over days

Example: Day with $T_{\text{avg}} = 15°$C, $T_{\text{base}} = 5°$C:

\[\text{GDD} = 15 - 5 = 10 \text{ degree-days}\]

Threshold for bud break: Typically 100–300 GDD accumulated since spring.

Photoperiod

Day length varies with latitude and day of year.

Approximation:

\[D = 12 + 2 \arcsin(\tan \phi \tan \delta) \times \frac{24}{2\pi}\]

Where:

$D$ = day length (hours)
$\phi$ = latitude (radians)
$\delta$ = solar declination (function of day of year)

Solar declination:

\[\delta = 23.45° \sin\left(\frac{360}{365}(d - 81)\right)\]

Where $d$ is day of year (1 = Jan 1).

Critical photoperiod: Some plants require day length > threshold to flower.

Long-day plants: Flower when days exceed ~14 hours (wheat, spinach)
Short-day plants: Flower when days fall below ~12 hours (soybeans, rice)
Day-neutral plants: Flower independent of photoperiod (tomatoes, corn)

Chilling Requirement

Dormancy: Many temperate plants require cold exposure before they can respond to spring warmth.

Chilling units: Days with temperature 0–7°C.

Example: Apple trees need ~1000 chilling hours (42 days at 0–7°C) before bud break.

Problem in warming climate: Insufficient winter chilling can delay spring phenology.

3. Building the Mathematical Model

Spring Phenology Model

Bud break timing:

Start date: January 1 (or after chilling requirement met)
Accumulate GDD: Sum daily degree-days
Trigger: When $\sum \text{GDD} > \text{GDD}_{\text{crit}}$

GDD threshold varies by species:

Early species (willow): 100 GDD
Mid-season (oak): 200 GDD
Late species (ash): 300 GDD

Leaf Area Index Dynamics

Spring green-up:

\[\frac{d\text{LAI}}{dt} = r_g \times (T - T_{\text{base}}) \times (LAI_{\text{max}} - \text{LAI})\]

Where:

$r_g$ = growth rate parameter
LAI asymptotes to $LAI_{\text{max}}$ (maximum leaf area)

Autumn senescence:

Triggered by:

Decreasing photoperiod (day length < threshold)
Frost events (temperature < 0°C)
Age (days since leaf emergence > 150)

\[\frac{d\text{LAI}}{dt} = -r_s \times \text{LAI}\]

Where $r_s$ is senescence rate (day⁻¹).

Growing Season Length

Start of season (SOS): When LAI > 0.5 or NDVI > 0.3

End of season (EOS): When LAI < 0.5 or NDVI < 0.3

Growing season length:

\[\text{GSL} = \text{EOS} - \text{SOS}\]

Typical values:

Tropics: GSL = 365 days (year-round)
Temperate: GSL = 150–200 days
Boreal: GSL = 90–120 days
Tundra: GSL = 60–90 days

Frost Risk

Last spring frost: Latest date with $T_{\min} < 0°$C

First fall frost: Earliest date with $T_{\min} < 0°$C

Frost-free period: Time between last spring and first fall frost.

Agricultural importance: Crops must mature within frost-free period.

Example: Corn needs 120–140 GDD from planting to maturity. In short-season climates, must plant as soon as last frost passes.

4. Worked Example by Hand

Problem: A deciduous tree at 45°N has:

Base temperature: $T_{\text{base}} = 5°$C
GDD requirement for bud break: 200 degree-days
Chilling requirement met by April 1

Temperature record (daily average):

April 1–10: 10°C
April 11–20: 12°C
April 21–30: 15°C

When does bud break occur?

Solution

GDD accumulation:

April 1–10 (10 days at 10°C):

\[\text{GDD} = 10 \times (10 - 5) = 10 \times 5 = 50 \text{ degree-days}\]

Cumulative: 50

April 11–20 (10 days at 12°C):

\[\text{GDD} = 10 \times (12 - 5) = 10 \times 7 = 70 \text{ degree-days}\]

Cumulative: 50 + 70 = 120

April 21–30 (10 days at 15°C):

\[\text{GDD} = 10 \times (15 - 5) = 10 \times 10 = 100 \text{ degree-days}\]

Cumulative: 120 + 100 = 220

Bud break occurs when cumulative GDD reaches 200.

Within April 21–30:

Need 200 - 120 = 80 more degree-days.

At 10 degree-days per day: 80 / 10 = 8 days.

Bud break date: April 21 + 8 = April 29

5. Computational Implementation

Below is an interactive phenology simulator.

Latitude: 45°N Species type: Climate scenario:

Bud break (SOS):

Peak LAI date:

Senescence start (EOS):

Growing season length: days

Annual GPP: kg C/m²

Try this:

Higher latitude: Later bud break (cooler spring), shorter growing season
Warming scenarios: Earlier bud break, longer growing season, higher annual GPP
Early species: Lower GDD requirement → earlier green-up
Late species: Higher GDD requirement → later green-up but avoid late frosts
Notice: Photoperiod (blue dotted) triggers senescence in autumn (falls below 12 hours)

Key insight: Climate warming extends growing season by triggering earlier spring phenology and delaying autumn senescence.

6. Interpretation

Climate Change Impacts

Observations:

Earlier spring: Bud break 2–3 weeks earlier than 50 years ago
Later autumn: Leaf fall 1–2 weeks later
Longer growing season: +10 to +20 days in temperate zones

Consequences:

Increased NPP: Longer photosynthesis season
Water demand: Earlier spring → earlier drought stress
Frost risk: Early bud break → vulnerable to late frosts
Phenological mismatch: Plants and pollinators/herbivores out of sync

Agricultural Applications

Crop maturity:

\[\text{Days to maturity} = \frac{\text{GDD}_{\text{required}}}{\text{Daily GDD average}}\]

Example: Corn requiring 1400 GDD in climate with 12 GDD/day average:

\[\text{Days} = \frac{1400}{12} = 117 \text{ days}\]

Planting date optimization:

Too early → frost risk
Too late → insufficient GDD before first fall frost

Variety selection:

Short-season varieties for high latitudes
Long-season varieties for warm climates

Remote Sensing Phenology

NDVI time series (from Model 25) tracks phenology:

Green-up date: When NDVI increases rapidly
Peak NDVI: Maximum LAI
Brown-down date: When NDVI decreases

Applications:

Crop monitoring (planting, harvest dates)
Forest health
Climate change detection

7. What Could Go Wrong?

Using Only Temperature

Photoperiod matters for:

Senescence timing (day length < 12 hours)
Flowering in long/short-day plants
Prevents false spring (warm January day doesn’t trigger growth)

Better models: Combine GDD + photoperiod thresholds.

Ignoring Chilling Requirement

Insufficient winter chill:

Delays or prevents bud break
Problem in warming climates (California cherries, wine grapes)
May require “chill hours” below 7°C, not just cumulative cold

Constant GDD Threshold

Real plants have dynamic thresholds:

Varies with chilling received
Varies with water stress
Genetics (cultivar differences)

Assuming Instantaneous Response

Actual phenology:

Bud break takes days to weeks
LAI increases gradually (not step function)
Senescence is progressive, not sudden

Better models: Use sigmoid curves for smooth transitions.

8. Extension: Frost Hardiness

Freeze tolerance varies with season:

Autumn: Plants acclimate as temperatures drop

Harden tissues
Accumulate sugars (antifreeze)
Tolerate −20 to −40°C

Spring: De-acclimate as warmth returns

Tissues more vulnerable
Tolerate only −2 to −5°C

Critical period: Early spring after de-acclimation.

Frost damage function:

\[D = \begin{cases} 0 & \text{if } T > T_{\text{kill}} \\ 1 - \frac{T}{T_{\text{kill}}} & \text{if } T < T_{\text{kill}} \end{cases}\]

Where $T_{\text{kill}}$ is killing temperature (depends on acclimation state).

9. Math Refresher: Accumulation and Thresholds

GDD as Integral

Continuous form:

\[\text{GDD}(t) = \int_0^t \max(0, T(\tau) - T_{\text{base}}) \, d\tau\]

Discrete (daily) approximation:

\[\text{GDD}(n) = \sum_{i=1}^n \max(0, T_i - T_{\text{base}})\]

Threshold Detection

Event occurs when cumulative variable crosses threshold:

\[t^* = \min\{t : \text{GDD}(t) > \text{GDD}_{\text{crit}}\}\]

Example: Find day when cumulative GDD exceeds 200:

Check daily until condition met.

Sigmoid Response

Gradual transition instead of step function:

\[\text{Response} = \frac{1}{1 + e^{-k(x - x_0)}}\]

Where:

$x_0$ = threshold
$k$ = steepness parameter

For LAI: Smooth green-up instead of instant jump.

Summary

Phenology = timing of biological events (bud break, flowering, senescence)
Growing Degree Days (GDD): Accumulated heat above base temperature
Bud break occurs after threshold GDD accumulation (100–300 degree-days)
Photoperiod (day length) triggers senescence and flowering in some species
Chilling requirement: Winter cold needed before spring response
Growing season length: Time between spring green-up and autumn senescence
Climate warming → earlier spring, later autumn, longer growing season
GDD used for crop maturity prediction and variety selection
NDVI time series from satellites track phenology globally
Phenological shifts affect productivity, water use, frost risk, and species interactions