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Stochasticity, invasions, and branching random walks.
Mark Kot1, Jan Medlock, Timothy Reluga
1Department of Applied Mathematics, Box 352420, University of Washington, Seattle, WA 98195-2420, USA. kot@amath.washington.edu
Theoretical Population Biology
|October 7, 2004
Summary
This study connects mathematical models to simulations of individual organisms. It finds that random variations can cause extinction but do not slow down invasion speeds.
Area of Science:
- Mathematical Biology
- Ecology
- Population Dynamics
Background:
- Deterministic integrodifference equations model population dynamics.
- Stochastic, individual-based simulations offer alternative perspectives.
- Linking these approaches is crucial for comprehensive ecological understanding.
Purpose of the Study:
- To bridge deterministic mathematical models with stochastic simulations.
- To analyze invasion speeds in ecological models.
- To investigate the impact of demographic stochasticity on invasion dynamics.
Main Methods:
- Utilizing branching random walks to connect integrodifference equations and individual-based models.
- Applying standard methods to calculate invasion speeds.
- Analyzing density-independent branching random walks.
Main Results:
- Invasion speeds were determined for both average population densities and the furthest-advancing individuals.
- Demographic stochasticity, in density-independent scenarios, can lead to population extinction.
- Stochasticity does not impede the overall asymptotic invasion speed or prevent accelerating invasions.
Conclusions:
- Branching random walks effectively link deterministic and stochastic ecological models.
- Demographic stochasticity introduces extinction risk but does not alter asymptotic invasion rates.
- Invasions can continue to accelerate even with random individual variations.