American Institute of Mathematical Sciences

Advanced
September  2012, 32(9): 3043-3058. doi: 10.3934/dcds.2012.32.3043

Monotone traveling waves for delayed Lotka-Volterra competition systems

Jian Fang 1, and  Jianhong Wu 2,

1. 

Department of Mathematics, Harbin Institute of Technology, Harbin, Heilongjiang, 150001, China
2. 

Centre for Disease Modelling and Department of Mathematics and Statistics, York University, Toronto, Ontario, M3J 1P3, Canada

Received  January 2012 Revised  March 2012 Published  April 2012

We consider a delayed reaction-diffusion Lotka-Volterra competition system which does not generate a monotone semiflow with respect to the standard ordering relation for competitive systems. We obtain a necessary and sufficient condition for the existence of traveling wave solutions connecting the extinction state to the coexistence state, and prove that such solutions are monotone and unique (up to translation).
Keywords: non-monotone systems, Delay, traveling waves, Lotka-Volterra competition system., time lags.
Mathematics Subject Classification: Primary: 35R10, 35K57; Secondary: 92D2.
Citation: Jian Fang, Jianhong Wu. Monotone traveling waves for delayed Lotka-Volterra competition systems. Discrete & Continuous Dynamical Systems - A, 2012, 32 (9) : 3043-3058. doi: 10.3934/dcds.2012.32.3043
References:
[1]

M. Aguerrea, C. Gomez and S. Trofimchuk, On uniqueness of semi-wavefronts: DiekmannKaper theory of a nonlinear convolution equation re-visited,, Math. Ann., ().

[2]

H. Berestycki, G. Nadin, B. Perthame and L. Ryzhik, The non-local Fisher-KPP equation: Travelling waves and steady states,, Nonlinearity, 22 (2009), 2813. doi: 10.1088/0951-7715/22/12/002.

[3]

N. Britton, Spatial structures and periodic travelling waves in an integro-differential reaction-diffusion population model,, SIAM J. Appl. Math., 50 (1990), 1663. doi: 10.1137/0150099.

[4]

J. Fang and X.-Q. Zhao, Existence and uniqueness of traveling waves for non-monotone integral equations with applicaitons,, J. Diff. Eqs., 248 (2010), 2199. doi: 10.1016/j.jde.2010.01.009.

[5]

J. Fang and X.-Q. Zhao, Monotone wavefronts of the nonlocal Fisher-KPP equation,, Nonlinearity, 24 (2011), 3043. doi: 10.1088/0951-7715/24/11/002.

[6]

T. Faria, W. Huang and J. Wu, Travelling waves for delayed reaction-diffusion equations with global response,, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 462 (2006), 229. doi: 10.1098/rspa.2005.1554.

[7]

T. Faria and J. Oliveira, Local and global stability for Lotka-Volterra systems with distributed delays and instantaneous negative feedbacks,, J. Diff. Eqs., 244 (2008), 1049. doi: 10.1016/j.jde.2007.12.005.

[8]

T. Faria and S. Trofimchuk, Positive travelling fronts for reaction-diffusion systems with distributed delay,, Nonlinearity, 23 (2010), 2457. doi: 10.1088/0951-7715/23/10/006.

[9]

T. Faria, Stability and extinction for Lotka-Volterra systems with infinite delay,, J. Dynam. Diff. Eqs., 22 (2010), 299. doi: 10.1007/s10884-010-9166-1.

[10]

G. Friesecke, Convergence to equilibrium for delay-diffusion equations with small delay,, J. Dynam. Diff. Eqs., 5 (1993), 89. doi: 10.1007/BF01063736.

[11]

A. Gomez and S. Trofimchuk, Monotone traveling wavefronts of the KPP-Fisher delayed equation,, J. Diff. Eqs., 250 (2011), 1767. doi: 10.1016/j.jde.2010.11.011.

[12]

S. Gourley, Travelling front solutions of a nonlocal Fisher equation,, J. Math. Biol., 41 (2000), 272. doi: 10.1007/s002850000047.

[13]

J.-S. Guo and X. Liang, The minimal speed of traveling fronts for the Lotka-Volterra competition system,, J. Dynam. Diff. Eqs., 23 (2011), 353. doi: 10.1007/s10884-011-9214-5.

