|
|
Pure Mathematics 2025
一类具有时空时滞的“食物有限”种群模型的波前解的局部稳定性
|
Abstract:
本文主要研究了一类具有时空时滞的“食物有限”种群模型的波前解的稳定性。利用线性化算子的谱分析，证明了波前解在有界一致连续函数空间中是不稳定的。但是，当波前解的初始扰动属于某个指数加权函数空间时，波前解是渐近稳定的。
This paper is concerned with the stability of traveling wavefronts for a “food-limited” population model with spatio-temporal delay. By the spectral analysis of the linearized operators, we show that the traveling wavefronts are essentially unstable in the space of bounded and uniformly continuous functions. However, if the initial perturbations of the traveling wavefronts belong to certain exponential weighted spaces, then we prove that the traveling wavefronts are asymptotically stable in the exponential weighted spaces.
| [1] | Britton, N.F. (1990) Spatial Structures and Periodic Travelling Waves in an Integro-Differential Reaction-Diffusion Population Model. SIAM Journal on Applied Mathematics, 50, 1663-1688. https://doi.org/10.1137/0150099 |
| [2] | Smith, H.L. and Thieme, H.R. (1991) Strongly Order Preserving Semiflows Generated by Functional Differential Equations. Journal of Differential Equations, 93, 332-363. https://doi.org/10.1016/0022-0396(91)90016-3 |
| [3] | Gourley, S.A. and Chaplain, M.A.J. (2002) Travelling Fronts in a Food-Limited Population Model with Time Delay. Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 132, 75-89. https://doi.org/10.1017/s0308210500001530 |
| [4] | Smith, F.E. (1963) Population Dynamics in Daphnia Magna and a New Model for Population Growth. Ecology, 44, 651-663. https://doi.org/10.2307/1933011 |
| [5] | Pielou, E.C. (1969) An Introduction to Mathematical Ecology. Wiley. |
| [6] | Hallam, T.G. and de Luna, J.T. (1984) Effects of Toxicants on Populations: A Qualitative. Journal of Theoretical Biology, 109, 411-429. https://doi.org/10.1016/s0022-5193(84)80090-9 |
| [7] | Gopalsamy, K., Kulenović, M.R.S. and Ladas, G. (1988) Time Lags in a “Food-Limited” Population. Applicable Analysis, 31, 225-237. https://doi.org/10.1080/00036818808839826 |
| [8] | Gopalsamy, K., Kulenović, M.R.S. and Ladas, G. (1990) Environmental Periodicity and Time Delays in a “Food-Limited” Population Model. Journal of Mathematical Analysis and Applications, 147, 545-555. https://doi.org/10.1016/0022-247x(90)90369-q |
| [9] | So, J.W. and Yu, J.S. (1995) On the Uniform Stability for a ‘Food-Limited’ Population Model with Time Delay. Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 125, 991-1002. https://doi.org/10.1017/s0308210500022605 |
| [10] | Gourley, S.A. (2001) Wave Front Solutions of a Diffusive Delay Model for Populations of Daphnia Magna. Computers & Mathematics with Applications, 42, 1421-1430. https://doi.org/10.1016/s0898-1221(01)00251-6 |
| [11] | Feng, W. and Lu, X. (1999) On Diffusive Population Models with Toxicants and Time Delays. Journal of Mathematical Analysis and Applications, 233, 373-386. https://doi.org/10.1006/jmaa.1999.6332 |
| [12] | Feng, W. and Lu, X. (2003) Global Periodicity in a Class of Reaction-Diffusion Systems with Time Delays. Discrete & Continuous Dynamical Systems—B, 3, 69-78. https://doi.org/10.3934/dcdsb.2003.3.69 |
| [13] | Wang, Z. and Li, W. (2007) Monotone Travelling Fronts of a Food-Limited Population Model with Nonlocal Delay. Nonlinear Analysis: Real World Applications, 8, 699-712. https://doi.org/10.1016/j.nonrwa.2006.03.001 |
| [14] | 叶其孝, 李正元, 王明新, 吴雅萍. 反应扩散方程引论[M]. 北京: 科学出版社, 2011. |
| [15] | Wu, J. and Zou, X. (2001) Traveling Wave Fronts of Reaction-Diffusion Systems with Delay. Journal of Dynamics and Differential Equations, 13, 651-687. |
| [16] | Wei, J., Tian, L., Zhou, J., Zhen, Z. and Xu, J. (2017) Existence and Asymptotic Behavior of Traveling Wave Fronts for a Food-Limited Population Model with Spatio-Temporal Delay. Japan Journal of Industrial and Applied Mathematics, 34, 305-320. https://doi.org/10.1007/s13160-017-0244-1 |
