Winter Visitor Model

Some species are present only during the winter months. They arrive in autumn, persist through winter, and depart again in spring — their period of presence spanning the end and beginning of the year.

This model describes that pattern: a seasonal presence that wraps around the year boundary, with a winter peak and near-absence through summer.

Concept

This model describes species that are present during the winter months, with activity spanning the end and beginning of the year.

It answers the question:

When is the species present?

Like the seasonal model, presence is limited to part of the year. Unlike it, the active period crosses the year boundary.

The model defines a seasonal target, representing expected activity through the year. The observed signal then adjusts towards this target over time.

The target combines:

• A winter component, representing peak presence
• An optional autumn component, representing arrival
• A summer suppression, reducing activity during the off-season

Together, these produce a cycle that rises through autumn, peaks in winter, and falls away into spring.

Model Parameters

A small number of parameters control the behaviour of the model:

Together, these parameters define:

• When the species arrives and peaks
• How concentrated the winter period is
• How gradually or abruptly the species appears and disappears
• How strongly the species is absent through summer

All timing parameters are expressed in months on a circular 12-month scale.

Mathematical Form

The model is a first-order system:

dy/dt = rate × (target(t) - y)

Where:

• y(t) is a relative, dimensionless measure of observable activity
• target(t) is the seasonal activity target
• rate is selected using GROWTH_RATE or DECAY_RATE depending on whether the signal is rising or falling

The target function is constructed from smooth periodic components:

target(t) = winter(t) + autumn(t) - summer(t) + BASELINE

Each component is a smooth function over a 12-month cycle, allowing continuous variation without discontinuities.

Model Behaviour

When applied over a full year, the model produces a winter-centred cycle:

• Activity rises through autumn as the species arrives
• Peaks in mid-winter
• Declines through late winter and early spring
• Remains close to zero through summer

The shape depends on:

• The timing and strength of the winter and autumn components
• The breadth of the winter peak
• The strength of summer suppression
• The rate at which the system responds to change

Unlike the seasonal presence model, the season is not bounded within a single part of the calendar year. Instead, it wraps across the year boundary.

Fitting to Observations

The model can be fitted to observed monthly data.

A parameter fitting process:

• Infers a plausible seasonal structure from the data
• Generates candidate parameter sets
• Runs the model
• Compares simulated and observed curves
• Scores the match
• Repeats to identify good solutions

This produces a set of parameters that describe the species’ seasonal behaviour.

These parameters are broadly interpretable:

• WINTER_PEAK → timing of peak presence
• AUTUMN_PEAK → timing of arrival
• WINTER_WIDTH → concentration of winter activity
• AUTUMN_WIDTH → spread of the arrival phase
• SUMMER_DIP / SUMMER_LOW → strength and timing of absence

Together, they provide a compact description of a species’ winter pattern.

As with the other models:

• Parameters are estimates rather than exact values
• Different combinations may produce similar curves
• Interpretation is most reliable when considered alongside the fitted curve

Normalisation

Model outputs are expressed as a relative measure of activity.

To allow comparison across species, results are normalised so that:

• 1.0 → peak activity
• 0.5 → half of peak activity
• 0.0 → effectively zero

This focuses attention on the timing and shape of seasonal variation.

Example

Redwing

Observed data show:

• Arrival in autumn
• A peak through mid-winter
• Departure in spring
• Absence through summer

The fitted model describes this pattern using:

• A winter peak centred early in the year
• A weaker autumn component representing arrival
• Strong summer suppression, reducing activity to near zero

The resulting curve captures:

• A seasonal cycle that wraps across the year boundary
• A concentrated winter presence with extended absence

Seasonal signature (modelled):

• Presence: November/December–March
• Peak: December–January
• Width: moderate
• Decline / absence: strong summer absence, with little residual activity

Interpretation

This model represents species that are present only during the winter period, with activity spanning the year boundary.

It provides a minimal explanation for patterns seen in the Seasonal Analyses, showing that a small number of simple processes can produce:

• Winter-centred presence
• Distinct arrival phases
• Extended absence through summer

The model does not attempt to describe detailed ecological mechanisms. Instead, it offers a way of understanding how seasonal structure and timing combine to produce the observed patterns.

In Context

Within the broader modelling framework, this model corresponds to species that are:

• Absent for much of the year
• Present across the year boundary
• Peaking in winter

These contrast with:

• Seasonal presence species, active within a bounded window
• Resident species, present throughout the year but variably detectable

Together, these models describe three distinct ways in which species occupy the year.