American Institute of Mathematical Sciences

Advanced
April  2021, 26(4): 2323-2342. doi: 10.3934/dcdsb.2021046

Asymptotic stability of two types of traveling waves for some predator-prey models

Hao Zhang 1, , Hirofumi Izuhara 2, and  Yaping Wu 1,,

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

* Corresponding author: Yaping Wu

Dedicated to Professor Sze-Bi Hsu on the occasion of his 70th birthday

Received  October 2020 Revised  January 2021 Published  April 2021 Early access  February 2021

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.

Keywords: Traveling waves, asymptotic stability, spectral analysis, Evans function.
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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

S. Dunbar, Travelling wave solutions of diffusive Lotka-Volterra equations, J. Math. Biol., 17 (1983), 11-32.  doi: 10.1007/BF00276112.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

P. Hartman, Ordinary Differential Equations, John Wiley & Sons, New York-London-Sydney, 1964.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

V. S. Ivlev, Experimental Ecology of the Feeding of Fishes, New Haven, CT: Yale University, 1961. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

J. D. Murray, Mathematical Biology. I. An Introduction, 3$^rd$ edition, Springer-Verlag, New York, 2002. doi: 10.1023/A:1022616217603.  Google Scholar

[21]

J. D. Murray, Mathematical Biology. II. Spatial Models and Biomedical Applications, 3$^rd$ edition, Springer-Verlag, New York, 2003.  Google Scholar

[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.  Google Scholar

[23]

A. Okubo and S. A. Levin, Diffusion and Ecological Problems, Modern Prespectives, Springer, Berlin, 2001. doi: 10.1007/978-1-4757-4978-6.  Google Scholar

[24]

B. P. Palka, An Introduction to Complex Function Theory, Springer-Verlag, New York, 1991.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27] L. F. Shampine, I. Gladwell and S. Thompson, Solving ODEs with MATLAB, Cambridge University Press, Cambridge, 2003.  doi: 10.1017/CBO9780511615542.  Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[31] Q. Ye, Z. Li, M. Wang and Y. Wu, Introduction to Reaction Diffusion Equation, (in Chinese), second edition, Science Press, Beijing, 2011.   Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

S. Dunbar, Travelling wave solutions of diffusive Lotka-Volterra equations, J. Math. Biol., 17 (1983), 11-32.  doi: 10.1007/BF00276112.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

P. Hartman, Ordinary Differential Equations, John Wiley & Sons, New York-London-Sydney, 1964.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

V. S. Ivlev, Experimental Ecology of the Feeding of Fishes, New Haven, CT: Yale University, 1961. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

J. D. Murray, Mathematical Biology. I. An Introduction, 3$^rd$ edition, Springer-Verlag, New York, 2002. doi: 10.1023/A:1022616217603.  Google Scholar

[21]

J. D. Murray, Mathematical Biology. II. Spatial Models and Biomedical Applications, 3$^rd$ edition, Springer-Verlag, New York, 2003.  Google Scholar

[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.  Google Scholar

[23]

A. Okubo and S. A. Levin, Diffusion and Ecological Problems, Modern Prespectives, Springer, Berlin, 2001. doi: 10.1007/978-1-4757-4978-6.  Google Scholar

[24]

B. P. Palka, An Introduction to Complex Function Theory, Springer-Verlag, New York, 1991.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27] L. F. Shampine, I. Gladwell and S. Thompson, Solving ODEs with MATLAB, Cambridge University Press, Cambridge, 2003.  doi: 10.1017/CBO9780511615542.  Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[31] Q. Ye, Z. Li, M. Wang and Y. Wu, Introduction to Reaction Diffusion Equation, (in Chinese), second edition, Science Press, Beijing, 2011.   Google Scholar
Figure 1.  The parameters $ (\beta,h) \subseteq [0,1]\times[0,1] $ with different $ \gamma $
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 2.   
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 3.  Numerical oscillatory wave profiles
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 4.  Oscillatory traveling waves for model HT1
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 5.  Oscillatory traveling waves for model HT2
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 6.  Unboundedness of solutions $ \widehat{\phi} $ (left) and $ \widehat{\psi} $ (right) of IVP (5.2) for models (4.1) and (4.2)
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 7.  $ Arg(D(\Gamma)) $ related to monotone waves for HT1
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 8.  $ D(\Gamma) $ and $ Arg(D(\Gamma)) $ related to oscillatory waves for HT1
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 9.  $ Arg(D(\Gamma)) $ related to monotone waves for HT2
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 10.  $ D(\Gamma) $ and $ Arg(D(\Gamma)) $ related to oscillatory waves for HT2
Figure Options

Download full-size image

Download as PowerPoint slide

[1]

Ramon Plaza, K. Zumbrun. An Evans function approach to spectral stability of small-amplitude shock profiles. Discrete & 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 & 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 & 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 & Continuous Dynamical Systems, 2004, 10 (4) : 857-869. doi: 10.3934/dcds.2004.10.857
[6]

Rui Huang, Ming Mei, Kaijun Zhang, Qifeng Zhang. Asymptotic stability of non-monotone traveling waves for time-delayed nonlocal dispersion equations. Discrete & Continuous Dynamical Systems, 2016, 36 (3) : 1331-1353. doi: 10.3934/dcds.2016.36.1331
[7]

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 & Applied Analysis, 2019, 18 (5) : 2457-2472. doi: 10.3934/cpaa.2019111
[8]

Yuri Latushkin, Alim Sukhtayev. The Evans function and the Weyl-Titchmarsh function. Discrete & Continuous Dynamical Systems - S, 2012, 5 (5) : 939-970. doi: 10.3934/dcdss.2012.5.939
[9]

Xiaojie Hou, Wei Feng. Traveling waves and their stability in a coupled reaction diffusion system. Communications on Pure & 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 & 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 & 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 & 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 & 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 & Continuous Dynamical Systems - B, 2021, 26 (9) : 5023-5045. doi: 10.3934/dcdsb.2020323
[15]

Salvador Cruz-García, Catherine García-Reimbert. On the spectral stability of standing waves of the one-dimensional $M^5$-model. Discrete & Continuous Dynamical Systems - B, 2016, 21 (4) : 1079-1099. doi: 10.3934/dcdsb.2016.21.1079
[16]

Björn Sandstede, Arnd Scheel. Evans function and blow-up methods in critical eigenvalue problems. Discrete & Continuous Dynamical Systems, 2004, 10 (4) : 941-964. doi: 10.3934/dcds.2004.10.941
[17]

Yaping Wu, Niannian Yan. Stability of traveling waves for autocatalytic reaction systems with strong decay. Discrete & 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 & 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 & Continuous Dynamical Systems, 2008, 21 (2) : 601-623. doi: 10.3934/dcds.2008.21.601

2020 Impact Factor: 1.327

Tools

Metrics

  • PDF downloads (204)
  • HTML views (68)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences | Privacy Policy