Amendment to: populations in environments with a soft carrying capacity are eventually extinct.

Area of Science:

• Mathematical Biology
• Population Dynamics
• Stochastic Processes

Background:

• Investigates population dynamics under discrete time changes.
• Builds upon existing models of population size regulation.
• Considers scenarios with arbitrary and random time points for population changes.

Purpose of the Study:

• To refine and sharpen previous results on population extinction.
• To establish conditions under which a population is guaranteed to die out.
• To analyze the impact of carrying capacity on population stability.

Main Methods:

• Utilizes conditional expectation to model population change.
• Incorporates conditional probability of population decrease.
• Analyzes discrete-time population models with stochastic elements.

Main Results:

• Demonstrates that a negative conditional expected change above carrying capacity leads to extinction.
• Shows that a high conditional probability of decrease below carrying capacity ensures population collapse.
• Provides a sharpened result for population extinction in discrete-time models.

Conclusions:

• The interplay between expected population change and extinction probability is critical.
• Populations with specific dynamics relative to carrying capacity are prone to extinction.
• The findings offer a robust condition for guaranteed population die-out.