| [17] | Fenichel, N. (1979) Geometric Singular Perturbation Theory for Ordinary Differential Equations. Journal of Differential Equations, 31, 53-98. https://doi.org/10.1016/0022-0396(79)90152-9 |
| [18] | 梁飞, 高洪俊. 非局部时滞反应扩散方程行波解的存在性[J]. 数学物理学报, 2011, 31(5): 135-143. |
| [19] | Wang, Z., Li, W. and Ruan, S. (2006) Travelling Wave Fronts in Reaction-Diffusion Systems with Spatio-Temporal Delays. Journal of Differential Equations, 222, 185-232. https://doi.org/10.1016/j.jde.2005.08.010 |
| [20] | 吴事良. 非局部时滯反应扩散方程的行波解和渐近传播速度[D]: [博士学位论文]. 西安: 西安电子科技大学, 2009. |
| [21] | 张建明, 彭亚红. 具有非局部反应的时滞扩散Nicholoson方程的行波解[J]. 数学年刊, 2006, 27(6): 771-778. |
| [22] | Alhasanat, A. and Ou, C. (2019) Stability of Traveling Waves to the Lotka‐Volterra Competition Model. Complexity, 2019, Article ID: 6569520. https://doi.org/10.1155/2019/6569520 |
| [23] | Chang, C. (2018) The Stability of Traveling Wave Solutions for a Diffusive Competition System of Three Species. Journal of Mathematical Analysis and Applications, 459, 564-576. https://doi.org/10.1016/j.jmaa.2017.10.013 |
| [24] | Chang, C., Hsu, C. and Yang, T. (2020) Traveling Wavefronts for a Lotka-Volterra Competition Model with Partially Nonlocal Interactions. Zeitschrift für angewandte Mathematik und Physik, 71, Article No. 70. https://doi.org/10.1007/s00033-020-1289-6 |
| [25] | Chen, X. (1997) Existence, Uniqueness, and Asymptotic Stability of Traveling Waves in Nonlocal Evolution Equations. Advances in Differential Equations, 2, 125-160. https://doi.org/10.57262/ade/1366809230 |
| [26] | Mei, M., So, J.W.-., Li, M.Y. and Shen, S.S.P. (2004) Asymptotic Stability of Travelling Waves for Nicholson’s Blowflies Equation with Diffusion. Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 134, 579-594. https://doi.org/10.1017/s0308210500003358 |
| [27] | Mei, M., Lin, C., Lin, C. and So, J.W.-. (2009) Traveling Wavefronts for Time-Delayed Reaction-Diffusion Equation: (II) Nonlocal Nonlinearity. Journal of Differential Equations, 247, 511-529. https://doi.org/10.1016/j.jde.2008.12.020 |
| [28] | Mei, M., Zhang, K.J. and Zhang, Q.F. (2019) Global Stability of Traveling Waves with Oscillations for Nicholsons Blowflies Equation. International Journal of Numerical Analysis and Modeling, 16, 375-397. |
| [29] | Sattinger, D.H. (1976) On the Stability of Waves of Nonlinear Parabolic Systems. Advances in Mathematics, 22, 312-355. https://doi.org/10.1016/0001-8708(76)90098-0 |
| [30] | Smith, H.L. and Zhao, X. (2000) Global Asymptotic Stability of Traveling Waves in Delayed Reaction-Diffusion Equations. SIAM Journal on Mathematical Analysis, 31, 514-534. https://doi.org/10.1137/s0036141098346785 |
| [31] | Wu, S., Zhao, H. and Liu, S. (2010) Asymptotic Stability of Traveling Waves for Delayed Reaction-Diffusion Equations with Crossing-monostability. Zeitschrift für angewandte Mathematik und Physik, 62, 377-397. https://doi.org/10.1007/s00033-010-0112-1 |
| [32] | Zhang, G. (2019) Global Stability of Non-Monotone Traveling Wave Solutions for a Nonlocal Dispersal Equation with Time Delay. Journal of Mathematical Analysis and Applications, 475, 605-627. https://doi.org/10.1016/j.jmaa.2019.02.058 |
| [33] | Lv, G. and Wang, M. (2010) Existence, Uniqueness and Asymptotic Behavior of Traveling Wave Fronts for a Vector Disease Model. Nonlinear Analysis: Real World Applications, 11, 2035-2043. https://doi.org/10.1016/j.nonrwa.2009.05.006 |
| [34] | Henry, D. (1981) Geometric Theory of Semilinear Parabolic Equations. Lecture Notes in Mathematics, vol. 840. Springer. |
| [35] | Volpert, A.I., Volpert, V.A. and Volpert, V.A. (1994) Traveling Wave Solutions of Parabolic Systems. Translations of Mathematical Monographs, Vol. 140. American Mathematical Society. |
