Winter Visitor Model

This model represents species that are present only during the winter period, with a winter peak, activity spanning the year boundary, and near-absence through spring and summer.

It provides a minimal explanation for patterns seen in the seasonal analysis of observations, 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.

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.

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.

Parameter Interpretation

After parameter fitting, the parameters are broadly interpretable as follows:

• 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

As with all simple models:

• Parameters should be treated as estimates rather than exact dates
• Different combinations may produce similar curves
• Interpretation is most reliable when considered alongside the fitted curve itself

In practice, each species can be described by both:

• Its fitted parameters
• The shape of its simulated seasonal curve

Together, these form a compact description of seasonal presence.