Populations in environments with a soft carrying capacity are eventually extinct.

Area of Science:

• Population dynamics
• Stochastic processes
• Mathematical biology

Background:

• Population sizes change through reproduction and death.
• Extinction is a final state where population size reaches zero.
• Models often consider Galton-Watson or birth-and-death processes.

Purpose of the Study:

• To investigate the long-term behavior of populations with stepwise size changes.
• To determine if populations modeled with a carrying capacity are guaranteed to go extinct.
• To explore the mathematical underpinnings of population extinction.

Main Methods:

• Modeling population size changes as a stepwise process.
• Incorporating a carrying capacity (K) where population growth is non-positive above K.
• Utilizing supermartingale convergence properties to analyze population extinction.
• Assuming a lower bound on the probability of individual death to avoid parity issues.

Main Results:

• The study proves that all populations modeled under these conditions will inevitably go extinct.
• A carrying capacity ensures that population growth is limited, contributing to eventual decline.
• The probability of reaching the extinction state is positive.

Conclusions:

• The mathematical framework demonstrates that extinction is an unavoidable outcome for these population models.
• The findings have implications for understanding population persistence and extinction risks in ecological and biological systems.
• The model provides a robust theoretical basis for population extinction, irrespective of specific life-span distributions or interdependencies.