# American Institute of Mathematical Sciences

doi: 10.3934/dcdsb.2022148
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## Bistable traveling waves in a delayed competitive system

 1 School of Science, Lanzhou University of Technology, Lanzhou, Gansu 730050, China 2 School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu 730000, China

*Corresponding author: Shuxia Pan

Received  January 2022 Revised  June 2022 Early access July 2022

Fund Project: The first author is supported by NSF of China grant 11971213

This paper is concerned with the existence, uniqueness and asymptotic stability of traveling wave solutions in a delayed competitive system with spatial diffusion. When the interspecific competition is strong, the corresponding functional differential system has two stable and two unstable steady states. Firstly, we establish the existence of bistable traveling wave solutions by the theory of monotone semiflows. With the help of squeezing technique based on comparison principle, the asymptotic stability of traveling wave solutions is proved, which further implies the uniqueness of wave speed and wave profile in the sense of phase shift.

Citation: Shuxia Pan, Shengnan Hao. Bistable traveling waves in a delayed competitive system. Discrete and Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2022148
##### References:
 [1] M. Alfaro, A. Ducrot and T. Giletti, Travelling waves for a non-monotone bistable equation with delay: Existence and oscillations, Proc. Lond. Math. Soc., 116 (2018), 729-759.  doi: 10.1112/plms.12092. [2] P. W. Bates, P. C. Fife, X. Ren and X. Wang, Traveling waves in a convolution model for phase transitions, Arch. Rational Mech. Anal., 138 (1997), 105-136.  doi: 10.1007/s002050050037. [3] W.-J. Bo, G. Lin and S. Ruan, Traveling wave solutions for time periodic reaction-diffusion systems, Discrete Contin. Dyn. Syst., 38 (2018), 4329-4351.  doi: 10.3934/dcds.2018189. [4] X. Chen, Existence, uniqueness, and asymptotic stability of traveling waves in nonlocal evolution equations, Adv. Differential Equations, 2 (1997), 125-160. [5] J. Fang and X.-Q. Zhao, Bistable traveling waves for monotone semiflows with application, J. Eur. Math. Soc., 17 (2015), 2243-2288.  doi: 10.4171/JEMS/556. [6] P. C. Fife and J. B. McLeod, The approach of solutions of nonlinear diffusion equations to travelling front solutions, Arch. Rational Mech. Anal., 65 (1977), 335-361.  doi: 10.1007/BF00250432. [7] K. Li and X. Li, Existence and stability of bistable wavefronts in a nonlocal delayed reaction-diffusion epidemic system, European J. Appl. Math., 32 (2021), 146-176.  doi: 10.1017/s0956792520000078. [8] X. Liang and X.-Q. Zhao, Asymptotic speeds of spread and traveling waves for monotone semiflows with applications, Comm. Pure Appl. Math., 60 (2007), 1-40.  doi: 10.1002/cpa.20154. [9] G. Lin and W.-T. Li, Bistable wavefronts in a diffusive and competitive Lotka-Volterra type system with nonlocal delays, J. Differential Equations, 244 (2008), 487-513.  doi: 10.1016/j.jde.2007.10.019. [10] S. Ma and J. Wu, Existence, uniqueness and asymptotic stability of traveling wavefronts in a non-local delayed diffusion equation, J. Dynam. Differential Equations, 19 (2007), 391-436.  