[14]

W. Huang and M. Han, Non-linear determinacy of minimum wave speed for a Lotka-Volterra competition model,, J. Diff. Eqs., 251 (2011), 1549. doi: 10.1016/j.jde.2011.05.012.

[15]

S.-B. Hsu and X.-Q. Zhao, Spreading speeds and traveling waves for nonmonotone integrodifference equations,, SIAM J. Math. Anal., 40 (2008), 776. doi: 10.1137/070703016.

[16]

Y. Kuang and H. Smith, Convergence in Lotka-Volterra type diffusive delay systems without dominating instantaneous negative feedbacks,, J. Austral. Math. Soc. Ser. B, 34 (1993), 471. doi: 10.1017/S0334270000009036.

[17]

Y. Kuang and H. Smith, Global stability for infinite delay Lotka-Volterra type systems,, J. Diff. Eqs., 103 (1993), 221. doi: 10.1006/jdeq.1993.1048.

[18]

Y. Kuang, Global stability in delay differential systems without dominating instantaneous negative feedbacks,, J. Diff. Eqs., 119 (1995), 503. doi: 10.1006/jdeq.1995.1100.

[19]

M. K. Kwong and C. Ou, Existence and nonexistence of monotone traveling waves for the delayed Fisher equation,, J. Diff. Eqs., 249 (2010), 728. doi: 10.1016/j.jde.2010.04.017.

[20]

M. Lewis, B. Li and H. Weinberger, Spreading speed and linear determinacy for two-species competition models,, J. Math. Biol., 45 (2002), 219. doi: 10.1007/s002850200144.

[21]

W.-T. Li, G. Lin and S. Ruan, Existence of traveling wave solutions in delayed reaction-diffusion systems with applications to diffusion-competition systems,, Nonlinearity, 19 (2006), 1253. doi: [10.1088/0951-7715/19/6/003].

[22]

G. Lin, W.-T. Li, and M. Ma, Traveling wave solutions in delayed reaction diffusion systems with applications to multi-species models,, Dis. Cont. Dyn. Syst. - Series B, 13 (2010), 393. doi: 10.3934/dcdsb.2010.13.393.

[23]

B. Li, H. Weinberger and M. Lewis, Spreading speeds as slowest wave speeds for cooperative systems,, Math. Biosci., 196 (2005), 82. doi: 10.1016/j.mbs.2005.03.008.

[24]

B. Li, M. Lewis and H. Weinberger, Existence of traveling waves for integral recursions with nonmonotone growth functions,, J. Math. Biol., 58 (2009), 323. doi: 10.1007/s00285-008-0175-1.

[25]

X. Liang and X.-Q. Zhao, Spreading speeds and traveling waves for abstract monostable evolution systems,, J. Funct. Anal., 259 (2010), 857. doi: 10.1016/j.jfa.2010.04.018.

[26]

R. Lui, Biological growth and spread modeled by systems of recursions. I. Mathematical theory,, Math. Biosci., 93 (1989), 269. doi: 10.1016/0025-5564(89)90026-6.

[27]

J. Mallet-Paret, The Fredholm alternative for functional-differential equations of mixed type,, J. Dynam. Diff. Eqs., 11 (1999), 1. doi: 10.1023/A:1021889401235.

[28]

R. H. Martin and H. L. Smith, Abstract functional-differential equations and reaction-diffusion systems,, Trans. Amer. Math. Soc., 321 (1990), 1. doi: 10.2307/2001590.

[29]

G. Nadin, B. Perthame and M. Tang, Can a traveling wave connect two unstable states? The case of the nonlocal Fisher equation,, C. R. Acad. Sci. Paris, 349 (2011), 553. doi: 10.1016/j.crma.2011.03.008.

[30]

C. Ou and J. Wu, Traveling wavefronts in a delayed food-limited population model,, SIAM J. Math. Anal., 39 (2007), 103. doi: 10.1137/050638011.

[31]

C. Ou and J. Wu, Persistence of wavefronts in delayed nonlocal reaction-diffusion equations,, J. Diff. Eqs., 235 (2007), 219. doi: 10.1016/j.jde.2006.12.010.

[32]

H. Smith, Systems of ordinary differential equations which generate an order preserving flow. A survey of results,, SIAM Rev., 30 (1988), 87. doi: 10.1137/1030003.

