Biodiversity and vulnerability in a 3D mutualistic system

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October 2014, 34(10): 4107-4126. doi: 10.3934/dcds.2014.34.4107

Biodiversity and vulnerability in a 3D mutualistic system

Giovanny Guerrero ^1, , José Antonio Langa ^2, and Antonio Suárez ^3,

1.	Universidad Central del Ecuador, Ciudadela Universitaria, Av. América S/N, Quito, Ecuador
2.	Departamento de Ecuaciones Diferenciales y Análisis Numérico, Facultad de Matemáticas, Universidad de Sevilla, Calle Tarfia s/n, 41012-Seville, Spain
3.	Dpto. de Ecuaciones Diferenciales y Análisis Numérico, Fac. de Matemáticas, Univ. de Sevilla, C/. Tarfia s/n, 41012 - Sevilla

Received January 2013 Revised February 2013 Published April 2014

Abstract
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In this paper we study a three dimensional mutualistic model of two plants in competition and a pollinator with cooperative relation with plants. We compare the dynamical properties of this system with the associated one under absence of the pollinator. We observe how cooperation is a common fact to increase biodiversity, which it is known that, generically, holds for general mutualistic dynamical systems in Ecology as introduced in [4]. We also give mathematical evidence on how a cooperative species induces an increased biodiversity, even if the species is push to extinction. For this fact, we propose a necessary change in the model formulation which could explain this kind of phenomenon.

Keywords: increasing biodiversity, 3D mutualistic model, global stability., permanence, vulnerability.

Mathematics Subject Classification: Primary: 92B05, 34A34, 34D05, 34D2.

Citation: Giovanny Guerrero, José Antonio Langa, Antonio Suárez. Biodiversity and vulnerability in a 3D mutualistic system. Discrete & Continuous Dynamical Systems - A, 2014, 34 (10) : 4107-4126. doi: 10.3934/dcds.2014.34.4107

References:

[1]	J. Bascompte, P. Jordano, C. J. Melián and J. M. Olesen, The nested assembly of plant-animal mutualistic networks,, Proc. Natl Acad. Sci. USA, 100 (2003), 9383. doi: 10.1073/pnas.1633576100. Google Scholar
[2]	J. Bascompte, P. Jordano and J. M. Olesen, Asymmetric coevolutionary networks facilitate biodiversity maintenance,, Science, 312 (2006), 431. doi: 10.1126/science.1123412. Google Scholar
[3]	J. Bascompte and P. Jordano, The structure of plant-animal mutualistic networks: The architecture of biodiversity,, Annu. Rev. Ecol. Evol. Syst., 38 (2007), 567. Google Scholar
[4]	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
[5]	R. S. Cantrell and C. Cosner, Spatial Ecology Via Reaction-Diffusion Equations,, Wiley Series in Mathematical and Computational Biology, (2003). doi: 10.1002/0470871296. Google Scholar
[6]	C. E. Clark and T. G. Hallam, The community matrix in three species community models,, J. Math. Biology, 16 (1982), 25. doi: 10.1007/BF00275158. Google Scholar
[7]	B. S. Goh, Stability in models of mutualism,, The American Naturalist, 111 (1977), 135. doi: 10.1086/283384. Google Scholar
[8]	B. S. Goh, Stability in models of mutualism,, The American Naturalist, 113 (1979), 261. doi: 10.1086/283384. Google Scholar
[9]	V. Hutson, A theorem on average Lyapunov functions,, Monatsch. Math., 98 (1984), 267. doi: 10.1007/BF01540776. Google Scholar
[10]	J. D. Murray, Mathematical Biology,, Springer, (1993). doi: 10.1007/b98869. Google Scholar
[11]	S. Saavedra, D. B. Stouffer, B. Uzzi and J. Bascompte, Strong contributors to network persistence are the most vulnerable to extinction,, Nature, 478 (2011), 233. doi: 10.1038/nature10433. Google Scholar
[12]	H. L. Smith, Competing subcommunities of mutualists and a generalized Kamke theorem,, SIAM J. Appl. Math., 46 (1986), 856. doi: 10.1137/0146052. Google Scholar
[13]	G. Sugihara and H. Ye, Cooperative network dynamics,, Nature (News and View), 458 (2009), 979. doi: 10.1038/458979a. Google Scholar
[14]	Y. Takeuchi, Global Dynamical Properties of Lotka-Volterra Systems,, World Scientific Publishing Co., (1996). doi: 10.1142/9789812830548. Google Scholar
[15]	Y. Wang and H. Wu, Dynamics of a cooperation-competition model for the WWW market,, Physica A, 339 (2004), 609. doi: 10.1016/j.physa.2004.03.067. Google Scholar

show all references

References:

[1]	J. Bascompte, P. Jordano, C. J. Melián and J. M. Olesen, The nested assembly of plant-animal mutualistic networks,, Proc. Natl Acad. Sci. USA, 100 (2003), 9383. doi: 10.1073/pnas.1633576100. Google Scholar
[2]	J. Bascompte, P. Jordano and J. M. Olesen, Asymmetric coevolutionary networks facilitate biodiversity maintenance,, Science, 312 (2006), 431. doi: 10.1126/science.1123412. Google Scholar
[3]	J. Bascompte and P. Jordano, The structure of plant-animal mutualistic networks: The architecture of biodiversity,, Annu. Rev. Ecol. Evol. Syst., 38 (2007), 567. Google Scholar
[4]	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
[5]	R. S. Cantrell and C. Cosner, Spatial Ecology Via Reaction-Diffusion Equations,, Wiley Series in Mathematical and Computational Biology, (2003). doi: 10.1002/0470871296. Google Scholar
[6]	C. E. Clark and T. G. Hallam, The community matrix in three species community models,, J. Math. Biology, 16 (1982), 25. doi: 10.1007/BF00275158. Google Scholar
[7]	B. S. Goh, Stability in models of mutualism,, The American Naturalist, 111 (1977), 135. doi: 10.1086/283384. Google Scholar
[8]	B. S. Goh, Stability in models of mutualism,, The American Naturalist, 113 (1979), 261. doi: 10.1086/283384. Google Scholar
[9]	V. Hutson, A theorem on average Lyapunov functions,, Monatsch. Math., 98 (1984), 267. doi: 10.1007/BF01540776. Google Scholar
[10]	J. D. Murray, Mathematical Biology,, Springer, (1993). doi: 10.1007/b98869. Google Scholar
[11]	S. Saavedra, D. B. Stouffer, B. Uzzi and J. Bascompte, Strong contributors to network persistence are the most vulnerable to extinction,, Nature, 478 (2011), 233. doi: 10.1038/nature10433. Google Scholar
[12]	H. L. Smith, Competing subcommunities of mutualists and a generalized Kamke theorem,, SIAM J. Appl. Math., 46 (1986), 856. doi: 10.1137/0146052. Google Scholar
[13]	G. Sugihara and H. Ye, Cooperative network dynamics,, Nature (News and View), 458 (2009), 979. doi: 10.1038/458979a. Google Scholar
[14]	Y. Takeuchi, Global Dynamical Properties of Lotka-Volterra Systems,, World Scientific Publishing Co., (1996). doi: 10.1142/9789812830548. Google Scholar
[15]	Y. Wang and H. Wu, Dynamics of a cooperation-competition model for the WWW market,, Physica A, 339 (2004), 609. doi: 10.1016/j.physa.2004.03.067. Google Scholar

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