Population extinction and quasi-stationary behavior in stochastic density-dependent structured models

Area of Science:

• Population dynamics modeling
• Mathematical biology
• Stochastic processes

Background:

• Structured population models are crucial for understanding population dynamics.
• Both density-independent and density-dependent factors influence population persistence.
• Stochasticity and determinism introduce different behaviors in population projections.

Purpose of the Study:

• To formulate and analyze discrete-time, structured population models.
• To compare deterministic and stochastic, density-independent and density-dependent models.
• To investigate the conditions for population extinction and persistence.

Main Methods:

• Formulation and numerical simulation of discrete-time, structured population models.
• Analysis of linear, density-independent models including Leslie matrix models and multitype branching processes.
• Generalization to nonlinear, density-dependent stochastic models and analysis of their behavior.

Main Results:

• In linear, density-independent models, extinction is certain if the dominant eigenvalue (lambda) is ≤ 1.
• For lambda > 1, deterministic models show exponential growth, while stochastic models retain a probability of extinction.
• Density-dependent stochastic models approximate density-independent ones for small populations, exhibiting similar extinction behaviors. For larger populations, they show a quasi-stationary distribution near the stable equilibrium.

Conclusions:

• Population extinction is a significant risk even in models with potential for growth, particularly for small populations.
• The behavior of nonlinear, density-dependent models can be understood by relating them to simpler linear, density-independent models.
• Population size is a critical factor influencing extinction probability and persistence time in stochastic models.