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

* 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.

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. ChengS. 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. HsuC. YangT. 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. HuangG. 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. ShampineI. 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. YeZ. LiM. 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. ChengS. 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. HsuC. YangT. 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. HuangG. 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. ShampineI. 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. YeZ. LiM. Wang and Y. Wu, Introduction to Reaction Diffusion Equation, (in Chinese), second edition, Science Press, Beijing, 2011. 
Figure 1.  The parameters $ (\beta,h) \subseteq [0,1]\times[0,1] $ with different $ \gamma $
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 8.  $ D(\Gamma) $ and $ Arg(D(\Gamma)) $ related to oscillatory waves for HT1
Figure 9.  $ Arg(D(\Gamma)) $ related to monotone waves for HT2
Figure 10.  $ D(\Gamma) $ and $ Arg(D(\Gamma)) $ related to oscillatory waves for HT2
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