# American Institute of Mathematical Sciences

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

## 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 Conservación, Dept. Biología y Geología, Universidad Rey Juan Carlos, 28933 Móstoles, 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.
Citation: Juan Manuel Pastor, Javier García-Algarra, Javier Galeano, José María Iriondo, José 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
##### 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. doi: 10.1073/pnas.0906910106. Google Scholar [2] J. Bascompte and P. Jordano, Plant-animal mutualistic networks: The architecture of biodiversity,, The Annual Review of Ecology, 38 (2007), 567. doi: 10.1146/annurev.ecolsys.38.091206.095818. Google Scholar [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. doi: 10.1038/nature07950. Google Scholar [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. doi: 10.1016/j.jtbi.2005.02.006. Google Scholar [5] W. Feller, On the logistic law of growth and its empirical verifications in biology,, Acta Biotheoretica, 5 (1940), 51. doi: 10.1007/BF01602862. Google Scholar [6] J. P. Gabriel, F. Saucy and L. F. Bersier, Paradoxes in the logistic equation?,, Ecological Modelling, 185 (2005), 147. doi: 10.1016/j.ecolmodel.2004.10.009. Google Scholar [7] J. R. Groff, Exploring dynamical systems and chaos using the logistic map model of population change,, American Journal of Physics, 81 (2013), 725. doi: 10.1119/1.4813114. Google Scholar [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. doi: 10.1016/j.mbs.2007.02.004. Google Scholar [9] C. A. Johnson and P. Amarasekare, Competitionfor benefits can promote the persistence of mutualistics interactions,, Journal of Theoretical Biology, 328 (2013), 54. doi: 10.1016/j.jtbi.2013.03.016. Google Scholar [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. Google Scholar [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, (1798). Google Scholar [12] R. May, Models for two interacting populations,, in Theoretical Ecology. Principles and Applications, (1981), 78. Google Scholar [13] J. D. Murray, Mathematical Biology I: An Introduction,, $3^{rd}$ edition, (2002). Google Scholar [14] R. Pearl, The biology of population growth,, Zeitschrift für Induktive Abstammungs- und Vererbungslehre, 49 (1929), 336. doi: 10.1007/BF01847581. Google Scholar [15] E. Stokstad, Will malthus continue to be wrong?,, Science, 309 (2005). doi: 10.1126/science.309.5731.102. Google Scholar [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. Google Scholar [17] V. Volterra, Fluctuations in the abundance of a species considered mathematically,, Nature, 118 (1926), 558. doi: 10.1038/118558a0. Google Scholar [18] D. H. Wright, A simple, stable model of mutualism incorporating handling time,, The American Naturalist, 134 (1989), 664. doi: 10.1086/285003. Google Scholar

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##### 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. doi: 10.1073/pnas.0906910106. Google Scholar [2] J. Bascompte and P. Jordano, Plant-animal mutualistic networks: The architecture of biodiversity,, The Annual Review of Ecology, 38 (2007), 567. doi: 10.1146/annurev.ecolsys.38.091206.095818. Google Scholar [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. doi: 10.1038/nature07950. Google Scholar [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. doi: 10.1016/j.jtbi.2005.02.006. Google Scholar [5] W. Feller, On the logistic law of growth and its empirical verifications in biology,, Acta Biotheoretica, 5 (1940), 51. doi: 10.1007/BF01602862. Google Scholar [6] J. P. Gabriel, F. Saucy and L. F. Bersier, Paradoxes in the logistic equation?,, Ecological Modelling, 185 (2005), 147. doi: 10.1016/j.ecolmodel.2004.10.009. Google Scholar [7] J. R. Groff, Exploring dynamical systems and chaos using the logistic map model of population change,, American Journal of Physics, 81 (2013), 725. doi: 10.1119/1.4813114. Google Scholar [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. doi: 10.1016/j.mbs.2007.02.004. Google Scholar [9] C. A. Johnson and P. Amarasekare, Competitionfor benefits can promote the persistence of mutualistics interactions,, Journal of Theoretical Biology, 328 (2013), 54. doi: 10.1016/j.jtbi.2013.03.016. Google Scholar [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. Google Scholar [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, (1798). Google Scholar [12] R. May, Models for two interacting populations,, in Theoretical Ecology. Principles and Applications, (1981), 78. Google Scholar [13] J. D. Murray, Mathematical Biology I: An Introduction,, $3^{rd}$ edition, (2002). Google Scholar [14] R. Pearl, The biology of population growth,, Zeitschrift für Induktive Abstammungs- und Vererbungslehre, 49 (1929), 336. doi: 10.1007/BF01847581. Google Scholar [15] E. Stokstad, Will malthus continue to be wrong?,, Science, 309 (2005). doi: 10.1126/science.309.5731.102. Google Scholar [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. Google Scholar [17] V. Volterra, Fluctuations in the abundance of a species considered mathematically,, Nature, 118 (1926), 558. doi: 10.1038/118558a0. Google Scholar [18] D. H. Wright, A simple, stable model of mutualism incorporating handling time,, The American Naturalist, 134 (1989), 664. doi: 10.1086/285003. Google Scholar
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