2016, 13(1): 101-118. doi: 10.3934/mbe.2016.13.101

The global stability of coexisting equilibria for three models of mutualism

1. 

Department of Mathematics, Technical University of Iaşi, Bd. Copou 11, 700506 Iaşi, Romania

2. 

Department of Financial Mathematics, Jiangsu University, ZhenJiang, Jiangsu, 212013, China, China

Received  January 2015 Revised  June 2015 Published  October 2015

We analyze the dynamics of three models of mutualism, establishing the global stability of coexisting equilibria by means of Lyapunov's second method. This further establishes the usefulness of certain Lyapunov functionals of an abstract nature introduced in an earlier paper. As a consequence, it is seen that the use of higher order self-limiting terms cures the shortcomings of Lotka-Volterra mutualisms, preventing unbounded growth and promoting global stability.
Citation: Paul Georgescu, Hong Zhang, Daniel Maxin. The global stability of coexisting equilibria for three models of mutualism. Mathematical Biosciences & Engineering, 2016, 13 (1) : 101-118. doi: 10.3934/mbe.2016.13.101
References:
[1]

N. Apreutesei, G. Dimitriu and R. Strugariu, An optimal control problem for a two-prey and one-predator model with diffusion,, Comput. Math. Appl., 67 (2014), 2127. doi: 10.1016/j.camwa.2014.02.020. Google Scholar

[2]

J. L. Bronstein, U. Dieckmann and R. Ferrière, Coevolutionary dynamics and the conservation of mutualisms,, in Evolutionary Conservation Biology (eds. R. Ferrière, (2004), 305. doi: 10.1017/CBO9780511542022.022. Google Scholar

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A. E. Douglas, The Symbiotic Habit,, Princeton University Press, (2010). Google Scholar

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P. Georgescu and Y.-H. Hsieh, Global stability for a virus dynamics model with nonlinear incidence of infection and removal,, SIAM J. Appl. Math., 67 (2006), 337. doi: 10.1137/060654876. Google Scholar

[5]

P. Georgescu, Y.-H. Hsieh and H. Zhang, A Lyapunov functional for a stage-structured predator-prey model with nonlinear predation rate,, Nonlinear Anal.: Real World Appl., 11 (2010), 3653. doi: 10.1016/j.nonrwa.2010.01.012. Google Scholar

[6]

P. Georgescu and H. Zhang, A Lyapunov functional for a SIRI model with nonlinear incidence of infection and relapse,, Appl. Math. Comput., 219 (2013), 8496. doi: 10.1016/j.amc.2013.02.044. Google Scholar

[7]

P. Georgescu and H. Zhang, Lyapunov functionals for two-species mutualisms,, Appl. Math. Comput., 226 (2014), 754. doi: 10.1016/j.amc.2013.10.061. Google Scholar

[8]

B. S. Goh, Stability in models of mutualism,, Am. Nat., 113 (1979), 261. doi: 10.1086/283384. Google Scholar

[9]

W. G. Graves, B. Peckham and J. Pastor, A bifurcation analysis of a differential equations model for mutualism,, Bull. Math. Biol., 68 (2006), 1851. doi: 10.1007/s11538-006-9070-3. Google Scholar

[10]

G. W. Harrison, Global stability of predator-prey interactions,, J. Math. Biol., 8 (1979), 159. doi: 10.1007/BF00279719. Google Scholar

[11]

Y.-H. Hsieh, Richards model: A simple procedure for real-time prediction of outbreak severity,, in Modeling and Dynamics of Infectious Diseases (eds. Z. Ma, 11 (2009), 216. Google Scholar

[12]

J. N. Holland and J.L. Bronstein, Mutualism,, in Population Dynamics, (2008), 2485. Google Scholar

[13]

J. N. Holland and D. L. DeAngelis, A consumer-resource approach to the density-dependent population dynamics of mutualism,, Ecology, 91 (2010), 1286. doi: 10.1890/09-1163.1. Google Scholar

[14]

A. Korobeinikov, Lyapunov functions and global properties for SEIR and SEIS epidemic models,, Math. Med. Biol., 21 (2004), 75. doi: 10.1093/imammb/21.2.75. Google Scholar

[15]

