Biodiversity and vulnerability in a 3D mutualistic system

Giovanny Guerrero; José Antonio Langa; Antonio Suárez

doi:10.3934/dcds.2014.34.4107

Article Contents

2014, Volume 34, Issue 10: 4107-4126. Doi: 10.3934/dcds.2014.34.4107

This issue Previous Article Attractors for a double time-delayed 2D-Navier-Stokes model Next Article Asymptotic behaviour for prey-predator systems and logistic equations with unbounded time-dependent coefficients

Biodiversity and vulnerability in a 3D mutualistic system

1.
Universidad Central del Ecuador, Ciudadela Universitaria, Av. América S/N, Quito
2.
Departamento de Ecuaciones Diferenciales y Análisis Numérico, Facultad de Matemáticas, Universidad de Sevilla, Calle Tarfia s/n, 41012-Seville
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: October 2014

Abstract / Introduction Related Papers Cited by

Abstract

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:

Mathematics Subject Classification: Primary: 92B05, 34A34, 34D05, 34D23.

Citation:

Related Papers

Cited by

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, 1993doi: 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.