[33]

H. Smith, "Monotone Dynamical Systems. An Introduction to the Theory of Competitive and Cooperative Systems,", Math. Surveys and Monographs, 41 (1995).

[34]

H. Smith and H. Thieme, Monotone semiflows in scalar non-quasi-monotone functional-differential equations,, J. Math. Anal. Appl., 150 (1990), 289. doi: 10.1016/0022-247X(90)90105-O.

[35]

H. Smith and H. Thieme, Strongly order preserving semiflows generated by functional-differential equations,, J. Diff. Eqs., 93 (1991), 332. doi: 10.1016/0022-0396(91)90016-3.

[36]

A. I. Volpert, V. A. Volpert and V. A. Volpert, "Traveling Wave Solutions of Parabolic Systems,", Translation of Mathematical Monographs, 140 (1994).

[37]

H. Wang, Spreading speeds and traveling waves for non-cooperative reaction-diffusion systems,, J. Non. Sci., 21 (2011), 747. doi: 10.1007/s00332-011-9099-9.

[38]

H. F. Weinberger, Long-time behavior of a class of biological models,, SIAM J. Math. Anal., 13 (1982), 353. doi: 10.1137/0513028.

[39]

H. F. Weinberger, On spreading speeds and traveling waves for growth and migration models in a periodic habitat,, J. Math. Biol., 45 (2002), 511. doi: 10.1007/s00285-002-0169-3.

[40]

J. Wu and X.-Q. Zhao, Diffusive monotonicity and threshold dynamics of delayed reaction diffusion equations,, J. Diff. Eqs., 186 (2002), 470. doi: 10.1016/S0022-0396(02)00012-8.

[41]

J. Wu and X. Zou, Traveling wave fronts of reaction-diffusion systems with delay,, J. Dyn. Diff. Eqs., 13 (2001), 651. doi: 10.1023/A:1016690424892.

show all references

References:
[1]

M. Aguerrea, C. Gomez and S. Trofimchuk, On uniqueness of semi-wavefronts: DiekmannKaper theory of a nonlinear convolution equation re-visited,, Math. Ann., ().

[2]

H. Berestycki, G. Nadin, B. Perthame and L. Ryzhik, The non-local Fisher-KPP equation: Travelling waves and steady states,, Nonlinearity, 22 (2009), 2813. doi: 10.1088/0951-7715/22/12/002.

[3]

N. Britton, Spatial structures and periodic travelling waves in an integro-differential reaction-diffusion population model,, SIAM J. Appl. Math., 50 (1990), 1663. doi: 10.1137/0150099.

[4]

J. Fang and X.-Q. Zhao, Existence and uniqueness of traveling waves for non-monotone integral equations with applicaitons,, J. Diff. Eqs., 248 (2010), 2199. doi: 10.1016/j.jde.2010.01.009.

[5]

J. Fang and X.-Q. Zhao, Monotone wavefronts of the nonlocal Fisher-KPP equation,, Nonlinearity, 24 (2011), 3043. doi: 10.1088/0951-7715/24/11/002.

[6]

T. Faria, W. Huang and J. Wu, Travelling waves for delayed reaction-diffusion equations with global response,, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 462 (2006), 229. doi: 10.1098/rspa.2005.1554.

[7]

T. Faria and J. Oliveira, Local and global stability for Lotka-Volterra systems with distributed delays and instantaneous negative feedbacks,, J. Diff. Eqs., 244 (2008), 1049. doi: 10.1016/j.jde.2007.12.005.

[8]

T. Faria and S. Trofimchuk, Positive travelling fronts for reaction-diffusion systems with distributed delay,, Nonlinearity, 23 (2010), 2457. doi: 10.1088/0951-7715/23/10/006.

[9]

T. Faria, Stability and extinction for Lotka-Volterra systems with infinite delay,, J. Dynam. Diff. Eqs., 22 (2010), 299. doi: 10.1007/s10884-010-9166-1.

[10]

G. Friesecke, Convergence to equilibrium for delay-diffusion equations with small delay,, J. Dynam. Diff. Eqs., 5 (1993), 89. doi: 10.1007/BF01063736.