A. Korobeinikov, Lyapunov functions and global stability for SIR and SIRS epidemiological models with non-linear transmission,, Bull. Math. Biol., 68 (2006), 615. doi: 10.1007/s11538-005-9037-9. Google Scholar

[16]

A. Korobeinikov, Stability of ecosystem: Global properties of a general predator-prey model,, Math. Med. Biol., 26 (2009), 309. doi: 10.1093/imammb/dqp009. Google Scholar

[17]

R. M. May, Models of two interacting populations,, in Theoretical Ecology: Principles and Application (ed. R. M. May), (1976), 78. Google Scholar

[18]

C. C. McCluskey, Global stability for an SEIR epidemiological model with varying infectivity and infinite delay,, Math. Biosci. Eng., 6 (2009), 603. doi: 10.3934/mbe.2009.6.603. Google Scholar

[19]

A. V. Melnik and A. Korobeinikov, Lyapunov functions and global stability for SIR and SEIR models with age-dependent susceptibility,, Math. Biosci. Eng., 10 (2013), 369. doi: 10.3934/mbe.2013.10.369. Google Scholar

[20]

T. M. Palmer and A. K. Brody, Mutualism as reciprocal exploitation: African plant-ants defend foliar but not reproductive structures,, Ecology, 88 (2007), 3004. doi: 10.1890/07-0133.1. Google Scholar

[21]

L. V. Pienaar and K. J. Turnbull, The Chapman-Richards generalization of von Bertalanffy's growth model for basal area growth and yield in even-aged stands,, Forest Science, 19 (1973), 2. Google Scholar

[22]

F. J. Richards, A flexible growth function for empirical use,, J. Exp. Bot., 10 (1959), 290. doi: 10.1093/jxb/10.2.290. Google Scholar

[23]

J. Vandermeer and D. Boucher, Varieties of mutualistic interaction in population models,, J. Theor. Biol., 74 (1978), 549. Google Scholar

[24]

C. Vargas-De-León, Lyapunov functions for two-species cooperative systems,, Appl. Math. Comput., 219 (2012), 2493. doi: 10.1016/j.amc.2012.08.084. Google Scholar

[25]

C. Vargas-De-León, On the global stability of infectious diseases models with relapse,, Abstraction & Application, 9 (2013), 50. Google Scholar

[26]

C. Vargas-De-León and G. Gómez-Alcaraz, Global stability in some ecological models of commensalism between two species,, Biomatemática, 23 (2013), 139. Google Scholar

[27]

C. Wolin and L. Lawlor, Models of facultative mutualism: Density effects,, Am. Nat., 124 (1984), 843. doi: 10.1086/284320. Google Scholar

show all references

References:
[1]

N. Apreutesei, G. Dimitriu and R. Strugariu, An optimal control problem for a two-prey and one-predator model with diffusion,, Comput. Math. Appl., 67 (2014), 2127. doi: 10.1016/j.camwa.2014.02.020. Google Scholar

[2]

J. L. Bronstein, U. Dieckmann and R. Ferrière, Coevolutionary dynamics and the conservation of mutualisms,, in Evolutionary Conservation Biology (eds. R. Ferrière, (2004), 305. doi: 10.1017/CBO9780511542022.022. Google Scholar

[3]

A. E. Douglas, The Symbiotic Habit,, Princeton University Press, (2010). Google Scholar

[4]

P. Georgescu and Y.-H. Hsieh, Global stability for a virus dynamics model with nonlinear incidence of infection and removal,, SIAM J. Appl. Math., 67 (2006), 337. doi: 10.1137/060654876. Google Scholar

[5]

P. Georgescu, Y.-H. Hsieh and H. Zhang, A Lyapunov functional for a stage-structured predator-prey model with nonlinear predation rate,, Nonlinear Anal.: Real World Appl., 11 (2010), 3653. doi: 10.1016/j.nonrwa.2010.01.012. Google Scholar

[6]

P. Georgescu and H. Zhang, A Lyapunov functional for a SIRI model with nonlinear incidence of infection and relapse,, Appl. Math. Comput., 219 (2013), 8496. doi: 10.1016/j.amc.2013.02.044. Google Scholar

[7]

