# American Institute of Mathematical Sciences

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), 2127-2143. doi: 10.1016/j.camwa.2014.02.020. [2] J. L. Bronstein, U. Dieckmann and R. Ferrière, Coevolutionary dynamics and the conservation of mutualisms, in Evolutionary Conservation Biology (eds. R. Ferrière, U. Dieckmann and D. Couvet), Cambridge University Press, (2004), 305-326. doi: 10.1017/CBO9780511542022.022. [3] A. E. Douglas, The Symbiotic Habit, Princeton University Press, Princeton, 2010. [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), 337-353. doi: 10.1137/060654876. [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), 3653-3665. doi: 10.1016/j.nonrwa.2010.01.012. [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), 8496-8507. doi: 10.1016/j.amc.2013.02.044. [7] P. Georgescu and H. Zhang, Lyapunov functionals for two-species mutualisms, Appl. Math. Comput., 226 (2014), 754-764. doi: 10.1016/j.amc.2013.10.061. [8] B. S. Goh, Stability in models of mutualism, Am. Nat., 113 (1979), 261-275. doi: 10.1086/283384. [9] W. G. Graves, B. Peckham and J. Pastor, A bifurcation analysis of a differential equations model for mutualism, Bull. Math. Biol., 68 (2006), 1851-1872. doi: 10.1007/s11538-006-9070-3. [10] G. W. Harrison, Global stability of predator-prey interactions, J. Math. Biol., 8 (1979), 159-171. doi: 10.1007/BF00279719. [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, J. Wu and Y. Zhou), Series in Contemporary Applied Mathematics (CAM), Higher Education Press, 11 (2009), 216-236. [12] J. N. Holland and J.L. Bronstein, Mutualism, in Population Dynamics, Vol 3 of Encyclopedia of Ecology (eds. S.E. Jorgensen and B.D. Fath), Elsevier, (2008), 2485-2491. [13] J. N. Holland and D. L. DeAngelis, A consumer-resource approach to the density-dependent population dynamics of mutualism, Ecology, 91 (2010), 1286-1295. doi: 10.1890/09-1163.1. [14] A. Korobeinikov, Lyapunov functions and global properties for SEIR and SEIS epidemic models, Math. Med. Biol., 21 (2004), 75-83. doi: 10.1093/imammb/21.2.75. [15] A. Korobeinikov, Lyapunov functions and global stability for SIR and SIRS epidemiological models with non-linear transmission, Bull. Math. Biol., 68 (2006), 615-626. doi: 10.1007/s11538-005-9037-9. [16] A. Korobeinikov, Stability of ecosystem: Global properties of a general predator-prey model, Math. Med. Biol., 26 (2009), 309-321. doi: 10.1093/imammb/dqp009. [17] R. M. May, Models of two interacting populations, in Theoretical Ecology: Principles and Application (ed. R. M. May), Saunders, (1976), 78-104. [18] C. C. McCluskey, Global stability for an SEIR epidemiological model with varying infectivity and infinite delay, Math. Biosci. Eng., 6 (2009), 603-610. doi: 10.3934/mbe.2009.6.603. [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), 369-378. doi: 10.3934/mbe.2013.10.369. [20] T. M. Palmer and A. K. Brody, Mutualism as reciprocal exploitation: African plant-ants defend foliar but not reproductive structures, Ecology, 88 (2007), 3004-3011. doi: 10.1890/07-0133.1. [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), 2-22. [22] F. J. Richards, A flexible growth function for empirical use, J. Exp. Bot., 10 (1959), 290-300. doi: 10.1093/jxb/10.2.290. [23] J. Vandermeer and D. Boucher, Varieties of mutualistic interaction in population models, J. Theor. Biol., 74 (1978), 549-558. [24] C. Vargas-De-León, Lyapunov functions for two-species cooperative systems, Appl. Math. Comput., 219 (2012), 2493-2497. doi: 10.1016/j.amc.2012.08.084. [25] C. Vargas-De-León, On the global stability of infectious diseases models with relapse, Abstraction & Application, 9 (2013), 50-61. [26] C. Vargas-De-León and G. Gómez-Alcaraz, Global stability in some ecological models of commensalism between two species, Biomatemática, 23 (2013), 139-146. [27] C. Wolin and L. Lawlor, Models of facultative mutualism: Density effects, Am. Nat., 124 (1984), 843-862. doi: 10.1086/284320.