[11]

A. Gomez and S. Trofimchuk, Monotone traveling wavefronts of the KPP-Fisher delayed equation,, J. Diff. Eqs., 250 (2011), 1767. doi: 10.1016/j.jde.2010.11.011.

[12]

S. Gourley, Travelling front solutions of a nonlocal Fisher equation,, J. Math. Biol., 41 (2000), 272. doi: 10.1007/s002850000047.

[13]

J.-S. Guo and X. Liang, The minimal speed of traveling fronts for the Lotka-Volterra competition system,, J. Dynam. Diff. Eqs., 23 (2011), 353. doi: 10.1007/s10884-011-9214-5.

[14]

W. Huang and M. Han, Non-linear determinacy of minimum wave speed for a Lotka-Volterra competition model,, J. Diff. Eqs., 251 (2011), 1549. doi: 10.1016/j.jde.2011.05.012.

[15]

S.-B. Hsu and X.-Q. Zhao, Spreading speeds and traveling waves for nonmonotone integrodifference equations,, SIAM J. Math. Anal., 40 (2008), 776. doi: 10.1137/070703016.

[16]

Y. Kuang and H. Smith, Convergence in Lotka-Volterra type diffusive delay systems without dominating instantaneous negative feedbacks,, J. Austral. Math. Soc. Ser. B, 34 (1993), 471. doi: 10.1017/S0334270000009036.

[17]

Y. Kuang and H. Smith, Global stability for infinite delay Lotka-Volterra type systems,, J. Diff. Eqs., 103 (1993), 221. doi: 10.1006/jdeq.1993.1048.

[18]

Y. Kuang, Global stability in delay differential systems without dominating instantaneous negative feedbacks,, J. Diff. Eqs., 119 (1995), 503. doi: 10.1006/jdeq.1995.1100.

[19]

M. K. Kwong and C. Ou, Existence and nonexistence of monotone traveling waves for the delayed Fisher equation,, J. Diff. Eqs., 249 (2010), 728. doi: 10.1016/j.jde.2010.04.017.

[20]

M. Lewis, B. Li and H. Weinberger, Spreading speed and linear determinacy for two-species competition models,, J. Math. Biol., 45 (2002), 219. doi: 10.1007/s002850200144.

[21]

W.-T. Li, G. Lin and S. Ruan, Existence of traveling wave solutions in delayed reaction-diffusion systems with applications to diffusion-competition systems,, Nonlinearity, 19 (2006), 1253. doi: [10.1088/0951-7715/19/6/003].

[22]

G. Lin, W.-T. Li, and M. Ma, Traveling wave solutions in delayed reaction diffusion systems with applications to multi-species models,, Dis. Cont. Dyn. Syst. - Series B, 13 (2010), 393. doi: 10.3934/dcdsb.2010.13.393.

[23]

B. Li, H. Weinberger and M. Lewis, Spreading speeds as slowest wave speeds for cooperative systems,, Math. Biosci., 196 (2005), 82. doi: 10.1016/j.mbs.2005.03.008.

[24]

B. Li, M. Lewis and H. Weinberger, Existence of traveling waves for integral recursions with nonmonotone growth functions,, J. Math. Biol., 58 (2009), 323. doi: 10.1007/s00285-008-0175-1.

[25]

X. Liang and X.-Q. Zhao, Spreading speeds and traveling waves for abstract monostable evolution systems,, J. Funct. Anal., 259 (2010), 857. doi: 10.1016/j.jfa.2010.04.018.

[26]

R. Lui, Biological growth and spread modeled by systems of recursions. I. Mathematical theory,, Math. Biosci., 93 (1989), 269. doi: 10.1016/0025-5564(89)90026-6.

[27]

J. Mallet-Paret, The Fredholm alternative for functional-differential equations of mixed type,, J. Dynam. Diff. Eqs., 11 (1999), 1. doi: 10.1023/A:1021889401235.

[28]

R. H. Martin and H. L. Smith, Abstract functional-differential equations and reaction-diffusion systems,, Trans. Amer. Math. Soc., 321 (1990), 1. doi: 10.2307/2001590.

[29]

G. Nadin, B. Perthame and M. Tang, Can a traveling wave connect two unstable states? The case of the nonlocal Fisher equation,, C. R. Acad. Sci. Paris, 349 (2011), 553. doi: 10.1016/j.crma.2011.03.008.

