- Previous Article
- DCDS-B Home
- This Issue
-
Next Article
Bifurcation analysis for an in-host Mycobacterium tuberculosis model
April
2021, 26(4): 2323-2342.
doi: 10.3934/dcdsb.2021046
Asymptotic stability of two types of traveling waves for some predator-prey models
| 1. | School of Mathematical Sciences, Capital Normal University, Xisanhuan Beilu 105, Beijing 100048, China |
| 2. | Faculty of Engineering, University of Miyazaki, 1-1, Gakuen Kibanadai-nishi, Miyazaki, 880-2192, Japan |
This paper is concerned with the asymptotic stability of wave fronts and oscillatory waves for some predator-prey models. By spectral analysis and applying Evans function method with some numerical simulations, we show that the two types of waves with noncritical speeds are spectrally stable and nonlinearly exponentially stable in some exponentially weighted spaces.
Mathematics Subject Classification:
Primary: 35C07, 35B35, Secondary: 35K57, 47A75, 34L16.
Citation:
Hao Zhang, Hirofumi Izuhara, Yaping Wu. Asymptotic stability of two types of traveling waves for some predator-prey models. Discrete and Continuous Dynamical Systems - B,
2021, 26
(4)
: 2323-2342.
doi: 10.3934/dcdsb.2021046
![]()
References:
| [1] |
L. Allen and T. J. Bridges,
Numerical exterior algebra and the compound matrix method, Numer. Math., 92 (2002), 197-232.
doi: 10.1007/s002110100365. |
| [2] |
K. Cheng, S. Hsu and S. Lin,
Some results on global stability of a predator-prey system, J. Math. Biol., 12 (1981), 115-126.
doi: 10.1007/BF00275207. |
| [3] |
W. Ding and W. Huang,
Traveling wave solutions for some classes of diffusive predator-prey models, J. Dyn. Diff. Equ., 28 (2016), 1293-1308.
doi: 10.1007/s10884-015-9472-8. |
| [4] |
S. Dunbar,
Travelling wave solutions of diffusive Lotka-Volterra equations, J. Math. Biol., 17 (1983), 11-32.
doi: 10.1007/BF00276112. |
| [5] |
S. Dunbar,
Traveling wave solutions of diffusive Lotka-Volterra equations: a heteroclinic connection in $\mathbb{R}^4$, Trans. Am. Math. Soc., 286 (1984), 557-594.
doi: 10.2307/1999810. |
| [6] |
S. Dunbar,
Traveling waves in diffusive predator-prey equations: Periodic orbits and pointto-periodic heteroclinic orbits, SIAM J. Appl. Math., 46 (1986), 1057-1078.
doi: 10.1137/0146063. |
| [7] |
S. Fu and J. Tsai,
The evolution of traveling waves in a simple isothermal chemical system modeling quadratic autocatalysis with strong decay, J. Differential Equations, 256 (2014), 3335-3364.
doi: 10.1016/j.jde.2014.02.009. |
| [8] |
S. Fu and J. Tsai,
Wave propagation in predator-prey systems, Nonlinearity, 28 (2015), 4389-4423.
doi: 10.1088/0951-7715/28/12/4389. |
| [9] |
R. Gadner and C. K. Jone,
Stability of traveling wave solutions of diffusive predator-prey systems, Trans. Amer. Math. Soc., 327 (1991), 465-524.
doi: 10.1090/S0002-9947-1991-1013331-0. |
| [10] |
P. Hartman, Ordinary Differential Equations, John Wiley & Sons, New York-London-Sydney, 1964. |
| [11] |
C. Hsu, C. Yang, T. Yang and T. Yang,
Existence of traveling wave solutions for diffusive predator-prey type systems, J. Differential Equations, 252 (2012), 3040-3075.
doi: 10.1016/j.jde.2011.11.008. |
| [12] |
S. B. Hsu,
On global stability of a predator-prey system, Math. Biosci., 39 (1978), 1-10.
doi: 10.1016/0025-5564(78)90025-1. |
| [13] |
S. Hsu and T. Hwang,
Global stability for a class of predator-prey systems, SIAM J. Appl. Math., 55 (1995), 763-783.
