Stability of Traveling Wave Fronts for Delayed Lotka-Volterra Competition Systems With Stage Structures
-
摘要: 主要研究了一类具有年龄结构的Lotka-Volterra竞争系统行波解的稳定性.在拟单调的情形下, 利用解析半群理论和抽象泛函微分方程理论,首先建立起系统初值问题的解在R上的存在性和比较原理.然后基于加权能量法、比较原理和嵌入定理, 建立起该系统在大初始扰动(即除去当x→-∞时在行波解附近的初始扰动是指数衰减的, 在其他位置的初始扰动可以任意大)下, 单稳大波速行波解的全局指数稳定性.研究结果表明, 行波解作为系统的稳态解, 通常决定着初值问题解的长时间渐近行为.其稳定性揭示了种间竞争的现象和结果能够被清晰地被观测到, 而不受外界因素的干扰.
-
关键词:
- Lotka-Volterra竞争模型 /
- 年龄结构 /
- 行波解 /
- 稳定性
Abstract: The stability of traveling wave solutions to a class of Lotka-Volterra competitive systems with age structures was studied. In the case of quasi-monotonicity, the existence and comparison theorems for the solutions to the initial value problems of the systems were first established on R with the analytic semigroup theory and the abstract functional differential equations. Then based on the weighted energy method, the comparison theorem as well as the embedding theorem, the global exponential stability of the monostable large-speed traveling wave solutions under the so-called large initial perturbation (i.e. the initial perturbation around the traveling wave decaying exponentially as x→-∞,but being arbitrarily large at other locations) was obtained for the systems in the weighted Sobolev space. The results show that, as the steady state solution of the system, the traveling wave solution usually determines the long-term asymptotic behavior of the solution to the initial value problem. Its stability reveals that the phenomena and results of inter-species competition systems can be clearly observed without interference by external factors.-
Key words:
- Lotka-Volterra competition model /
- stage structure /
- traveling wave solution /
- stability
-
[1] HOSONO Y. Singular perturbation analysis of traveling waves for diffusive Lotka-Volterra competitive models[J]. Numerical and Applied Mathematics, Part Ⅱ,1989: 687-692. [2] HOSONO Y. The minimal speed of traveling fronts for a diffusion Lotka-Volterra competition model[J]. Bulletin of Mathematical Biology,1998,60(3): 435-448. [3] KAN-ON Y. Fisher wave fronts for the Lotka-Volterra competition model with diffusion[J]. Nonlinear Analysis Theory Methods & Applications,1997,28(1): 145-164. [4] KAN-ON Y, FANG Q. Stability of monotone traveling waves for competition-diffusion equations[J]. Japan Journal of Industrial and Applied Mathematics,1996,13(2): 343-349. [5] VOLPERT A I, VOLPERT V A, VOLPERT V A. Traveling Wave Solutions of Parabolic Systems [M]. Providence: American Mathematical Society, 1994. [6] AL-OMARI J F M, GOURLEY S A. Stabililty and traveling fronts in Lotka-Volterra competition models with stage structure[J]. SIAM Journal on Applied Mathematics,2003,63(6): 2063-2086. [7] MEI M, OU C, ZHAO X Q. Global stability of monostable traveling waves for nonlocal time-delayed reaction-diffusion equations[J]. SIAM Journal on Mathematical Analysis,2010,42(6): 2762-2790. [8] HUANG R, MEI M, WANG Y. Planar traveling waves for nonlocal dispersion equation with monostable nonlinearity[J]. Discrete and Continuous Dynamical Systems,2012,32(10): 3621-3649. [9] LI B, ZHANG L. Travelling wave solutions in delayed cooperative systems[J]. Nonlinearity,2011,24(6): 1759-1776. [10] ZHANG L, LI B, SHANG J. Stability and travelling waves for a time-delayed population system with stage structure[J]. Nonlinear Analysis Real World Applications,2012,13(3): 1429-1440. [11] LIN C K, LIN C T, LIN Y P, et al. Exponential stability of nonmonotone traveling waves for Nicholson’s blowflies equation[J]. SIAM Journal on Mathematical Analysis,2014,46(2): 1053-1084. [12] LEUNG A W, HOU X, LI Y. Exclusive traveling waves for competitive reaction-diffusion systems and their stabilities[J]. Journal of Mathematical Analysis and Applications,2008,338(2): 902-924. [13] LIN G, LI W T. Bistable wavefronts in a diffusive and competitive Lotka-Volterra type system with nonlocal delays[J]. Journal of Differential Equations,2008,244(3): 487-513. [14] 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. [15] GARDNER R A. Existence and stability of traveling wave solutions of competition models: a degree theorem approach[J]. Journal of Differential Equations,1982,44(3): 343-364. [16] WU S L, LI W T. Global asymptotic stability of bistable traveling fronts in reaction-diffusion systems and their applications to biological models[J]. Chaos Solitons and Fractals,2009,40(3): 1229-1239. [17] L G Y, WANG M X. Nonlinear stability of traveling wave fronts for delayed reaction diffusion systems[J].Journal of Mathematical Analysis and Applications,2012,13(4): 1854-1865. [18] MA Z H, WU X, YUAN R. Nonlinear stability of traveling wavefronts for competitive-cooperative Lotka-Volterra systems of three species[J]. Applied Mathematics and Computation,2017,315: 331-346. [19] TIAN G, ZHANG G B. Stability of traveling wavefronts for a discrete diffusive Lotka-Volterra competition system[J]. Journal of Mathematical Analysis and Applications,2017,447(1): 222-242. [20] ZHAO G Y, RUAN S G. Existence, uniqueness and asymptotic stability of time periodic traveling waves for a periodic Lotka-Volterra competition system with diffusion[J]. Journal De Mathematiques Pures et Appliquees,2011,95(6): 627-671. [21] BAO X X, WANG Z C. Existence and stability of time periodic traveling waves for a periodic bistable Lotka-Volterra competition system[J]. Journal of Differential Equations,2013,255(8): 2402-2435. [22] SHENG W J. Stability of planar traveling fronts in bistable reaction-diffusion systems[J]. Nonlinear Analysis,2017,156: 42-60. [23] WANG X H. Stability of planar waves in a Lotka-Volterra system[J]. Applied Mathematics and Computation,2015,259(C): 313-326. [24] LIANG X, ZHAO X Q. Asymptotic speeds of spread and traveling waves for monotone semiflows with applications[J]. Communications on Pure and Applied Mathematics,2007,61(1): 1-40. [25] GUO J S, WU C H. Traveling wave front for a two-component lattice dynamical system arising in competition models[J]. Journal of Differential Equations,2012,252(8): 4357-4391. [26] MARTIN R H, SMITH H L. Abstract functional-differential equations and reaction-diffusion systems[J]. Transactions of the American Mathematical Society,1990,321(1): 1-44.
点击查看大图
计量
- 文章访问数: 1036
- HTML全文浏览量: 115
- PDF下载量: 465
- 被引次数: 0