[30]

C. Ou and J. Wu, Traveling wavefronts in a delayed food-limited population model,, SIAM J. Math. Anal., 39 (2007), 103. doi: 10.1137/050638011.

[31]

C. Ou and J. Wu, Persistence of wavefronts in delayed nonlocal reaction-diffusion equations,, J. Diff. Eqs., 235 (2007), 219. doi: 10.1016/j.jde.2006.12.010.

[32]

H. Smith, Systems of ordinary differential equations which generate an order preserving flow. A survey of results,, SIAM Rev., 30 (1988), 87. doi: 10.1137/1030003.

[33]

H. Smith, "Monotone Dynamical Systems. An Introduction to the Theory of Competitive and Cooperative Systems,", Math. Surveys and Monographs, 41 (1995).

[34]

H. Smith and H. Thieme, Monotone semiflows in scalar non-quasi-monotone functional-differential equations,, J. Math. Anal. Appl., 150 (1990), 289. doi: 10.1016/0022-247X(90)90105-O.

[35]

H. Smith and H. Thieme, Strongly order preserving semiflows generated by functional-differential equations,, J. Diff. Eqs., 93 (1991), 332. doi: 10.1016/0022-0396(91)90016-3.

[36]

A. I. Volpert, V. A. Volpert and V. A. Volpert, "Traveling Wave Solutions of Parabolic Systems,", Translation of Mathematical Monographs, 140 (1994).

[37]

H. Wang, Spreading speeds and traveling waves for non-cooperative reaction-diffusion systems,, J. Non. Sci., 21 (2011), 747. doi: 10.1007/s00332-011-9099-9.

[38]

H. F. Weinberger, Long-time behavior of a class of biological models,, SIAM J. Math. Anal., 13 (1982), 353. doi: 10.1137/0513028.

[39]

H. F. Weinberger, On spreading speeds and traveling waves for growth and migration models in a periodic habitat,, J. Math. Biol., 45 (2002), 511. doi: 10.1007/s00285-002-0169-3.

[40]

J. Wu and X.-Q. Zhao, Diffusive monotonicity and threshold dynamics of delayed reaction diffusion equations,, J. Diff. Eqs., 186 (2002), 470. doi: 10.1016/S0022-0396(02)00012-8.

[41]

J. Wu and X. Zou, Traveling wave fronts of reaction-diffusion systems with delay,, J. Dyn. Diff. Eqs., 13 (2001), 651. doi: 10.1023/A:1016690424892.

[1]

Guo-Bao Zhang, Ruyun Ma, Xue-Shi Li. Traveling waves of a Lotka-Volterra strong competition system with nonlocal dispersal. Discrete & Continuous Dynamical Systems - B, 2018, 23 (2) : 587-608. doi: 10.3934/dcdsb.2018035
[2]

Zhao-Xing Yang, Guo-Bao Zhang, Ge Tian, Zhaosheng Feng. Stability of non-monotone non-critical traveling waves in discrete reaction-diffusion equations with time delay. Discrete & Continuous Dynamical Systems - S, 2017, 10 (3) : 581-603. doi: 10.3934/dcdss.2017029
[3]

Yuzo Hosono. Traveling waves for a diffusive Lotka-Volterra competition model I: singular perturbations. Discrete & Continuous Dynamical Systems - B, 2003, 3 (1) : 79-95. doi: 10.3934/dcdsb.2003.3.79
[4]

Yang Wang, Xiong Li. Uniqueness of traveling front solutions for the Lotka-Volterra system in the weak competition case. Discrete & Continuous Dynamical Systems - B, 2019, 24 (7) : 3067-3075. doi: 10.3934/dcdsb.2018300
[5]

Zhi-Cheng Wang, Hui-Ling Niu, Shigui Ruan. On the existence of axisymmetric traveling fronts in Lotka-Volterra competition-diffusion systems in ℝ3. Discrete & Continuous Dynamical Systems - B, 2017, 22 (3) : 1111-1144. doi: 10.3934/dcdsb.2017055
[6]

Rui Huang, Ming Mei, Kaijun Zhang, Qifeng Zhang. Asymptotic stability of non-monotone traveling waves for time-delayed nonlocal dispersion equations. Discrete & Continuous Dynamical Systems - A, 2016, 36 (3) : 1331-1353. doi: 10.3934/dcds.2016.36.1331
[7]

