Reduction of slow-fast discrete models coupling migration and demography

Area of Science:

• Mathematical Biology
• Dynamical Systems Theory
• Population Ecology

Background:

• Many population dynamics models involve processes occurring at different time scales.
• Simplifying these multi-time scale systems is crucial for analytical tractability.

Purpose of the Study:

• To apply the aggregation of variables method to reduce two-time scale discrete nonlinear dynamical systems.
• To analyze population dynamics models with slow and fast demographic and migratory processes.

Main Methods:

• Developed a reduced system using the aggregation of variables method.
• Applied the reduction technique to discrete nonlinear dynamical systems.
• Investigated population models with demographic and migratory processes at distinct time scales.

Main Results:

• When demography (Leslie type) is faster than migration, the reduced dynamics exhibit a multi-attractor scenario.
• When migration is faster than demography, the reduction yields novel interpretations of established discrete models.
• This includes the emergence of Allee effect scenarios.

Conclusions:

• The aggregation of variables method effectively simplifies complex two-time scale population dynamics.
• The relative speeds of demographic and migratory processes significantly influence the emergent dynamics and model interpretations.
• This approach offers new insights into ecological dynamics, including Allee effects.