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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

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 and Continuous Dynamical Systems, 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), 9383-9387. doi: 10.1073/pnas.1633576100.

[2]

J. Bascompte, P. Jordano and J. M. Olesen, Asymmetric coevolutionary networks facilitate biodiversity maintenance, Science, 312 (2006), 431-433. doi: 10.1126/science.1123412.

[3]

J. Bascompte and P. Jordano, The structure of plant-animal mutualistic networks: The architecture of biodiversity, Annu. Rev. Ecol. Evol. Syst., 38 (2007), 567-593.

[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), 1018-1020. doi: 10.1038/nature07950.

[5]

R. S. Cantrell and C. Cosner, Spatial Ecology Via Reaction-Diffusion Equations, Wiley Series in Mathematical and Computational Biology, John Wiley & Sons, Ltd., Chichester, 2003. doi: 10.1002/0470871296.

[6]

C. E. Clark and T. G. Hallam, The community matrix in three species community models, J. Math. Biology, 16 (1982), 25-31. doi: 10.1007/BF00275158.

[7]

B. S. Goh, Stability in models of mutualism, The American Naturalist, 111 (1977), 135-143. doi: 10.1086/283384.

[8]

B. S. Goh, Stability in models of mutualism, The American Naturalist, 113 (1979), 261-275. doi: 10.1086/283384.

[9]

V. Hutson, A theorem on average Lyapunov functions, Monatsch. Math., 98 (1984), 267-275. doi: 10.1007/BF01540776.

[10]

J. D. Murray, Mathematical Biology, Springer, New York, 1993 doi: 10.1007/b98869.

[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), 233-235. doi: 10.1038/nature10433.

[12]

H. L. Smith, Competing subcommunities of mutualists and a generalized Kamke theorem, SIAM J. Appl. Math., 46 (1986), 856-874. doi: 10.1137/0146052.

[13]

G. Sugihara and H. Ye, Cooperative network dynamics, Nature (News and View), 458 (2009), 979-980. doi: 10.1038/458979a.

[14]

Y. Takeuchi, Global Dynamical Properties of Lotka-Volterra Systems, World Scientific Publishing Co., Inc., River Edge, NJ, 1996. doi: 10.1142/9789812830548.

[15]

Y. Wang and H. Wu, Dynamics of a cooperation-competition model for the WWW market, Physica A, 339 (2004), 609-620. doi: 10.1016/j.physa.2004.03.067.

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), 9383-9387. doi: 10.1073/pnas.1633576100.

[2]

J. Bascompte, P. Jordano and J. M. Olesen, Asymmetric coevolutionary networks facilitate biodiversity maintenance, Science, 312 (2006), 431-433. doi: 10.1126/science.1123412.

[3]

J. Bascompte and P. Jordano, The structure of plant-animal mutualistic networks: The architecture of biodiversity, Annu. Rev. Ecol. Evol. Syst., 38 (2007), 567-593.

[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), 1018-1020. doi: 10.1038/nature07950.

[5]

R. S. Cantrell and C. Cosner, Spatial Ecology Via Reaction-Diffusion Equations, Wiley Series in Mathematical and Computational Biology, John Wiley & Sons, Ltd., Chichester, 2003. doi: 10.1002/0470871296.

[6]

C. E. Clark and T. G. Hallam, The community matrix in three species community models, J. Math. Biology, 16 (1982), 25-31. doi: 10.1007/BF00275158.

[7]

B. S. Goh, Stability in models of mutualism, The American Naturalist, 111 (1977), 135-143. doi: 10.1086/283384.

[8]

B. S. Goh, Stability in models of mutualism, The American Naturalist, 113 (1979), 261-275. doi: 10.1086/283384.

[9]

V. Hutson, A theorem on average Lyapunov functions, Monatsch. Math., 98 (1984), 267-275. doi: 10.1007/BF01540776.

[10]

J. D. Murray, Mathematical Biology, Springer, New York, 1993 doi: 10.1007/b98869.

[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), 233-235. doi: 10.1038/nature10433.

[12]

H. L. Smith, Competing subcommunities of mutualists and a generalized Kamke theorem, SIAM J. Appl. Math., 46 (1986), 856-874. doi: 10.1137/0146052.

[13]

G. Sugihara and H. Ye, Cooperative network dynamics, Nature (News and View), 458 (2009), 979-980. doi: 10.1038/458979a.

[14]

Y. Takeuchi, Global Dynamical Properties of Lotka-Volterra Systems, World Scientific Publishing Co., Inc., River Edge, NJ, 1996. doi: 10.1142/9789812830548.

[15]

Y. Wang and H. Wu, Dynamics of a cooperation-competition model for the WWW market, Physica A, 339 (2004), 609-620. doi: 10.1016/j.physa.2004.03.067.

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