Yuzo Hosono. Traveling waves for the Lotka-Volterra predator-prey system without diffusion of the predator. Discrete & Continuous Dynamical Systems - B, 2015, 20 (1) : 161-171. doi: 10.3934/dcdsb.2015.20.161
[8]

Zhaohai Ma, Rong Yuan, Yang Wang, Xin Wu. Multidimensional stability of planar traveling waves for the delayed nonlocal dispersal competitive Lotka-Volterra system. Communications on Pure & Applied Analysis, 2019, 18 (4) : 2069-2092. doi: 10.3934/cpaa.2019093
[9]

Li-Jun Du, Wan-Tong Li, Jia-Bing Wang. Invasion entire solutions in a time periodic Lotka-Volterra competition system with diffusion. Mathematical Biosciences & Engineering, 2017, 14 (5&6) : 1187-1213. doi: 10.3934/mbe.2017061
[10]

Dejun Fan, Xiaoyu Yi, Ling Xia, Jingliang Lv. Dynamical behaviors of stochastic type K monotone Lotka-Volterra systems. Discrete & Continuous Dynamical Systems - B, 2019, 24 (6) : 2901-2922. doi: 10.3934/dcdsb.2018291
[11]

Lih-Ing W. Roeger, Razvan Gelca. Dynamically consistent discrete-time Lotka-Volterra competition models. Conference Publications, 2009, 2009 (Special) : 650-658. doi: 10.3934/proc.2009.2009.650
[12]

Yuan Lou, Dongmei Xiao, Peng Zhou. Qualitative analysis for a Lotka-Volterra competition system in advective homogeneous environment. Discrete & Continuous Dynamical Systems - A, 2016, 36 (2) : 953-969. doi: 10.3934/dcds.2016.36.953
[13]

Jong-Shenq Guo, Ying-Chih Lin. The sign of the wave speed for the Lotka-Volterra competition-diffusion system. Communications on Pure & Applied Analysis, 2013, 12 (5) : 2083-2090. doi: 10.3934/cpaa.2013.12.2083
[14]

Qi Wang, Chunyi Gai, Jingda Yan. Qualitative analysis of a Lotka-Volterra competition system with advection. Discrete & Continuous Dynamical Systems - A, 2015, 35 (3) : 1239-1284. doi: 10.3934/dcds.2015.35.1239
[15]

Qi Wang, Yang Song, Lingjie Shao. Boundedness and persistence of populations in advective Lotka-Volterra competition system. Discrete & Continuous Dynamical Systems - B, 2018, 23 (6) : 2245-2263. doi: 10.3934/dcdsb.2018195
[16]

Anthony W. Leung, Xiaojie Hou, Wei Feng. Traveling wave solutions for Lotka-Volterra system re-visited. Discrete & Continuous Dynamical Systems - B, 2011, 15 (1) : 171-196. doi: 10.3934/dcdsb.2011.15.171
[17]

Fuke Wu, Yangzi Hu. Stochastic Lotka-Volterra system with unbounded distributed delay. Discrete & Continuous Dynamical Systems - B, 2010, 14 (1) : 275-288. doi: 10.3934/dcdsb.2010.14.275
[18]

Xiaoli Liu, Dongmei Xiao. Bifurcations in a discrete time Lotka-Volterra predator-prey system. Discrete & Continuous Dynamical Systems - B, 2006, 6 (3) : 559-572. doi: 10.3934/dcdsb.2006.6.559
[19]

De Tang. Dynamical behavior for a Lotka-Volterra weak competition system in advective homogeneous environment. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 1-16. doi: 10.3934/dcdsb.2019037
[20]

Qihuai Liu, Dingbian Qian, Zhiguo Wang. Quasi-periodic solutions of the Lotka-Volterra competition systems with quasi-periodic perturbations. Discrete & Continuous Dynamical Systems - B, 2012, 17 (5) : 1537-1550. doi: 10.3934/dcdsb.2012.17.1537

2017 Impact Factor: 1.179

Tools

Metrics

  • PDF downloads (18)
  • HTML views (0)
  • Cited by (9)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences