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April  2021, 26(4): 2161-2172. doi: 10.3934/dcdsb.2021014

Carrying simplex in the Lotka-Volterra competition model with seasonal succession with applications

Lin Niu 1, , Yi Wang 1, and  Xizhuang Xie 1,2,,

1. 

School of Mathematical Sciences, University of Science and Technology of China, Hefei, Anhui, 230026, China
2. 

School of Mathematical Sciences, Huaqiao University, Quanzhou, Fujian, 362021, China

* Corresponding author: Xizhuang Xie

Received  October 2020 Revised  December 2020 Published  April 2021 Early access  December 2020

Fund Project: This work is supported by NSF of China No. 11825106, 11871231 and 11771414, CAS Wu Wen-Tsun Key Laboratory of Mathematics, University of Science and Technology of China

We investigate the dynamics of the Poincar$ \acute{\rm e} $-map for an $ n $-dimensional Lotka-Volterra competitive model with seasonal succession. It is proved that there exists an $ (n-1) $-dimensional carrying simplex $ \Sigma $ which attracts every nontrivial orbit in $ \mathbb{R}^n_+ $. By using the theory of the carrying simplex, we simplify the approach for the complete classification of global dynamics for the two-dimensional Lotka-Volterra competitive model with seasonal succession proposed in [Hsu and Zhao, J. Math. Biology 64(2012), 109-130]. Our approach avoids the complicated estimates for the Floquet multipliers of the positive periodic solutions.

Keywords: Seasonal succession, Lotka-Volterra model, Carrying simplex, Competition, Poincaré-map.
Mathematics Subject Classification: Primary: 34C25, 37C65; Secondary: 92D25.
Citation: Lin Niu, Yi Wang, Xizhuang Xie. Carrying simplex in the Lotka-Volterra competition model with seasonal succession with applications. Discrete & Continuous Dynamical Systems - B, 2021, 26 (4) : 2161-2172. doi: 10.3934/dcdsb.2021014
References:
[1]

E. N. Dancer and P. Hess, Stability of fixed points for order-preserving discrete-time dynamical systems, J. Reine Angew. Math., 419 (1991), 125-139.   Google Scholar

[2]

D. J. D. Earn, P. Rohani, B. M. Bolker and B. T. Grenfell, A simple model for complex dynamical transitions in epidemics, Science, 287 (2000), 667-670.  doi: 10.1126/science.287.5453.667.  Google Scholar

[3]

O. Diekmann, Y. Wang and P. Yan, Carrying simplices in discrete competitive systems and age-structured semelparous populations, Discrete Contin. Dyn. Syst., 20 (2008), 37-52.  doi: 10.3934/dcds.2008.20.37.  Google Scholar

[4]

M. Gyllenberg, J. Jiang, L. Niu and P. Yan, On the classification of generalized competitive Atkinson-Allen models via the dynamics on the boundary of the carrying simplex, Discrete Contin. Dyn. Syst., 38 (2018), 615-650.  doi: 10.3934/dcds.2018027.  Google Scholar

[5]

M. W. Hirsch, On existence and uniqueness of the carrying simplex for competitive dynamical systems, J. Biol. Dynam., 2 (2008), 169-179.  doi: 10.1080/17513750801939236.  Google Scholar

[6]

A. Huppert, B. Blasius, R. Olinky and L. Stone, A model for seasonal phytoplankton blooms, J. Theor. Biol., 236 (2005), 276-290.  doi: 10.1016/j.jtbi.2005.03.012.  Google Scholar

[7]

S. B. Hsu, A competition model for a seasonally fluctuating nutrient, J. Math. Biol., 9 (1980), 115-132.  doi: 10.1007/BF00275917.  Google Scholar

[8]

S. B. Hsu and X. Q. Zhao, A Lotka-Volterra competition model with seasonal succession, J. Math. Biol., 64 (2012), 109-130.  doi: 10.1007/s00285-011-0408-6.  Google Scholar

[9]

S. B. Hsu, H. L. Smith and P. Waltman, Competitive exclusion and coexistence for competitive systems on ordered Banach spaces, Trans. Amer. Math. Soc., 348 (1996), 4083-4094.  doi: 10.1090/S0002-9947-96-01724-2.  Google Scholar

