A simple and bounded model of population dynamics for mutualistic networks

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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

Abstract
Related Papers

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

References:

[1]	D. Balcan, V. Colizza, B. Gonçalves, H. Hu, J. J. Ramasco and A. Vespignani, Multiscale mobility networks and the spatial spreading of infectious diseases, Proceedings of the National Academy of Sciences USA, 106 (2009), 21484-21489. doi: 10.1073/pnas.0906910106.
[2]	J. Bascompte and P. Jordano, Plant-animal mutualistic networks: The architecture of biodiversity, The Annual Review of Ecology, Evolution, and Systematics, 38 (2007), 567-593. doi: 10.1146/annurev.ecolsys.38.091206.095818.
[3]	U. Bastolla, M. A. Fortuna, A. Pascual-García, A. Ferrera, B. Luque and J. Bascompte, The architecture of mutualistic networks minimizes competition and increases biodiversity, Nature, 458 (2009), 1018-1020. doi: 10.1038/nature07950.
[4]	U. Bastolla, M. Lässig, S. C. Manrubia and A. Valleriani, Biodiversity in model ecosystems, II: Species assembly and food web structure, Journal of Theoretical Biology, 235 (2005), 531-539. doi: 10.1016/j.jtbi.2005.02.006.
[5]	W. Feller, On the logistic law of growth and its empirical verifications in biology, Acta Biotheoretica, 5 (1940), 51-66. doi: 10.1007/BF01602862.
[6]	J. P. Gabriel, F. Saucy and L. F. Bersier, Paradoxes in the logistic equation?, Ecological Modelling, 185 (2005), 147-151. doi: 10.1016/j.ecolmodel.2004.10.009.
[7]	J. R. Groff, Exploring dynamical systems and chaos using the logistic map model of population change, American Journal of Physics, 81 (2013), 725-732. doi: 10.1119/1.4813114.
[8]	L. Gustafsson and M. Sternad, Bringing consistency to simulation of population models-Poisson simulation as a bridge between micro and macro simulation, Mathematical Biosciences, 209 (2007), 361-385. doi: 10.1016/j.mbs.2007.02.004.
[9]	C. A. Johnson and P. Amarasekare, Competitionfor benefits can promote the persistence of mutualistics interactions, Journal of Theoretical Biology, 328 (2013), 54-64. doi: 10.1016/j.jtbi.2013.03.016.
[10]	E. Kuno, Some strange properties of the logistic equation defined with r and k: Inherent defects or artifacts?, Researches on population ecology, 14 (1991), 33-39.
[11]	T. R. Malthus, An Essay on the Principle of Population or a View of Its Past and Present Effects on Human Happiness; with an Inquiry into Our Prospects Respecting the Future Removal on Mitigation of the Evils which It Occasions, 1st edition, Roger Chew Weightman, Washington, 1798. Available from: http://opac.newsbank.com/select/shaw/17975.
[12]	R. May, Models for two interacting populations, in Theoretical Ecology. Principles and Applications, $2^{nd}$ edition (ed. R. May), 1981, 78-104.
[13]	J. D. Murray, Mathematical Biology I: An Introduction, $3^{rd}$ edition, Springer-Verlag, New York, 2002.
[14]	R. Pearl, The biology of population growth, Zeitschrift für Induktive Abstammungs- und Vererbungslehre, 49 (1929), 336-338. doi: 10.1007/BF01847581.
[15]	E. Stokstad, Will malthus continue to be wrong?, Science, 309 (2005), p102. doi: 10.1126/science.309.5731.102.
[16]	P. F. Verhulst, Recherches mathematiques sur la loi d'accroissement de la population [Mathematical researches into the law of population growth increase], Nouveaux Memoires de l'Academie Royale des Sciences et Belles-Lettres de Bruxelles, 18 (1845), 1-42.
[17]	V. Volterra, Fluctuations in the abundance of a species considered mathematically, Nature, 118 (1926), 558-560. doi: 10.1038/118558a0.
[18]	D. H. Wright, A simple, stable model of mutualism incorporating handling time, The American Naturalist, 134 (1989), 664-667. doi: 10.1086/285003.

