Volume 42 Issue 12
Dec.  2021
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ZHENG Jingpan. The Wave Speed Signs for Bistable Traveling Wave Solutions in 3-Species Competition-Diffusion Systems[J]. Applied Mathematics and Mechanics, 2021, 42(12): 1296-1305. doi: 10.21656/1000-0887.420093
 Citation: ZHENG Jingpan. The Wave Speed Signs for Bistable Traveling Wave Solutions in 3-Species Competition-Diffusion Systems[J]. Applied Mathematics and Mechanics, 2021, 42(12): 1296-1305.

# The Wave Speed Signs for Bistable Traveling Wave Solutions in 3-Species Competition-Diffusion Systems

##### doi: 10.21656/1000-0887.420093
• Rev Recd Date: 2021-06-02
• Available Online: 2021-12-31
• In the bistable competition-diffusion model, the wave speed signs for the traveling waves can predict which species are more dominant and will eventually occupy the whole habitat. Therefore, it is of great biological significance to study the speed signs for the traveling waves. Firstly, the Lotka-Volterra competition-diffusion system was transformed into a cooperative system. Under the comparison principle, the comparison theorem for the bistable wave speed and the specific upper-lower solution wave speeds of wave profile equations was obtained. Then, according to the comparison theorem and through construction of suitable upper-lower solutions, some sufficient conditions for determining the bistable traveling wave speed signs were obtained. The results help predict and control the competition results of biological populations.
•  GUO J S, LIN Y C. The sign of the wave speed for the Lotka-Volterra competition-diffusion system[J].Communications on Pure and Applied Analysis,2013,12(5): 2083-2090. [2]MA M, HUANG Z, OU C. Speed of the traveling wave for the bistable Lotka-Volterra competition model[J].Nonlinearity,2019,32(9): 3143-3162. [3]MA M, ZHANG Q, YUE J, et al. Bistable wave speed of the Lotka-Volterra competition model[J].Journal of Biological Dynamics,2020,14(1): 608-620. [4]WANG H, OU C. Propagation speed of the bistable traveling wave to the Lotka-Volterra competition system in a periodic habitat[J].Journal of Nonlinear Science,2020,30: 3129-3159. [5]WANG H, OU C. Propagation direction of the traveling wave for the Lotka-Volterra competitive lattice system[J].Journal of Dynamics and Differential Equation,2021,33: 1153-1174. [6]马满军, 岳缘希, OU Chunhua. 具非局部扩散Lotka-Volterra系统的双稳波速[J].中国科学: 数学, 2021,51: 1-16.（MA Manjun, YUE Yuanxi, OU Chunhua. Bistable wave velocities with nonlocally diffused Lotka-Volterra systems[J].Science China: Mathematics,2021,51: 1-16.(in Chinese)） [7]张国宝, 何娟. 时滞非局部扩散方程的双稳波速[J].西北师范大学学报(自然科学版), 2021,57(3): 7-12.(ZHANG Guobao, HE Juan. Bistable wave speed for delayed nonlocal dispersal equations[J].Journal of Northwest Normal University(Natural Science),2021,57(3): 7-12.(in Chinese)) [8]GUO J S, NAKAMURA K I, OGIWARA T, et al. The sign of traveling wave speed in bistable dynamics[J].Discrete and Continuous Dynamical Systems,2020,40(6): 3451-3466. [9]CHANG C H. The stability of traveling wave solutions for a diffusive competition system of three species[J].Journal of Mathematical Analysis and Applications,2018,459(1): 564-576. [10]CHEN X F. Existence, uniqueness, and asymptotic stability of traveling waves in nonlocal evolution equations[J].Advances in Differential Equations,1997,2(1): 125-160. [11] CHEN G S, WU S L, HSU S H. Stability of traveling wavefronts for a discrete diffusive competition system with three species[J]. Journal of Mathematical Analysis and Applications,2019,474(2): 909-930. [12]FANG J, ZHAO X Q. Bistable traveling wave for monotone semiflows with applications[J].Journal of the European Mathematical Society,2015,17: 2243-2288. [13]GARDNER R A. Existence and stability of travelling wave solutions of competition models: a degree theoretic approach[J].Journal of Differential Equations,1982,44(3): 343-364. [14]GUO J S, WU C C. The existence of traveling wave solutions for a bistable three-component lattice dynamical system[J].Journal of Differential Equations,2016,260(12): 1445-1455. [15]SU T, ZHANG G B. Stability of traveling wavefronts for a three-component Lotka-Volterra competition system on a lattice[J].Electronic Journal of Differential Equations,2018,57: 1-16. [16]VOLPERT A I, VOLPERT V A, VOLPERT V A. Traveling Wave Solutions of Parabolic Systems[M].American Mathematical Society, 1994. [17]WU C C. Existence of traveling wavefront for discrete bistable competition model[J]. Discrete and Continuous Dynamical Systems(Series B),2013,16: 973-984. [18]MA M, YUE J, OU C. Propagation direction of the bistable travelling wavefront for delayed non-local reaction diffusion equations[J].Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences,2019,475(2223): 20180898.

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