doi: 10.1137/S0036139993253201. |
| [14] |
J. Huang, G. Lu and S. Ruan,
Existence of traveling wave solutions in a diffusive predator-prey model, J. Math. Biol., 46 (2003), 132-152.
doi: 10.1007/s00285-002-0171-9. |
| [15] |
W. Huang,
Traveling wave solutions for a class of predator-prey systems, J. Dyn. Diff. Equat., 24 (2012), 633-644.
doi: 10.1007/s10884-012-9255-4. |
| [16] |
V. S. Ivlev, Experimental Ecology of the Feeding of Fishes, New Haven, CT: Yale University, 1961. |
| [17] |
W. Li and S. Wu,
Traveling waves in a diffusive predator-prey model with holling type-III functional response, Chaos Solitons Fract., 37 (2008), 476-486.
doi: 10.1016/j.chaos.2006.09.039. |
| [18] |
Y. Li and Y. Wu,
Stability of traveling front solutions with algebraic spatial decay for some autocatalytic chemical reaction systems, SIAM J. Math. Anal., 44 (2012), 1474-1521.
doi: 10.1137/100814974. |
| [19] |
Y. Li and Y. Wu,
Existence and stability of travelling front solutions for general auto-catalytic chemical reaction systems, Math. Model. Nat. Phenom., 8 (2013), 104-132.
doi: 10.1051/mmnp/20138308. |
| [20] |
J. D. Murray, Mathematical Biology. I. An Introduction, 3$^rd$ edition, Springer-Verlag, New York, 2002.
doi: 10.1023/A:1022616217603. |
| [21] |
J. D. Murray, Mathematical Biology. II. Spatial Models and Biomedical Applications, 3$^rd$ edition, Springer-Verlag, New York, 2003. |
| [22] |
B. S. Ng and W. H. Reid,
The compound matrix method for ordinary differential systems, J. Comput. Phys., 58 (1985), 209-228.
doi: 10.1016/0021-9991(85)90177-9. |
| [23] |
A. Okubo and S. A. Levin, Diffusion and Ecological Problems, Modern Prespectives, Springer, Berlin, 2001.
doi: 10.1007/978-1-4757-4978-6. |
| [24] |
B. P. Palka, An Introduction to Complex Function Theory, Springer-Verlag, New York, 1991. |
| [25] |
S. Petrovskii and H. Malchow,
Critical phenomena in plankton communities: KISS model revisited, Nonlinear Analysis, 1 (2000), 37-51.
doi: 10.1016/S0362-546X(99)00392-2. |
| [26] |
S. Ruan and D. Xiao,
Global analysis in a predator-prey system with nonmonotonic functional response, SIAM J. Appl. Math., 61 (2000/01), 1445-1472.
doi: 10.1137/s0036139999361896. |
| [27] |
L. F. Shampine, I. Gladwell and S. Thompson, Solving ODEs with MATLAB, Cambridge University Press, Cambridge, 2003.
doi: 10.1017/CBO9780511615542.
|
| [28] |
A. I. Volpert, V. A. Volpert and V. A. Volpert, Traveling Wave Solutions of Parabolic Systems, American Mathematical Society, Providence, 1994.
doi: 10.1090/mmono/140. |
| [29] |
Y. Wu and N. Yan,
Stability of travelling waves for autocatalytic reaction systems with strong decay, Discrete Continuous Dynam. Systems - B, 22 (2017), 1601-1633.
doi: 10.3934/dcdsb.2017033. |
| [30] |
Y. Wu and X. Zhao,
The existence and stability of travelling waves with transition layers for some singular cross-diffusion systems, Phys. D, 200 (2005), 325-358.
doi: 10.1016/j.physd.2004.11.010. |
| [31] |
Q. Ye, Z. Li, M. Wang and Y. Wu, Introduction to Reaction Diffusion Equation, (in Chinese), second edition, Science Press, Beijing, 2011.
|
show all references
References:
| [1] |
L. Allen and T. J. Bridges,
Numerical exterior algebra and the compound matrix method, Numer. Math., 92 (2002), 197-232.
doi: 10.1007/s002110100365. |
| [2] |
K. Cheng, S. Hsu and S. Lin,
Some results on global stability of a predator-prey system, J. Math. Biol., 12 (1981), 115-126.
