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Winnerless competition in coupled Lotka-Volterra maps
L A González-Díaz1, E D Gutiérrez, P Varona
1Laboratorio de Dinámica Estocástica, Centro de Física, Instituto Venezolano de Investigaciones Científicas, Caracas 1020-A, Venezuela. lagonzalezd@gmail.com
Summary
Winnerless competition dynamics in Lotka-Volterra type coupled maps are fully exhibited under specific parameter conditions. These conditions reveal structurally stable heteroclinic cycles, with analytical expressions for residence times derived for N-dimensional cases.
Area of Science:
- Dynamical Systems
- Theoretical Ecology
- Mathematical Biology
Background:
- Winnerless competition describes ecological scenarios where no single species consistently dominates.
- Lotka-Volterra models are fundamental in population dynamics.
- Coupled map lattices offer a framework for studying complex spatio-temporal dynamics.
Purpose of the Study:
- To analyze winnerless competition in discrete-time Lotka-Volterra type coupled maps of arbitrary dimensions.
- To deduce conditions for the emergence of structurally stable heteroclinic cycles.
- To derive an analytical expression for residence times in N-dimensional systems.
Main Methods:
- Analysis of coupled maps with discrete temporal evolution.
- Deduction of necessary and sufficient conditions for heteroclinic cycles.
- Computational recreation of dynamics for low, intermediate, and high dimensions.
- Derivation of an analytical expression for residence times.
Main Results:
- Identified necessary and sufficient conditions for structurally stable heteroclinic cycles.
- Demonstrated that these conditions fully exhibit winnerless competition dynamics.
- Successfully recreated dynamics across various dimensions (low, intermediate, high).
- Derived an analytical expression for residence times validated by simulations.
Conclusions:
- Winnerless competition in these systems is governed by specific parameter-dependent conditions leading to heteroclinic cycles.
- The derived analytical expression for residence times is accurate for N-dimensional systems.
- The study provides a comprehensive understanding of winnerless competition dynamics in a generalized Lotka-Volterra framework.
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