A simple and bounded model of population dynamics for mutualistic networks

Juan Manuel Pastor; Javier García-Algarra; Javier Galeano; José María Iriondo; José J. Ramasco; Juan Manuel Pastor; Javier García-Algarra; Javier Galeano; José María Iriondo; José J. Ramasco

doi:10.3934/nhm.2015.10.53

Networks and Heterogeneous Media

2015, Volume 10, Issue 1: 53-70. doi: 10.3934/nhm.2015.10.53

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A simple and bounded model of population dynamics for mutualistic networks

1.
Complex System Group, Technical University of Madrid, Av. Puerta Hierro 4, 28040-Madrid
2.
Área de Biodiversidad y Conservación, Dept. Biología y Geología, Universidad Rey Juan Carlos, 28933 Móstoles
3.
Instituto de Física Interdisciplinar y Sistemas Complejos IFISC (CSIC-UIB), 07122 Palma de Mallorca

Received: 01 July 2014 Revised: 01 November 2014
Primary: 92D25, 92C42; Secondary: 92D40.

Dynamic population models are based on the Verhulst's equation (logisitic equation), where the classic Malthusian growth rate is damped by intraspecific competition terms. Mainstream population models for mutualism are modifications of the logistic equation with additional terms to account for the benefits produced by the interspecies interactions. These models have shortcomings as the population divergence under some conditions (May's equations) or a mathematical complexity that difficults their analytical treatment (Wright's type II models). In this work, we introduce a model for the population dynamics in mutualism inspired by the logistic equation but cured of divergences. The model is also mathematically more simple than the type II. We use numerical simulations to study the model stability in more general interaction scenarios. Despite its simplicity, our results suggest that the model dynamics are rich and may be used to gain further insights in the dynamics of mutualistic interactions.
- Population dynamics,
- Verhulst equation,
- extinction,
- carrying capacity,
- mutualistic network,
- Type II functional
Citation: Juan Manuel Pastor, Javier García-Algarra, Javier Galeano, José María Iriondo, José J. Ramasco. A simple and bounded model of population dynamics for mutualistic networks[J]. Networks and Heterogeneous Media, 2015, 10(1): 53-70. doi: 10.3934/nhm.2015.10.53

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Abstract

Dynamic population models are based on the Verhulst's equation (logisitic equation), where the classic Malthusian growth rate is damped by intraspecific competition terms. Mainstream population models for mutualism are modifications of the logistic equation with additional terms to account for the benefits produced by the interspecies interactions. These models have shortcomings as the population divergence under some conditions (May's equations) or a mathematical complexity that difficults their analytical treatment (Wright's type II models). In this work, we introduce a model for the population dynamics in mutualism inspired by the logistic equation but cured of divergences. The model is also mathematically more simple than the type II. We use numerical simulations to study the model stability in more general interaction scenarios. Despite its simplicity, our results suggest that the model dynamics are rich and may be used to gain further insights in the dynamics of mutualistic interactions.

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