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

Hao Zhang; Hirofumi Izuhara; Yaping Wu

doi:10.3934/dcdsb.2021046

Article Contents

2021, Volume 26, Issue 4: 2323-2342. Doi: 10.3934/dcdsb.2021046

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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
^* Corresponding author: Yaping Wu

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

Received: September 30, 2020

Revised: December 31, 2020

Early access: January 2021

Published: April 2021

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Abstract

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.

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Mathematics Subject Classification: Primary: 35C07, 35B35, Secondary: 35K57, 47A75, 34L16.

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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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