doi: 10.1007/s10884-006-9065-7. [11] M. C. Mackey and L. Glass, Oscillation and chaos in physiological control systems, Science, 197 (1977), 287-289.  doi: 10.1126/science.267326. [12] R. H. Martin and H. L. Smith, Abstract functional-differential equations and reaction-diffusion systems, Trans. Amer. Math. Soc., 321 (1990), 1-44.  doi: 10.2307/2001590. [13] A. Ruiz-Herrera, Delay reaction-diffusion systems via discrete dynamics, SIAM J. Math. Anal., 52 (2020), 6297-6312.  doi: 10.1137/19M1304477. [14] K. W. Schaaf, Asymptotic behavior and traveling wave solutions for parabolic functional-differential equations, Trans. Amer. Math. Soc., 302 (1987), 587-615.  doi: 10.2307/2000859. [15] H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, AMS, Providence, RI, 1995. [16] H. L. Smith and X.-Q. Zhao, Global asymptotic stability of traveling waves in delayed reaction-diffusion equations, SIAM J. Math. Anal., 31 (2000), 514-534.  doi: 10.1137/S0036141098346785. [17] H. R. Thieme and X.-Q. Zhao, Asymptotic speeds of spread and traveling waves for integral equations and delayed reaction-diffusion models, J. Differential Equations, 195 (2003), 430-470.  doi: 10.1016/S0022-0396(03)00175-X. [18] Y.-L. Tian and X.-Q. Zhao, Bistable traveling waves for a competitive-cooperative system with nonlocal delays, J. Differential Equations, 264 (2018), 5263-5299.  doi: 10.1016/j.jde.2018.01.010. [19] S. Trofimchuk and V. Volpert, Global continuation of monotone waves for bistable delayed equations with unimodal nonlinearities, Nonlinearity, 32 (2019), 2593-2632.  doi: 10.1088/1361-6544/ab0e23. [20] Z.-C. Wang, W.-T. Li and S. Ruan, Existence and stability of traveling wave fronts in reaction advection diffusion equations with nonlocal delay, J. Differential Equations, 238 (2007), 153-200.  doi: 10.1016/j.jde.2007.03.025. [21] J. Wu, Theory and Applications of Partial Functional Differential Equations, Springer-Verlag, New York, 1996. doi: 10.1007/978-1-4612-4050-1. [22] S.-L. Wu and C.-H. Hsu, Entire solutions with annihilating fronts to a nonlocal dispersal equation with bistable nonlinearity and spatio-temporal delay, J. Dynam. Differential Equations, 29 (2017), 409-430.  doi: 10.1007/s10884-015-9450-1. [23] S.-L. Wu and C.-H. Hsu, Periodic traveling fronts for partially degenerate reaction-diffusion systems with bistable and time-periodic nonlinearity, Adv. Nonlinear Anal., 9 (2020), 923-957.  doi: 10.1515/anona-2020-0033. [24] Y. Xing and G. Lin, Traveling wave solutions in a delayed competitive model, J. Math. Anal. Appl., 507 (2022), Paper No. 125766, 20 pp. doi: 10.1016/j.jmaa.2021.125766. [25] Y.-R. Yang, W.-T. Li and S.-L. Wu, Global exponential stability of traveling fronts in delayed bistable systems. (Chinese), Chinese J. Contemp. Math., 31 (2010), 249-262. [26] Y.-R. Yang and N.-W. Liu, Monotonicity and uniqueness of traveling waves in bistable systems with delay, Electron. J. Differential Equations, 2014 (2014), 12 pp. [27] Y. Zhang and X.-Q. Zhao, Uniqueness and stability of bistable waves for monotone semiflows, Proc. Amer. Math. Soc., 149 (2021), 4287-4302.  doi: 10.1090/proc/15506.

