Field Notes Journal Entry
Seasonal Patterns and Simple Models
Exploring how simple mathematical models can reproduce the seasonal patterns observed in wildlife records
In the Seasonal Analyses, I’ve described the year through repeated encounters with the same species, recorded over time and examined for pattern.
Those patterns are often striking. Some species appear briefly and then vanish. Others are present throughout the year, but vary in how readily they are seen. Over time, these differences become familiar: the short, intense burst of bluebells in spring; the sudden arrival and departure of swifts; the quieter but persistent presence of robins through winter.
Having spent some time describing those patterns, I became interested in a slightly different question:
Could simple processes produce the shapes seen in the data?
From Observation to Model
The idea was not to build a detailed ecological model, but something much simpler — a way of testing whether a small number of assumptions might be sufficient to reproduce the observed curves.
Two distinct patterns suggested two different approaches.
Some species are only present for part of the year. They appear within a seasonal window, persist for a time, and then disappear again. Others are always present, but vary in detectability, becoming more or less visible as behaviour and conditions change.
These differences led to two small models.
Seasonal Presence
For species such as bluebell or swift, the pattern is one of appearance and disappearance.
The model for these species combines three elements:
- A smooth seasonal driver, representing changing conditions
- A window limiting when presence is possible
- A decay term allowing activity to fall away outside that window
Together, these produce a single seasonal pulse — a rise into the season, a peak, and a decline.
When compared with observed data, the model captures the brevity and timing of strongly seasonal species. The exact details vary, but the overall shape — the presence of a defined window and a relatively rapid decline — emerges from these simple assumptions.
Resident Detectability
Other species follow a different pattern. The robin, for example, is present throughout the year, but its visibility changes with season and behaviour.
Here, the model does not attempt to create or remove presence. Instead, it defines a seasonal pattern of expected detectability, and allows the observed signal to adjust towards it over time.
The result is a continuous curve:
- Higher in winter and early spring
- Lower through summer
- Rising again towards the end of the year
This reflects a change not in presence, but in how readily the species is observed.
A Way of Thinking
These models are deliberately simple. They do not attempt to describe ecological processes in detail, nor are they intended to predict future observations.
Instead, they act as a way of thinking — a means of asking whether the patterns seen in the data might arise from a small number of underlying processes.
In both cases, the answer seems to be: often, yes.
A seasonal window, combined with growth and decline, is enough to produce the sharply bounded patterns seen in flowering plants or migratory species. A continuous baseline, combined with seasonal variation in detectability, produces the more gradual cycles seen in resident species.
In Context
The modelling sits alongside the Seasonal Analyses. The analyses describe how species occupy the year; the models offer a simple account of how those patterns might arise.
They are not a replacement for observation, and they do not attempt to explain everything. But they provide a different perspective — one that connects pattern with process, even if only in the most minimal sense.