P. Georgescu and H. Zhang, Lyapunov functionals for two-species mutualisms,, Appl. Math. Comput., 226 (2014), 754. doi: 10.1016/j.amc.2013.10.061. Google Scholar

[8]

B. S. Goh, Stability in models of mutualism,, Am. Nat., 113 (1979), 261. doi: 10.1086/283384. Google Scholar

[9]

W. G. Graves, B. Peckham and J. Pastor, A bifurcation analysis of a differential equations model for mutualism,, Bull. Math. Biol., 68 (2006), 1851. doi: 10.1007/s11538-006-9070-3. Google Scholar

[10]

G. W. Harrison, Global stability of predator-prey interactions,, J. Math. Biol., 8 (1979), 159. doi: 10.1007/BF00279719. Google Scholar

[11]

Y.-H. Hsieh, Richards model: A simple procedure for real-time prediction of outbreak severity,, in Modeling and Dynamics of Infectious Diseases (eds. Z. Ma, 11 (2009), 216. Google Scholar

[12]

J. N. Holland and J.L. Bronstein, Mutualism,, in Population Dynamics, (2008), 2485. Google Scholar

[13]

J. N. Holland and D. L. DeAngelis, A consumer-resource approach to the density-dependent population dynamics of mutualism,, Ecology, 91 (2010), 1286. doi: 10.1890/09-1163.1. Google Scholar

[14]

A. Korobeinikov, Lyapunov functions and global properties for SEIR and SEIS epidemic models,, Math. Med. Biol., 21 (2004), 75. doi: 10.1093/imammb/21.2.75. Google Scholar

[15]

A. Korobeinikov, Lyapunov functions and global stability for SIR and SIRS epidemiological models with non-linear transmission,, Bull. Math. Biol., 68 (2006), 615. doi: 10.1007/s11538-005-9037-9. Google Scholar

[16]

A. Korobeinikov, Stability of ecosystem: Global properties of a general predator-prey model,, Math. Med. Biol., 26 (2009), 309. doi: 10.1093/imammb/dqp009. Google Scholar

[17]

R. M. May, Models of two interacting populations,, in Theoretical Ecology: Principles and Application (ed. R. M. May), (1976), 78. Google Scholar

[18]

C. C. McCluskey, Global stability for an SEIR epidemiological model with varying infectivity and infinite delay,, Math. Biosci. Eng., 6 (2009), 603. doi: 10.3934/mbe.2009.6.603. Google Scholar

[19]

A. V. Melnik and A. Korobeinikov, Lyapunov functions and global stability for SIR and SEIR models with age-dependent susceptibility,, Math. Biosci. Eng., 10 (2013), 369. doi: 10.3934/mbe.2013.10.369. Google Scholar

[20]

T. M. Palmer and A. K. Brody, Mutualism as reciprocal exploitation: African plant-ants defend foliar but not reproductive structures,, Ecology, 88 (2007), 3004. doi: 10.1890/07-0133.1. Google Scholar

[21]

L. V. Pienaar and K. J. Turnbull, The Chapman-Richards generalization of von Bertalanffy's growth model for basal area growth and yield in even-aged stands,, Forest Science, 19 (1973), 2. Google Scholar

[22]

F. J. Richards, A flexible growth function for empirical use,, J. Exp. Bot., 10 (1959), 290. doi: 10.1093/jxb/10.2.290. Google Scholar

[23]

J. Vandermeer and D. Boucher, Varieties of mutualistic interaction in population models,, J. Theor. Biol., 74 (1978), 549. Google Scholar

[24]

C. Vargas-De-León, Lyapunov functions for two-species cooperative systems,, Appl. Math. Comput., 219 (2012), 2493. doi: 10.1016/j.amc.2012.08.084. Google Scholar

[25]

C. Vargas-De-León, On the global stability of infectious diseases models with relapse,, Abstraction & Application, 9 (2013), 50. Google Scholar

[26]

C. Vargas-De-León and G. Gómez-Alcaraz, Global stability in some ecological models of commensalism between two species,, Biomatemática, 23 (2013), 139. Google Scholar

[27]

C. Wolin and L. Lawlor, Models of facultative mutualism: Density effects,, Am. Nat., 124 (1984), 843. doi: 10.1086/284320. Google Scholar

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