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Mathematical modeling of an immune checkpoint inhibitor and its synergy with an immunostimulant
Carrying simplex in the Lotka-Volterra competition model with seasonal succession with applications
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 |
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.
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.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[7] |
S. B. Hsu,
A competition model for a seasonally fluctuating nutrient, J. Math. Biol., 9 (1980), 115-132.
doi: 10.1007/BF00275917. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[15] |
E. Litchman and C. A. Klausmeier,
Competition of phytoplankton under fluctuating light, American Naturalist, 157 (2001), 170-187.
doi: 10.1086/318628. |
[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. |
[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. |
[18] |
M. Shub, Global Stability of Dynamical Systems, Springer, New-York/Berlin, 1987.
doi: 10.1007/978-1-4757-1947-5. |
[19] |
L. Stone, R. Olinky and A. Huppert,
Seasonal dynamics of recurrent epidemics, Nature, 446 (2007), 533-536.
doi: 10.1038/nature05638. |
[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. |
[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. |
[22] |
H. L. Smith,
Periodic solutions of periodic competitive and cooperative systems, SIAM J. Math. Anal., 17 (1986), 1289-1318.
doi: 10.1137/0517091. |
[23] |
H. L. Smith and P. Waltman, The Theory of the Chemostat, Cambridge University Press, 1995.
doi: 10.1017/CBO9780511530043. |
[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. |
[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. |
[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. |
[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. |
[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. |
[29] |
M. L. Zeeman,
Hopf bifurcations in competitive three-dimensional Lotka-Volterra systems, Dynam. Stability Systems, 8 (1993), 189-217.
doi: 10.1080/02681119308806158. |
[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. |
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.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[7] |
S. B. Hsu,
A competition model for a seasonally fluctuating nutrient, J. Math. Biol., 9 (1980), 115-132.
doi: 10.1007/BF00275917. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[15] |
E. Litchman and C. A. Klausmeier,
Competition of phytoplankton under fluctuating light, American Naturalist, 157 (2001), 170-187.
doi: 10.1086/318628. |
[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. |
[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. |
[18] |
M. Shub, Global Stability of Dynamical Systems, Springer, New-York/Berlin, 1987.
doi: 10.1007/978-1-4757-1947-5. |
[19] |
L. Stone, R. Olinky and A. Huppert,
Seasonal dynamics of recurrent epidemics, Nature, 446 (2007), 533-536.
doi: 10.1038/nature05638. |
[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. |
[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. |
[22] |
H. L. Smith,
Periodic solutions of periodic competitive and cooperative systems, SIAM J. Math. Anal., 17 (1986), 1289-1318.
doi: 10.1137/0517091. |
[23] |
H. L. Smith and P. Waltman, The Theory of the Chemostat, Cambridge University Press, 1995.
doi: 10.1017/CBO9780511530043. |
[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. |
[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. |
[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. |
[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. |
[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. |
[29] |
M. L. Zeeman,
Hopf bifurcations in competitive three-dimensional Lotka-Volterra systems, Dynam. Stability Systems, 8 (1993), 189-217.
doi: 10.1080/02681119308806158. |
[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. |
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