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March  2015, 10(1): 53-70. doi: 10.3934/nhm.2015.10.53

A simple and bounded model of population dynamics for mutualistic networks

Juan Manuel Pastor 1, , Javier García-Algarra 1, , Javier Galeano 1, , José María Iriondo 2, and  José J. Ramasco 3,

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, Spain
3. 

Instituto de Física Interdisciplinar y Sistemas Complejos IFISC (CSIC-UIB), 07122 Palma de Mallorca

Received  July 2014 Revised  November 2014 Published  February 2015

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.
Keywords: Verhulst equation, carrying capacity, Population dynamics, extinction, mutualistic network, Type II functional..
Mathematics Subject Classification: Primary: 92D25, 92C42; Secondary: 92D4.
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. Networks & Heterogeneous Media, 2015, 10 (1) : 53-70. doi: 10.3934/nhm.2015.10.53
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show all references

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Proceedings of the National Academy of Sciences USA, 106 (2009), 21484-21489. doi: 10.1073/pnas.0906910106.  Google Scholar

[2]

The Annual Review of Ecology, Evolution, and Systematics, 38 (2007), 567-593. doi: 10.1146/annurev.ecolsys.38.091206.095818.  Google Scholar

[3]

Nature, 458 (2009), 1018-1020. doi: 10.1038/nature07950.  Google Scholar

[4]

Journal of Theoretical Biology, 235 (2005), 531-539. doi: 10.1016/j.jtbi.2005.02.006.  Google Scholar

[5]

Acta Biotheoretica, 5 (1940), 51-66. doi: 10.1007/BF01602862.  Google Scholar

[6]

Ecological Modelling, 185 (2005), 147-151. doi: 10.1016/j.ecolmodel.2004.10.009.  Google Scholar

[7]

American Journal of Physics, 81 (2013), 725-732. doi: 10.1119/1.4813114.  Google Scholar

[8]

Mathematical Biosciences, 209 (2007), 361-385. doi: 10.1016/j.mbs.2007.02.004.  Google Scholar

[9]

Journal of Theoretical Biology, 328 (2013), 54-64. doi: 10.1016/j.jtbi.2013.03.016.  Google Scholar

[10]

Researches on population ecology, 14 (1991), 33-39. Google Scholar

[11]

1st edition, Roger Chew Weightman, Washington, 1798. Available from: http://opac.newsbank.com/select/shaw/17975. Google Scholar

[12]

in Theoretical Ecology. Principles and Applications, $2^{nd}$ edition (ed. R. May), 1981, 78-104. Google Scholar

[13]

$3^{rd}$ edition, Springer-Verlag, New York, 2002.  Google Scholar

[14]

Zeitschrift für Induktive Abstammungs- und Vererbungslehre, 49 (1929), 336-338. doi: 10.1007/BF01847581.  Google Scholar

[15]

Science, 309 (2005), p102. doi: 10.1126/science.309.5731.102.  Google Scholar

[16]

Nouveaux Memoires de l'Academie Royale des Sciences et Belles-Lettres de Bruxelles, 18 (1845), 1-42. Google Scholar

[17]

Nature, 118 (1926), 558-560. doi: 10.1038/118558a0.  Google Scholar

[18]

The American Naturalist, 134 (1989), 664-667. doi: 10.1086/285003.  Google Scholar

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