[10]

J. Jiang and L. Niu, On the equivalent classification of three-dimensional competitive Atkinson/Allen models relative to the boundary fixed points, Discrete Contin. Dyn. Syst., 36 (2016), 217-244.  doi: 10.3934/dcds.2016.36.217.  Google Scholar

[11]

J. Jiang and L. Niu, On the equivalent classification of three-dimensional competitive Leslie/Gower models via the boundary dynamics on the carrying simplex, J. Math. Biol., 74 (2017), 1223-1261.  doi: 10.1007/s00285-016-1052-y.  Google Scholar

[12]

J. Jiang, X. Liang and X. Q. Zhao, Saddle-point behavior for monotone semiflows and reaction-diffusion models, J. Differential Equations, 203 (2004), 313-330.  doi: 10.1016/j.jde.2004.05.002.  Google Scholar

[13]

J. Jiang, J. Mierczyński and Y. Wang, Smoothness of the carrying simplex for discrete-time competitive dynamical systems: A characterization of neat embedding, J. Differential Equations, 246 (2009), 1623-1672.  doi: 10.1016/j.jde.2008.10.008.  Google Scholar

[14]

C. A. Klausmeier, Successional state dynamics: A novel approach to modeling nonequilibrium foodweb dynamics, J. Theor. Biol., 262 (2010), 584-595.  doi: 10.1016/j.jtbi.2009.10.018.  Google Scholar

[15]

E. Litchman and C. A. Klausmeier, Competition of phytoplankton under fluctuating light, American Naturalist, 157 (2001), 170-187.  doi: 10.1086/318628.  Google Scholar

[16]

L. Niu and A. Ruiz-Herrera, Trivial dynamics in discrete-time systems: Carrying simplex and translation arcs, Nonlinearity, 31 (2018), 2633-2650.  doi: 10.1088/1361-6544/aab46e.  Google Scholar

[17]

A. Ruiz-Herrera, Exclusion and dominance in discrete population models via the carrying simplex, Journal of Difference Equations and Applications, 19 (2013), 96-113.  doi: 10.1080/10236198.2011.628663.  Google Scholar

[18]

M. Shub, Global Stability of Dynamical Systems, Springer, New-York/Berlin, 1987. doi: 10.1007/978-1-4757-1947-5.  Google Scholar

[19]

L. Stone, R. Olinky and A. Huppert, Seasonal dynamics of recurrent epidemics, Nature, 446 (2007), 533-536.  doi: 10.1038/nature05638.  Google Scholar

[20]

H. L. Smith, Monotone Dynamical Systems. An Introduction to the Theory of Competitive and Cooperative Systems, Math. Surveys Monogr. 41, AMS, Providence, RI, 1995.  Google Scholar

[21]

H. L. Smith, Periodic competitive differential equations and the discrete dynamics of competitive maps, J. Differential Equations, 64 (1986), 165-194.  doi: 10.1016/0022-0396(86)90086-0.  Google Scholar

[22]

H. L. Smith, Periodic solutions of periodic competitive and cooperative systems, SIAM J. Math. Anal., 17 (1986), 1289-1318.  doi: 10.1137/0517091.  Google Scholar

[23]

H. L. Smith and P. Waltman, The Theory of the Chemostat, Cambridge University Press, 1995. doi: 10.1017/CBO9780511530043.  Google Scholar

[24]

H. L. Smith and H. R. Thieme, Stable coexistence and bi-stability for competitive systems on ordered Banach spaces, J. Differential Equations, 176 (2001), 195-222.  doi: 10.1006/jdeq.2001.3981.  Google Scholar

[25]

C. F. Steiner, A. S. Schwaderer, V. Huber, C. A. Klausmeier and E. Litchman, Periodically forced food-chain dynamics: Model predictions and experimental validation, Ecology, 90 (2009), 3099-3107.  doi: 10.1890/08-2377.1.  Google Scholar

[26]

U. Sommer, Z. M. Gliwicz, W. Lampert and A. Duncan, The PEG-model of seasonal succession of planktonic events in fresh waters, Archiv für Hydrobiologie, 106 (1986), 433–471. Google Scholar

[27]

