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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  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 $
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Figure 2.   
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Figure 3.  Numerical oscillatory wave profiles
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Figure 4.  Oscillatory traveling waves for model HT1
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Figure 5.  Oscillatory traveling waves for model HT2
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Figure 6.  Unboundedness of solutions $ \widehat{\phi} $ (left) and $ \widehat{\psi} $ (right) of IVP (5.2) for models (4.1) and (4.2)
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Figure 7.  $ Arg(D(\Gamma)) $ related to monotone waves for HT1
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Figure 8.  $ D(\Gamma) $ and $ Arg(D(\Gamma)) $ related to oscillatory waves for HT1
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Figure 9.  $ Arg(D(\Gamma)) $ related to monotone waves for HT2
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Figure 10.  $ D(\Gamma) $ and $ Arg(D(\Gamma)) $ related to oscillatory waves for HT2
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