show all references

References:

[1]	D. Balcan, V. Colizza, B. Gonçalves, H. Hu, J. J. Ramasco and A. Vespignani, Multiscale mobility networks and the spatial spreading of infectious diseases, Proceedings of the National Academy of Sciences USA, 106 (2009), 21484-21489. doi: 10.1073/pnas.0906910106.
[2]	J. Bascompte and P. Jordano, Plant-animal mutualistic networks: The architecture of biodiversity, The Annual Review of Ecology, Evolution, and Systematics, 38 (2007), 567-593. doi: 10.1146/annurev.ecolsys.38.091206.095818.
[3]	U. Bastolla, M. A. Fortuna, A. Pascual-García, A. Ferrera, B. Luque and J. Bascompte, The architecture of mutualistic networks minimizes competition and increases biodiversity, Nature, 458 (2009), 1018-1020. doi: 10.1038/nature07950.
[4]	U. Bastolla, M. Lässig, S. C. Manrubia and A. Valleriani, Biodiversity in model ecosystems, II: Species assembly and food web structure, Journal of Theoretical Biology, 235 (2005), 531-539. doi: 10.1016/j.jtbi.2005.02.006.
[5]	W. Feller, On the logistic law of growth and its empirical verifications in biology, Acta Biotheoretica, 5 (1940), 51-66. doi: 10.1007/BF01602862.
[6]	J. P. Gabriel, F. Saucy and L. F. Bersier, Paradoxes in the logistic equation?, Ecological Modelling, 185 (2005), 147-151. doi: 10.1016/j.ecolmodel.2004.10.009.
[7]	J. R. Groff, Exploring dynamical systems and chaos using the logistic map model of population change, American Journal of Physics, 81 (2013), 725-732. doi: 10.1119/1.4813114.
[8]	L. Gustafsson and M. Sternad, Bringing consistency to simulation of population models-Poisson simulation as a bridge between micro and macro simulation, Mathematical Biosciences, 209 (2007), 361-385. doi: 10.1016/j.mbs.2007.02.004.
[9]	C. A. Johnson and P. Amarasekare, Competitionfor benefits can promote the persistence of mutualistics interactions, Journal of Theoretical Biology, 328 (2013), 54-64. doi: 10.1016/j.jtbi.2013.03.016.
[10]	E. Kuno, Some strange properties of the logistic equation defined with r and k: Inherent defects or artifacts?, Researches on population ecology, 14 (1991), 33-39.
[11]	T. R. Malthus, An Essay on the Principle of Population or a View of Its Past and Present Effects on Human Happiness; with an Inquiry into Our Prospects Respecting the Future Removal on Mitigation of the Evils which It Occasions, 1st edition, Roger Chew Weightman, Washington, 1798. Available from: http://opac.newsbank.com/select/shaw/17975.
[12]	R. May, Models for two interacting populations, in Theoretical Ecology. Principles and Applications, $2^{nd}$ edition (ed. R. May), 1981, 78-104.
[13]	J. D. Murray, Mathematical Biology I: An Introduction, $3^{rd}$ edition, Springer-Verlag, New York, 2002.
[14]	R. Pearl, The biology of population growth, Zeitschrift für Induktive Abstammungs- und Vererbungslehre, 49 (1929), 336-338. doi: 10.1007/BF01847581.
[15]	E. Stokstad, Will malthus continue to be wrong?, Science, 309 (2005), p102. doi: 10.1126/science.309.5731.102.
[16]	P. F. Verhulst, Recherches mathematiques sur la loi d'accroissement de la population [Mathematical researches into the law of population growth increase], Nouveaux Memoires de l'Academie Royale des Sciences et Belles-Lettres de Bruxelles, 18 (1845), 1-42.
[17]	V. Volterra, Fluctuations in the abundance of a species considered mathematically, Nature, 118 (1926), 558-560. doi: 10.1038/118558a0.
[18]	D. H. Wright, A simple, stable model of mutualism incorporating handling time, The American Naturalist, 134 (1989), 664-667. doi: 10.1086/285003.

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