doi: 10.1007/BF00275207. |
| [3] |
W. Ding and W. Huang,
Traveling wave solutions for some classes of diffusive predator-prey models, J. Dyn. Diff. Equ., 28 (2016), 1293-1308.
doi: 10.1007/s10884-015-9472-8. |
| [4] |
S. Dunbar,
Travelling wave solutions of diffusive Lotka-Volterra equations, J. Math. Biol., 17 (1983), 11-32.
doi: 10.1007/BF00276112. |
| [5] |
S. Dunbar,
Traveling wave solutions of diffusive Lotka-Volterra equations: a heteroclinic connection in $\mathbb{R}^4$, Trans. Am. Math. Soc., 286 (1984), 557-594.
doi: 10.2307/1999810. |
| [6] |
S. Dunbar,
Traveling waves in diffusive predator-prey equations: Periodic orbits and pointto-periodic heteroclinic orbits, SIAM J. Appl. Math., 46 (1986), 1057-1078.
doi: 10.1137/0146063. |
| [7] |
S. Fu and J. Tsai,
The evolution of traveling waves in a simple isothermal chemical system modeling quadratic autocatalysis with strong decay, J. Differential Equations, 256 (2014), 3335-3364.
doi: 10.1016/j.jde.2014.02.009. |
| [8] |
S. Fu and J. Tsai,
Wave propagation in predator-prey systems, Nonlinearity, 28 (2015), 4389-4423.
doi: 10.1088/0951-7715/28/12/4389. |
| [9] |
R. Gadner and C. K. Jone,
Stability of traveling wave solutions of diffusive predator-prey systems, Trans. Amer. Math. Soc., 327 (1991), 465-524.
doi: 10.1090/S0002-9947-1991-1013331-0. |
| [10] |
P. Hartman, Ordinary Differential Equations, John Wiley & Sons, New York-London-Sydney, 1964. |
| [11] |
C. Hsu, C. Yang, T. Yang and T. Yang,
Existence of traveling wave solutions for diffusive predator-prey type systems, J. Differential Equations, 252 (2012), 3040-3075.
doi: 10.1016/j.jde.2011.11.008. |
| [12] |
S. B. Hsu,
On global stability of a predator-prey system, Math. Biosci., 39 (1978), 1-10.
doi: 10.1016/0025-5564(78)90025-1. |
| [13] |
S. Hsu and T. Hwang,
Global stability for a class of predator-prey systems, SIAM J. Appl. Math., 55 (1995), 763-783.
doi: 10.1137/S0036139993253201. |
| [14] |
J. Huang, G. Lu and S. Ruan,
Existence of traveling wave solutions in a diffusive predator-prey model, J. Math. Biol., 46 (2003), 132-152.
doi: 10.1007/s00285-002-0171-9. |
| [15] |
W. Huang,
Traveling wave solutions for a class of predator-prey systems, J. Dyn. Diff. Equat., 24 (2012), 633-644.
doi: 10.1007/s10884-012-9255-4. |
| [16] |
V. S. Ivlev, Experimental Ecology of the Feeding of Fishes, New Haven, CT: Yale University, 1961. |
| [17] |
W. Li and S. Wu,
Traveling waves in a diffusive predator-prey model with holling type-III functional response, Chaos Solitons Fract., 37 (2008), 476-486.
doi: 10.1016/j.chaos.2006.09.039. |
| [18] |
Y. Li and Y. Wu,
Stability of traveling front solutions with algebraic spatial decay for some autocatalytic chemical reaction systems, SIAM J. Math. Anal., 44 (2012), 1474-1521.
doi: 10.1137/100814974. |
| [19] |
Y. Li and Y. Wu,
Existence and stability of travelling front solutions for general auto-catalytic chemical reaction systems, Math. Model. Nat. Phenom., 8 (2013), 104-132.
doi: 10.1051/mmnp/20138308. |
| [20] |
J. D. Murray, Mathematical Biology. I. An Introduction, 3$^rd$ edition, Springer-Verlag, New York, 2002.
doi: 10.1023/A:1022616217603. |
| [21] |
J. D. Murray, Mathematical Biology. II. Spatial Models and Biomedical Applications, 3$^rd$ edition, Springer-Verlag, New York, 2003. |
| [22] |
B. S. Ng and W. H. Reid,
The compound matrix method for ordinary differential systems, J. Comput. Phys., 58 (1985), 209-228.