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##### 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), 2127-2143. doi: 10.1016/j.camwa.2014.02.020. [2] J. L. Bronstein, U. Dieckmann and R. Ferrière, Coevolutionary dynamics and the conservation of mutualisms, in Evolutionary Conservation Biology (eds. R. Ferrière, U. Dieckmann and D. Couvet), Cambridge University Press, (2004), 305-326. doi: 10.1017/CBO9780511542022.022. [3] A. E. Douglas, The Symbiotic Habit, Princeton University Press, Princeton, 2010. [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), 337-353. doi: 10.1137/060654876. [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), 3653-3665. doi: 10.1016/j.nonrwa.2010.01.012. [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), 8496-8507. doi: 10.1016/j.amc.2013.02.044. [7] P. Georgescu and H. Zhang, Lyapunov functionals for two-species mutualisms, Appl. Math. Comput., 226 (2014), 754-764. doi: 10.1016/j.amc.2013.10.061. [8] B. S. Goh, Stability in models of mutualism, Am. Nat., 113 (1979), 261-275. doi: 10.1086/283384. [9] W. G. Graves, B. Peckham and J. Pastor, A bifurcation analysis of a differential equations model for mutualism, Bull. Math. Biol., 68 (2006), 1851-1872. doi: 10.1007/s11538-006-9070-3. [10] G. W. Harrison, Global stability of predator-prey interactions, J. Math. Biol., 8 (1979), 159-171. doi: 10.1007/BF00279719. [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, J. Wu and Y. Zhou), Series in Contemporary Applied Mathematics (CAM), Higher Education Press, 11 (2009), 216-236. [12] J. N. Holland and J.L. Bronstein, Mutualism, in Population Dynamics, Vol 3 of Encyclopedia of Ecology (eds. S.E. Jorgensen and B.D. Fath), Elsevier, (2008), 2485-2491. [13] J. N. Holland and D. L. DeAngelis, A consumer-resource approach to the density-dependent population dynamics of mutualism, Ecology, 91 (2010), 1286-1295. doi: 10.1890/09-1163.1. [14] A. Korobeinikov, Lyapunov functions and global properties for SEIR and SEIS epidemic models, Math. Med. Biol., 21 (2004), 75-83. doi: 10.1093/imammb/21.2.75. [15] A. Korobeinikov, Lyapunov functions and global stability for SIR and SIRS epidemiological models with non-linear transmission, Bull. Math. Biol., 68 (2006), 615-626. doi: 10.1007/s11538-005-9037-9. [16] A. Korobeinikov, Stability of ecosystem: Global properties of a general predator-prey model, Math. Med. Biol., 26 (2009), 309-321. doi: 10.1093/imammb/dqp009. [17] R. M. May, Models of two interacting populations, in Theoretical Ecology: Principles and Application (ed. R. M. May), Saunders, (1976), 78-104. [18] C. C. McCluskey, Global stability for an SEIR epidemiological model with varying infectivity and infinite delay, Math. Biosci. Eng., 6 (2009), 603-610. doi: 10.3934/mbe.2009.6.603. [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), 369-378. doi: 10.3934/mbe.2013.10.369. [20] T. M. Palmer and A. K. Brody, Mutualism as reciprocal exploitation: African plant-ants defend foliar but not reproductive structures, Ecology, 88 (2007), 3004-3011. doi: 10.1890/07-0133.1. [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), 2-22. [22] F. J. Richards, A flexible growth function for empirical use, J. Exp. Bot., 10 (1959), 290-300. doi: 10.1093/jxb/10.2.290. [23] J. Vandermeer and D. Boucher, Varieties of mutualistic interaction in population models, J. Theor. Biol., 74 (1978), 549-558. [24] C. Vargas-De-León, Lyapunov functions for two-species cooperative systems, Appl. Math. Comput., 219 (2012), 2493-2497. doi: 10.1016/j.amc.2012.08.084. [25] C. Vargas-De-León, On the global stability of infectious diseases models with relapse, Abstraction & Application, 9 (2013), 50-61. [26] C. Vargas-De-León and G. Gómez-Alcaraz, Global stability in some ecological models of commensalism between two species, Biomatemática, 23 (2013), 139-146. [27] C. Wolin and L. Lawlor, Models of facultative mutualism: Density effects, Am. Nat., 124 (1984), 843-862. doi: 10.1086/284320.
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