Wave speed and critical patch size for integro-difference equations with a strong Allee effect.

Area of Science:

• Mathematical Biology
• Ecological Modeling
• Dynamical Systems

Background:

• Integro-difference equations model population dynamics with spatial structure.
• The Allee effect describes reduced per capita fitness at low population densities.
• Habitat size and dispersal influence population persistence.

Purpose of the Study:

• To determine conditions for the existence and positivity of wave speed in an integro-difference equation with a strong Allee effect.
• To establish the existence of a critical patch size for population persistence in bounded habitats.
• To analyze the relationship between wave speed, critical patch size, and Allee threshold.

Main Methods:

• Analysis of integro-difference equations with strong Allee effects.
• Derivation of simplified conditions for wave speed existence and positivity.
• Development of an analytical integral formula for critical patch size using a Laplace dispersal kernel.
• Numerical simulations to validate theoretical findings.

Main Results:

• Simplified conditions for wave speed existence and positivity were established for unbounded habitats.
• The existence of a critical patch size was proven for bounded habitats.
• A positive wave speed implies persistence above a critical habitat size; negative wave speed leads to extinction.
• An analytical formula for critical patch size revealed multiple equilibrium solutions.

Conclusions:

• Wave speed and critical patch size are key factors in determining population persistence in spatially structured environments with Allee effects.
• The study provides a theoretical framework and numerical evidence for understanding ecological dynamics influenced by dispersal and density-dependent effects.
• The findings are applicable to conservation biology and habitat management strategies.