Demographic stochasticity and extinction in populations with Allee effect

Area of Science:

• Ecology
• Mathematical Biology
• Population Dynamics

Background:

• The Allee effect describes reduced per capita growth rates at low population densities.
• Demographic stochasticity plays a crucial role in population dynamics, especially for small populations.
• Deterministic models often simplify population dynamics, potentially overlooking stochastic effects.

Purpose of the Study:

• To investigate stochastic population models incorporating the Allee effect.
• To analyze the role of demographic stochasticity in population extinction.
• To compare approximate and exact solutions for the mean time to extinction.

Main Methods:

• Utilizing simple stochastic scenarios based on birth-and-death Markovian processes.
• Deriving mean-field deterministic equations from stochastic models.
• Applying the Wentzel-Kramers-Brillouin (WKB) approximation for extinction time analysis.
• Developing and using a recursive equation for exact solutions in special cases.
• Conducting numerical simulations to validate analytical findings.

Main Results:

• Recovered well-known deterministic evolution equations in the mean-field limit.
• Obtained the mean time to extinction using the WKB approximation for both strong and weak Allee effects.
• Found exact solutions for the mean time to extinction in specific scenarios.
• Identified conditions for the validity of the WKB approximation.
• Analyzed the transition boundary between weak and strong Allee effects through simulations.

Conclusions:

• Stochastic models provide a more comprehensive understanding of populations with the Allee effect.
• The WKB approximation offers a reliable method for estimating extinction times in many Allee effect scenarios.
• Exact solutions and numerical simulations are essential for validating approximations and understanding model boundaries.