P. Takáč, Domains of attraction of generic $\omega$-limit sets for strongly monotone discrete-time semigroups, J. Reine Angew. Math., 423 (1992), 101-173.  doi: 10.1515/crll.1992.423.101.  Google Scholar

[28]

Y. Wang and J. Jiang, Uniqueness and attractivity of the carrying simplex for discrete-time competitive dynamical systems, J. Differential Equations, 186 (2002), 611-632.  doi: 10.1016/S0022-0396(02)00025-6.  Google Scholar

[29]

M. L. Zeeman, Hopf bifurcations in competitive three-dimensional Lotka-Volterra systems, Dynam. Stability Systems, 8 (1993), 189-217.  doi: 10.1080/02681119308806158.  Google Scholar

[30]

Y. X. Zhang and X. Q. Zhao, Bistable travelling waves for reaction and diffusion model with seasonal succession, Nonlinearity, 26 (2013), 691-709.  doi: 10.1088/0951-7715/26/3/691.  Google Scholar

show all references

References:
[1]

E. N. Dancer and P. Hess, Stability of fixed points for order-preserving discrete-time dynamical systems, J. Reine Angew. Math., 419 (1991), 125-139.   Google Scholar

[2]

D. J. D. Earn, P. Rohani, B. M. Bolker and B. T. Grenfell, A simple model for complex dynamical transitions in epidemics, Science, 287 (2000), 667-670.  doi: 10.1126/science.287.5453.667.  Google Scholar

[3]

O. Diekmann, Y. Wang and P. Yan, Carrying simplices in discrete competitive systems and age-structured semelparous populations, Discrete Contin. Dyn. Syst., 20 (2008), 37-52.  doi: 10.3934/dcds.2008.20.37.  Google Scholar

[4]

M. Gyllenberg, J. Jiang, L. Niu and P. Yan, On the classification of generalized competitive Atkinson-Allen models via the dynamics on the boundary of the carrying simplex, Discrete Contin. Dyn. Syst., 38 (2018), 615-650.  doi: 10.3934/dcds.2018027.  Google Scholar

[5]

M. W. Hirsch, On existence and uniqueness of the carrying simplex for competitive dynamical systems, J. Biol. Dynam., 2 (2008), 169-179.  doi: 10.1080/17513750801939236.  Google Scholar

[6]

A. Huppert, B. Blasius, R. Olinky and L. Stone, A model for seasonal phytoplankton blooms, J. Theor. Biol., 236 (2005), 276-290.  doi: 10.1016/j.jtbi.2005.03.012.  Google Scholar

[7]

S. B. Hsu, A competition model for a seasonally fluctuating nutrient, J. Math. Biol., 9 (1980), 115-132.  doi: 10.1007/BF00275917.  Google Scholar

[8]

S. B. Hsu and X. Q. Zhao, A Lotka-Volterra competition model with seasonal succession, J. Math. Biol., 64 (2012), 109-130.  doi: 10.1007/s00285-011-0408-6.  Google Scholar

[9]

S. B. Hsu, H. L. Smith and P. Waltman, Competitive exclusion and coexistence for competitive systems on ordered Banach spaces, Trans. Amer. Math. Soc., 348 (1996), 4083-4094.  doi: 10.1090/S0002-9947-96-01724-2.  Google Scholar

[10]

J. Jiang and L. Niu, On the equivalent classification of three-dimensional competitive Atkinson/Allen models relative to the boundary fixed points, Discrete Contin. Dyn. Syst., 36 (2016), 217-244.  doi: 10.3934/dcds.2016.36.217.  Google Scholar

[11]

J. Jiang and L. Niu, On the equivalent classification of three-dimensional competitive Leslie/Gower models via the boundary dynamics on the carrying simplex, J. Math. Biol., 74 (2017), 1223-1261.  doi: 10.1007/s00285-016-1052-y.  Google Scholar

[12]

J. Jiang, X. Liang and X. Q. Zhao, Saddle-point behavior for monotone semiflows and reaction-diffusion models, J. Differential Equations, 203 (2004), 313-330.  doi: 10.1016/j.jde.2004.05.002.  Google Scholar

[13]