doi: 10.1016/0021-9991(85)90177-9. |
| [23] |
A. Okubo and S. A. Levin, Diffusion and Ecological Problems, Modern Prespectives, Springer, Berlin, 2001.
doi: 10.1007/978-1-4757-4978-6. |
| [24] |
B. P. Palka, An Introduction to Complex Function Theory, Springer-Verlag, New York, 1991. |
| [25] |
S. Petrovskii and H. Malchow,
Critical phenomena in plankton communities: KISS model revisited, Nonlinear Analysis, 1 (2000), 37-51.
doi: 10.1016/S0362-546X(99)00392-2. |
| [26] |
S. Ruan and D. Xiao,
Global analysis in a predator-prey system with nonmonotonic functional response, SIAM J. Appl. Math., 61 (2000/01), 1445-1472.
doi: 10.1137/s0036139999361896. |
| [27] |
L. F. Shampine, I. Gladwell and S. Thompson, Solving ODEs with MATLAB, Cambridge University Press, Cambridge, 2003.
doi: 10.1017/CBO9780511615542.
|
| [28] |
A. I. Volpert, V. A. Volpert and V. A. Volpert, Traveling Wave Solutions of Parabolic Systems, American Mathematical Society, Providence, 1994.
doi: 10.1090/mmono/140. |
| [29] |
Y. Wu and N. Yan,
Stability of travelling waves for autocatalytic reaction systems with strong decay, Discrete Continuous Dynam. Systems - B, 22 (2017), 1601-1633.
doi: 10.3934/dcdsb.2017033. |
| [30] |
Y. Wu and X. Zhao,
The existence and stability of travelling waves with transition layers for some singular cross-diffusion systems, Phys. D, 200 (2005), 325-358.
doi: 10.1016/j.physd.2004.11.010. |
| [31] |
Q. Ye, Z. Li, M. Wang and Y. Wu, Introduction to Reaction Diffusion Equation, (in Chinese), second edition, Science Press, Beijing, 2011.
|
Figure 3.
Numerical oscillatory wave profiles
Figure 4.
Oscillatory traveling waves for model HT1
Figure 5.
Oscillatory traveling waves for model HT2
Figure 6.
Unboundedness of solutions $ \widehat{\phi} $ (left) and $ \widehat{\psi} $ (right) of IVP (5.2) for models (4.1) and (4.2)
Figure 7.
$ Arg(D(\Gamma)) $ related to monotone waves for HT1
Figure 9.
$ Arg(D(\Gamma)) $ related to monotone waves for HT2
| [1] |
Ramon Plaza, K. Zumbrun. An Evans function approach to spectral stability of small-amplitude shock profiles. Discrete and Continuous Dynamical Systems, 2004, 10 (4) : 885-924. doi: 10.3934/dcds.2004.10.885 |
| [2] |
Peter Howard, K. Zumbrun. The Evans function and stability criteria for degenerate viscous shock waves. Discrete and Continuous Dynamical Systems, 2004, 10 (4) : 837-855. doi: 10.3934/dcds.2004.10.837 |
| [3] |
Aslihan Demirkaya, Panayotis G. Kevrekidis, Milena Stanislavova, Atanas Stefanov. Spectral stability analysis for standing waves of a perturbed Klein-Gordon equation. Conference Publications, 2015, 2015 (special) : 359-368. doi: 10.3934/proc.2015.0359 |
| [4] |
Hua Chen, Ling-Jun Wang. A perturbation approach for the transverse spectral stability of small periodic traveling waves of the ZK equation. Kinetic and Related Models, 2012, 5 (2) : 261-281. doi: 10.3934/krm.2012.5.261 |
| [5] |
Todd Kapitula, Björn Sandstede. Eigenvalues and resonances using the Evans function. Discrete and Continuous Dynamical Systems, 2004, 10 (4) : 857-869. doi: 10.3934/dcds.2004.10.857 |
| [6] |
Yuri Latushkin, Alim Sukhtayev. The Evans function and the Weyl-Titchmarsh function. Discrete and Continuous Dynamical Systems - S, 2012, 5 (5) : 939-970. doi: 10.3934/dcdss.2012.5.939 |
