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Multiple attractors and boundary crises in a tri-trophic food chain
M P Boer1, B W Kooi, S A Kooijman
1Department of Theoretical Biology, Faculty of Biology, Institute of Ecological Science, Vrije Universiteit, De Boelelaan 1087, 1081 HV, Amsterdam, The Netherlands. m.p.boer@plant.wag-ur.nl
Mathematical Biosciences
|February 13, 2001
Summary
This study analyzes a tri-trophic food chain model, revealing how environmental enrichment can lead to the extinction of the top predator through complex bifurcations and chaotic dynamics.
Area of Science:
- Ecology
- Mathematical Biology
- Dynamical Systems
Background:
- Ecological models, particularly food chains, exhibit complex dynamics.
- The Monod growth model is a standard for describing microbial growth.
- Understanding population dynamics is crucial for ecosystem stability.
Purpose of the Study:
- To analyze the asymptotic behavior of a tri-trophic food chain model.
- To relate the model's bifurcation structure to a simplified one-dimensional map.
- To investigate the impact of environmental enrichment on population stability.
Main Methods:
- Analysis of a non-linear dynamical system of four ordinary differential equations.
- Reduction of system dimension using mass conservation.
- Construction and analysis of a two-dimensional Poincaré next-return map.
- Study of a one-dimensional bi-modal, non-invertible map to understand bifurcations.
Main Results:
- The food chain model's dynamics can be understood through the bifurcations of a simplified one-dimensional map.
- Coexistence of two attractors: a stable limit cycle (top predator absent) and an interior attractor.
- The stable manifold of the predator-absent limit cycle forms a complex basin boundary.
- Homoclinic bifurcations are linked to boundary crises, leading to sudden chaotic attractor disappearance and top-predator extinction.
Conclusions:
- Environmental enrichment can destabilize tri-trophic food chains, leading to top predator extinction.
- Complex dynamics, including chaos and bifurcations, play a significant role in ecosystem stability.
- The study provides insights into the 'Paradox of Enrichment' in more complex food webs.
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