J. Jiang, J. Mierczyński and Y. Wang, Smoothness of the carrying simplex for discrete-time competitive dynamical systems: A characterization of neat embedding, J. Differential Equations, 246 (2009), 1623-1672.  doi: 10.1016/j.jde.2008.10.008.  Google Scholar

[14]

C. A. Klausmeier, Successional state dynamics: A novel approach to modeling nonequilibrium foodweb dynamics, J. Theor. Biol., 262 (2010), 584-595.  doi: 10.1016/j.jtbi.2009.10.018.  Google Scholar

[15]

E. Litchman and C. A. Klausmeier, Competition of phytoplankton under fluctuating light, American Naturalist, 157 (2001), 170-187.  doi: 10.1086/318628.  Google Scholar

[16]

L. Niu and A. Ruiz-Herrera, Trivial dynamics in discrete-time systems: Carrying simplex and translation arcs, Nonlinearity, 31 (2018), 2633-2650.  doi: 10.1088/1361-6544/aab46e.  Google Scholar

[17]

A. Ruiz-Herrera, Exclusion and dominance in discrete population models via the carrying simplex, Journal of Difference Equations and Applications, 19 (2013), 96-113.  doi: 10.1080/10236198.2011.628663.  Google Scholar

[18]

M. Shub, Global Stability of Dynamical Systems, Springer, New-York/Berlin, 1987. doi: 10.1007/978-1-4757-1947-5.  Google Scholar

[19]

L. Stone, R. Olinky and A. Huppert, Seasonal dynamics of recurrent epidemics, Nature, 446 (2007), 533-536.  doi: 10.1038/nature05638.  Google Scholar

[20]

H. L. Smith, Monotone Dynamical Systems. An Introduction to the Theory of Competitive and Cooperative Systems, Math. Surveys Monogr. 41, AMS, Providence, RI, 1995.  Google Scholar

[21]

H. L. Smith, Periodic competitive differential equations and the discrete dynamics of competitive maps, J. Differential Equations, 64 (1986), 165-194.  doi: 10.1016/0022-0396(86)90086-0.  Google Scholar

[22]

H. L. Smith, Periodic solutions of periodic competitive and cooperative systems, SIAM J. Math. Anal., 17 (1986), 1289-1318.  doi: 10.1137/0517091.  Google Scholar

[23]

H. L. Smith and P. Waltman, The Theory of the Chemostat, Cambridge University Press, 1995. doi: 10.1017/CBO9780511530043.  Google Scholar

[24]

H. L. Smith and H. R. Thieme, Stable coexistence and bi-stability for competitive systems on ordered Banach spaces, J. Differential Equations, 176 (2001), 195-222.  doi: 10.1006/jdeq.2001.3981.  Google Scholar

[25]

C. F. Steiner, A. S. Schwaderer, V. Huber, C. A. Klausmeier and E. Litchman, Periodically forced food-chain dynamics: Model predictions and experimental validation, Ecology, 90 (2009), 3099-3107.  doi: 10.1890/08-2377.1.  Google Scholar

[26]

U. Sommer, Z. M. Gliwicz, W. Lampert and A. Duncan, The PEG-model of seasonal succession of planktonic events in fresh waters, Archiv für Hydrobiologie, 106 (1986), 433–471. Google Scholar

[27]

P. Takáč, Domains of attraction of generic $\omega$-limit sets for strongly monotone discrete-time semigroups, J. Reine Angew. Math., 423 (1992), 101-173.  doi: 10.1515/crll.1992.423.101.  Google Scholar

[28]

Y. Wang and J. Jiang, Uniqueness and attractivity of the carrying simplex for discrete-time competitive dynamical systems, J. Differential Equations, 186 (2002), 611-632.  doi: 10.1016/S0022-0396(02)00025-6.  Google Scholar

[29]

M. L. Zeeman, Hopf bifurcations in competitive three-dimensional Lotka-Volterra systems, Dynam. Stability Systems, 8 (1993), 189-217.  doi: 10.1080/02681119308806158.  Google Scholar

[30]

Y. X. Zhang and X. Q. Zhao, Bistable travelling waves for reaction and diffusion model with seasonal succession, Nonlinearity, 26 (2013), 691-709.  doi: 10.1088/0951-7715/26/3/691.  Google Scholar

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