[Paper Review] On the principle of competitive exclusion in metapopulation models

Davide Belocchio, Roberto Cavoretto+3 more|arXiv (Cornell University)|Mar 5, 2014

Stochastic processes and statistical mechanics20 references3 citations

TL;DR

This paper proposes a two-patch metapopulation model with asymmetric migration to study competitive exclusion in fragmented habitats. It demonstrates that while competitive exclusion generally holds at the metapopulation level, bidirectional migration can be overridden by unidirectional or one-way migration, enabling coexistence of competing species under specific conditions, challenging classical assumptions in ecological modeling.

ABSTRACT

In this paper we present and analyse a simple two populations model with migrations among two different environments. The populations interact by competing for resources. Equilibria are investigated. A proof for the boundedness of the populations is provided. A kind of competitive exclusion principle for metapopulation systems is obtained. At the same time we show that the competitive exclusion principle at the local patch level may be prevented to hold by the migration phenomenon, i.e. two competing populations may coexist, provided that only one of them is allowed to freely move or that migrations for both occur just in one direction.

Motivation & Objective

• To investigate how migration dynamics affect competitive exclusion in metapopulation systems with two competing species.
• To challenge the classical assumption that superior competitors always exclude inferior ones in patchy environments.
• To analyze conditions under which coexistence of competing populations can occur despite local competitive exclusion.
• To explore the impact of unidirectional or one-way migration on population persistence and equilibrium stability.

Proposed method

• Formulates a system of four coupled nonlinear differential equations modeling population dynamics in two patches with migration.
• Introduces distinct carrying capacities and intrinsic growth rates for each species in each patch, relaxing assumptions from prior models.
• Applies stability analysis and equilibrium classification to determine feasible and stable population states.
• Uses numerical simulations and basin of attraction analysis to visualize dynamic behavior under varying migration rates.
• Employs an algorithm from [4] to compute and compare basins of attraction for different migration scenarios.
• Considers special cases: one-way migration, unidirectional migration, and asymmetric migration rates to isolate effects on coexistence.

Experimental results

Research questions

• RQ1 Under what conditions does competitive exclusion fail to occur at the local patch level due to migration dynamics?
• RQ2 Can two competing species coexist in a metapopulation system when only one is capable of migration?
• RQ3 How does unidirectional migration influence the stability and feasibility of coexistence equilibria in a two-patch system?
• RQ4 What role do migration rates play in altering the basins of attraction for competing populations?
• RQ5 Can the classical competitive exclusion principle be extended to the metapopulation level under asymmetric migration?

Key findings

• When only one population is allowed to migrate, stable equilibria such as (8.0057, 0, 8.1962, 0) and (0, 5.7, 0, 6.5) are observed, indicating local exclusion but global persistence.
• With increased migration rate from patch 2 to patch 1, the equilibrium shifts to (9.3399, 0, 5.4726, 0), showing altered population distribution due to migration asymmetry.
• In unidirectional migration scenarios, equilibria like (7.875, 0, 8.3408, 0) and (0, 5.5860, 0, 6.6192) emerge, with coexistence possible in the destination patch if stability and feasibility conditions are met.
• Higher migration rates for a population increase its basin of attraction in the destination patch and reduce it in the source patch, consistent with intuitive expectations.
• Transcritical bifurcations occur when migration rates cross critical thresholds, leading to the emergence of new stable equilibria such as Ẽ6 and Ẽ11.
• The model reveals that classical assumptions of bidirectional migration and inevitable exclusion may not hold, enabling coexistence under unidirectional or one-way migration.