Persistence versus extinction for a class of discrete-time structured population models.

Area of Science:

• Mathematical Biology
• Dynamical Systems Theory
• Population Ecology

Background:

• Population dynamics are often modeled using discrete-time dynamical systems.
• These systems frequently involve linear (survival, development) and nonlinear (reproduction) components.
• Distinguishing population persistence from extinction is crucial in ecological studies.

Purpose of the Study:

• To establish precise conditions for determining population persistence or extinction.
• To analyze discrete-time dynamical systems generated by sums of linear and nonlinear operators.
• To extend previous persistence results by relaxing conditions on the nonlinear component.

Main Methods:

• Analysis of discrete-time dynamical systems on the positive cone of a Banach space.
• Utilizing a map composed of a linear contraction (A) and a compact, differentiable nonlinear perturbation (G).
• Application of the Krein-Rutman Theorem to find the principal eigenvalue of (II−A)(−1)G'(0).

Main Results:

• Sharp conditions for persistence and extinction were derived.
• The principal eigenvalue of (II−A)(−1)G'(0) serves as the critical threshold.
• Persistence is characterized by associated eigenfunctionals.

Conclusions:

• The study provides a robust mathematical framework for population persistence analysis.
• Results are broadly applicable to population models, including those with age, stage, or size structure.
• Demonstrated applicability to a plant model incorporating a seed bank, extending prior findings.