Stochastic population growth in spatially heterogeneous environments: the density-dependent case

Area of Science:

• Mathematical Biology
• Ecology
• Stochastic Processes

Background:

• Population dynamics are influenced by environmental stochasticity, competition, and dispersal.
• Understanding long-term population behavior requires analyzing complex ecological models.

Purpose of the Study:

• To investigate the dynamics of a structured population under combined environmental and ecological pressures.
• To determine the role of the stochastic growth rate in population persistence and convergence.
• To generalize and extend existing models of population dynamics with stochasticity.

Main Methods:

• Modeling population abundances using nonlinear stochastic differential equations across n patches.
• Analyzing the stochastic growth rate as a Lyapunov exponent of a linearized system.
• Investigating convergence properties under different conditions of the stochastic growth rate.

Main Results:

• The stochastic growth rate (r) dictates long-term population behavior: positive r leads to convergence to a stable measure, negative r to extinction.
• Persistence is robust to perturbations in growth rates, dispersal, and noise.
• Degenerate environmental noise, including correlated patch environments, is analyzed.

Conclusions:

• The stochastic growth rate is a critical determinant of population persistence and stability.
• The model provides a generalized framework for understanding structured populations in stochastic environments.
• Specific analysis of the two-patch case reveals that coupling sink patches does not guarantee persistence.