show all references

##### References:
 [1] M. Alfaro, A. Ducrot and T. Giletti, Travelling waves for a non-monotone bistable equation with delay: Existence and oscillations, Proc. Lond. Math. Soc., 116 (2018), 729-759.  doi: 10.1112/plms.12092. [2] P. W. Bates, P. C. Fife, X. Ren and X. Wang, Traveling waves in a convolution model for phase transitions, Arch. Rational Mech. Anal., 138 (1997), 105-136.  doi: 10.1007/s002050050037. [3] W.-J. Bo, G. Lin and S. Ruan, Traveling wave solutions for time periodic reaction-diffusion systems, Discrete Contin. Dyn. Syst., 38 (2018), 4329-4351.  doi: 10.3934/dcds.2018189. [4] X. Chen, Existence, uniqueness, and asymptotic stability of traveling waves in nonlocal evolution equations, Adv. Differential Equations, 2 (1997), 125-160. [5] J. Fang and X.-Q. Zhao, Bistable traveling waves for monotone semiflows with application, J. Eur. Math. Soc., 17 (2015), 2243-2288.  doi: 10.4171/JEMS/556. [6] P. C. Fife and J. B. McLeod, The approach of solutions of nonlinear diffusion equations to travelling front solutions, Arch. Rational Mech. Anal., 65 (1977), 335-361.  doi: 10.1007/BF00250432. [7] K. Li and X. Li, Existence and stability of bistable wavefronts in a nonlocal delayed reaction-diffusion epidemic system, European J. Appl. Math., 32 (2021), 146-176.  doi: 10.1017/s0956792520000078. [8] X. Liang and X.-Q. Zhao, Asymptotic speeds of spread and traveling waves for monotone semiflows with applications, Comm. Pure Appl. Math., 60 (2007), 1-40.  doi: 10.1002/cpa.20154. [9] G. Lin and W.-T. Li, Bistable wavefronts in a diffusive and competitive Lotka-Volterra type system with nonlocal delays, J. Differential Equations, 244 (2008), 487-513.  doi: 10.1016/j.jde.2007.10.019. [10] S. Ma and J. Wu, Existence, uniqueness and asymptotic stability of traveling wavefronts in a non-local delayed diffusion equation, J. Dynam. Differential Equations, 19 (2007), 391-436.  doi: 10.1007/s10884-006-9065-7. [11] M. C. Mackey and L. Glass, Oscillation and chaos in physiological control systems, Science, 197 (1977), 287-289.  doi: 10.1126/science.267326. [12] R. H. Martin and H. L. Smith, Abstract functional-differential equations and reaction-diffusion systems, Trans. Amer. Math. Soc., 321 (1990), 1-44.  doi: 10.2307/2001590. [13] A. Ruiz-Herrera, Delay reaction-diffusion systems via discrete dynamics, SIAM J. Math. Anal., 52 (2020), 6297-6312.  doi: 10.1137/19M1304477. [14] K. W. Schaaf, Asymptotic behavior and traveling wave solutions for parabolic functional-differential equations, Trans. Amer. Math. Soc., 302 (1987), 587-615.  doi: 10.2307/2000859. [15] H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, AMS, Providence, RI, 1995. [16] H. L. Smith and X.-Q. Zhao, Global asymptotic stability of traveling waves in delayed reaction-diffusion equations, SIAM J. Math. Anal., 31 (2000), 514-534.  doi: 10.1137/S0036141098346785. [17] H. R. Thieme and X.-Q. Zhao, Asymptotic speeds of spread and traveling waves for integral equations and delayed reaction-diffusion models, J. Differential Equations, 195 (2003), 430-470.  doi: 10.1016/S0022-0396(03)00175-X. [18] Y.-L. Tian and X.-Q. Zhao, Bistable traveling waves for a competitive-cooperative system with nonlocal delays, J. Differential Equations, 264 (2018), 5263-5299.  doi: 10.1016/j.jde.2018.01.010. [19] S. Trofimchuk and V. Volpert, Global continuation of monotone waves for bistable delayed equations with unimodal nonlinearities, Nonlinearity, 32 (2019), 2593-2632.  doi: 10.1088/1361-6544/ab0e23. [20] Z.-C. Wang, W.-T. Li and S. Ruan, Existence and stability of traveling wave fronts in reaction advection diffusion equations with nonlocal delay, J. Differential Equations, 238 (2007), 153-200.  doi: 10.1016/j.jde.2007.03.025. [21] J. Wu, Theory and Applications of Partial Functional Differential Equations, Springer-Verlag, New York, 1996. doi: 10.1007/978-1-4612-4050-1. [22] S.-L. Wu and C.-H. Hsu, Entire solutions with annihilating fronts to a nonlocal dispersal equation with bistable nonlinearity and spatio-temporal delay, J. Dynam. Differential Equations, 29 (2017), 409-430.  doi: 10.1007/s10884-015-9450-1. [23] S.-L. Wu and C.-H. Hsu, Periodic traveling fronts for partially degenerate reaction-diffusion systems with bistable and time-periodic nonlinearity, Adv. Nonlinear Anal., 9 (2020), 923-957.  doi: 10.1515/anona-2020-0033. [24] Y. Xing and G. Lin, Traveling wave solutions in a delayed competitive model, J. Math. Anal. Appl., 507 (2022), Paper No. 125766, 20 pp. doi: 10.1016/j.jmaa.2021.125766. [25] Y.-R. Yang, W.-T. Li and S.-L. Wu, Global exponential stability of traveling fronts in delayed bistable systems. (Chinese), Chinese J. Contemp. Math., 31 (2010), 249-262. [26] Y.-R. Yang and N.-W. Liu, Monotonicity and uniqueness of traveling waves in bistable systems with delay, Electron. J. Differential Equations, 2014 (2014), 12 pp. [27] Y. Zhang and X.-Q. Zhao, Uniqueness and stability of bistable waves for monotone semiflows, Proc. Amer. Math. Soc., 149 (2021), 4287-4302.  doi: 10.1090/proc/15506.
The evolution of (13) with $r = 0.5$. From the viewpoint of population dynamics, $u_2$ almost invades the habitat of $u_1$ at a constant speed $0.17$
The evolution of (13) with $r = 2.0$. From the viewpoint of population dynamics, $u_1$ almost invades the habitat of $u_2$ at a constant speed $0.24$
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