Boundedness, persistence and stability for classes of forced difference equations arising in population ecology

Area of Science:

• Mathematical Biology
• Population Dynamics
• Nonlinear Dynamics

Background:

• Nonlinear difference equations are crucial in modeling ecological and biological systems.
• Forcing terms represent external factors like harvesting, environmental changes, or migration.
• Understanding model stability and persistence is vital for predicting population behavior.

Purpose of the Study:

• To investigate boundedness and persistence properties of forced nonlinear difference equations.
• To develop stability conditions for non-zero equilibria that account for persistent forcing.
• To extend the analysis to infinite-dimensional models, including integral projection models (IPMs).

Main Methods:

• Analysis of nonlinear, possibly infinite-dimensional, forced difference equations.
• Development of sufficient conditions for uniform boundedness and persistence.
• Adaptation of input-to-state stability concepts for discrete dynamical systems.
• Application to population dynamics models.

Main Results:

• Sufficient conditions are provided for states to remain bounded and persistent uniformly with respect to forcing.
• A novel stability concept is introduced for non-zero equilibria, robust to persistent forcing.
• The framework accommodates infinite-dimensional models like integral projection models (IPMs).

Conclusions:

• The developed theory offers a robust framework for analyzing forced dynamical systems in ecology and biology.
• The stability conditions ensure predictable population dynamics despite external disturbances.
• The approach is applicable to a range of population dynamics examples, including IPMs.