| [7] |
Rui Huang, Ming Mei, Kaijun Zhang, Qifeng Zhang. Asymptotic stability of non-monotone traveling waves for time-delayed nonlocal dispersion equations. Discrete and Continuous Dynamical Systems, 2016, 36 (3) : 1331-1353. doi: 10.3934/dcds.2016.36.1331 |
| [8] |
Luyi Ma, Hong-Tao Niu, Zhi-Cheng Wang. Global asymptotic stability of traveling waves to the Allen-Cahn equation with a fractional Laplacian. Communications on Pure and Applied Analysis, 2019, 18 (5) : 2457-2472. doi: 10.3934/cpaa.2019111 |
| [9] |
Xiaojie Hou, Wei Feng. Traveling waves and their stability in a coupled reaction diffusion system. Communications on Pure and Applied Analysis, 2011, 10 (1) : 141-160. doi: 10.3934/cpaa.2011.10.141 |
| [10] |
Judith R. Miller, Huihui Zeng. Multidimensional stability of planar traveling waves for an integrodifference model. Discrete and Continuous Dynamical Systems - B, 2013, 18 (3) : 741-751. doi: 10.3934/dcdsb.2013.18.741 |
| [11] |
Wei Feng, Michael Freeze, Xin Lu. On competition models under allee effect: Asymptotic behavior and traveling waves. Communications on Pure and Applied Analysis, 2020, 19 (12) : 5609-5626. doi: 10.3934/cpaa.2020256 |
| [12] |
Zengji Du, Zhaosheng Feng. Existence and asymptotic behaviors of traveling waves of a modified vector-disease model. Communications on Pure and Applied Analysis, 2018, 17 (5) : 1899-1920. doi: 10.3934/cpaa.2018090 |
| [13] |
Dashun Xu, Xiao-Qiang Zhao. Asymptotic speed of spread and traveling waves for a nonlocal epidemic model. Discrete and Continuous Dynamical Systems - B, 2005, 5 (4) : 1043-1056. doi: 10.3934/dcdsb.2005.5.1043 |
| [14] |
Zhenzhen Wang, Tianshou Zhou. Asymptotic behaviors and stochastic traveling waves in stochastic Fisher-KPP equations. Discrete and Continuous Dynamical Systems - B, 2021, 26 (9) : 5023-5045. doi: 10.3934/dcdsb.2020323 |
| [15] |
Björn Sandstede, Arnd Scheel. Evans function and blow-up methods in critical eigenvalue problems. Discrete and Continuous Dynamical Systems, 2004, 10 (4) : 941-964. doi: 10.3934/dcds.2004.10.941 |
| [16] |
Salvador Cruz-García, Catherine García-Reimbert. On the spectral stability of standing waves of the one-dimensional $M^5$-model. Discrete and Continuous Dynamical Systems - B, 2016, 21 (4) : 1079-1099. doi: 10.3934/dcdsb.2016.21.1079 |
| [17] |
Yaping Wu, Niannian Yan. Stability of traveling waves for autocatalytic reaction systems with strong decay. Discrete and Continuous Dynamical Systems - B, 2017, 22 (4) : 1601-1633. doi: 10.3934/dcdsb.2017033 |
| [18] |
Fengxin Chen. Stability and uniqueness of traveling waves for system of nonlocal evolution equations with bistable nonlinearity. Discrete and Continuous Dynamical Systems, 2009, 24 (3) : 659-673. doi: 10.3934/dcds.2009.24.659 |
| [19] |
Grigori Chapiro, Lucas Furtado, Dan Marchesin, Stephen Schecter. Stability of interacting traveling waves in reaction-convection-diffusion systems. Conference Publications, 2015, 2015 (special) : 258-266. doi: 10.3934/proc.2015.0258 |
| [20] |
Je-Chiang Tsai. Global exponential stability of traveling waves in monotone bistable systems. Discrete and Continuous Dynamical Systems, 2008, 21 (2) : 601-623. doi: 10.3934/dcds.2008.21.601 |
2020 Impact Factor: 1.327
Tools
Metrics
Other articles
by authors
[Back to Top]