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., ().   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

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

[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.  Google Scholar

[12]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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].  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[33]

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

[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.  Google Scholar

[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.  Google Scholar

[36]

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

[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.  Google Scholar

[38]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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., ().   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

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

[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.  Google Scholar

[12]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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].  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[33]

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

[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.  Google Scholar

[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.  Google Scholar

[36]

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

[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.  Google Scholar

[38]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[1]

Xianyong Chen, Weihua Jiang. Multiple spatiotemporal coexistence states and Turing-Hopf bifurcation in a Lotka-Volterra competition system with nonlocal delays. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2021013
[2]

Lin Niu, Yi Wang, Xizhuang Xie. Carrying simplex in the Lotka-Volterra competition model with seasonal succession with applications. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2021014
[3]

Ting Liu, Guo-Bao Zhang. Global stability of traveling waves for a spatially discrete diffusion system with time delay. Electronic Research Archive, , () : -. doi: 10.3934/era.2021003
[4]

Wei Feng, Michael Freeze, Xin Lu. On competition models under allee effect: Asymptotic behavior and traveling waves. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5609-5626. doi: 10.3934/cpaa.2020256
[5]

Wei-Chieh Chen, Bogdan Kazmierczak. Traveling waves in quadratic autocatalytic systems with complexing agent. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020364
[6]

Jonathan J. Wylie, Robert M. Miura, Huaxiong Huang. Systems of coupled diffusion equations with degenerate nonlinear source terms: Linear stability and traveling waves. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 561-569. doi: 10.3934/dcds.2009.23.561
[7]

Maoli Chen, Xiao Wang, Yicheng Liu. Collision-free flocking for a time-delay system. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 1223-1241. doi: 10.3934/dcdsb.2020251
[8]

Chongyang Liu, Meijia Han, Zhaohua Gong, Kok Lay Teo. Robust parameter estimation for constrained time-delay systems with inexact measurements. Journal of Industrial & Management Optimization, 2021, 17 (1) : 317-337. doi: 10.3934/jimo.2019113
[9]

Jianquan Li, Xin Xie, Dian Zhang, Jia Li, Xiaolin Lin. Qualitative analysis of a simple tumor-immune system with time delay of tumor action. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020341
[10]

Zhenzhen Wang, Tianshou Zhou. Asymptotic behaviors and stochastic traveling waves in stochastic Fisher-KPP equations. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020323
[11]

Simone Göttlich, Elisa Iacomini, Thomas Jung. Properties of the LWR model with time delay. Networks & Heterogeneous Media, 2020  doi: 10.3934/nhm.2020032
[12]

Pan Zheng. Asymptotic stability in a chemotaxis-competition system with indirect signal production. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1207-1223. doi: 10.3934/dcds.2020315
[13]

Hirofumi Izuhara, Shunsuke Kobayashi. Spatio-temporal coexistence in the cross-diffusion competition system. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 919-933. doi: 10.3934/dcdss.2020228
[14]

Hai Huang, Xianlong Fu. Optimal control problems for a neutral integro-differential system with infinite delay. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020107
[15]

Jerry L. Bona, Angel Durán, Dimitrios Mitsotakis. Solitary-wave solutions of Benjamin-Ono and other systems for internal waves. I. approximations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 87-111. doi: 10.3934/dcds.2020215
[16]

Zhongbao Zhou, Yanfei Bai, Helu Xiao, Xu Chen. A non-zero-sum reinsurance-investment game with delay and asymmetric information. Journal of Industrial & Management Optimization, 2021, 17 (2) : 909-936. doi: 10.3934/jimo.2020004
[17]

Sergey Rashkovskiy. Hamilton-Jacobi theory for Hamiltonian and non-Hamiltonian systems. Journal of Geometric Mechanics, 2020, 12 (4) : 563-583. doi: 10.3934/jgm.2020024
[18]

Xiaoping Zhai, Yongsheng Li. Global large solutions and optimal time-decay estimates to the Korteweg system. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1387-1413. doi: 10.3934/dcds.2020322
[19]

Fathalla A. Rihan, Hebatallah J. Alsakaji. Stochastic delay differential equations of three-species prey-predator system with cooperation among prey species. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020468
[20]

Soniya Singh, Sumit Arora, Manil T. Mohan, Jaydev Dabas. Approximate controllability of second order impulsive systems with state-dependent delay in Banach spaces. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020103

2019 Impact Factor: 1.338

Tools

Metrics

  • PDF downloads (56)
  • HTML views (0)
  • Cited by (13)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences