Browsing by Author "Lin, Chi-Tien"

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EXPONENTIAL STABILITY OF NONMONOTONE TRAVELING WAVES FOR NICHOLSON'S BLOWFLIES EQUATION
(2014-01-01)
Lin, Chi-Kun
;
Lin, Chi-Tien
;
Lin, Yanping
;
Mei, Ming
;
應用數學系
This paper is concerned with Nicholson's blowflies equation, a kind of time-delayed reaction-diffusion equation. It is known that when the ratio of birth rate coefficient and death rate coefficient satisfies 1 < p/d <= e, the equation is monotone and possesses monotone traveling wavefronts, which have been intensively studied in previous research. However, when p/d > e, the equation losses its monotonicity, and its traveling waves are oscillatory when the time-delay r or the wave speed c is large, which causes the study of stability of these nonmonotone traveling waves to be challenging. In this paper, we use the technical weighted energy method to prove that when e < p/d <= e(2), all noncritical traveling waves phi(x + ct) with c > c(*) > 0 are exponentially stable, where c(*) > 0 is the minimum wave speed. Here, we allow the traveling wave to be either monotone or nonmonotone with any speed c > c(*) and any size of the time-delay r > 0; however, when p/d > e(2) with a small time-delay r < [pi-arctan root ln p/d(ln p/d -2)]/d root ln p/d(ln p/d -2), all noncritical traveling waves phi(x + ct) with c > c(*) > 0 are exponentially stable, too. As a corollary, we also prove the uniqueness of traveling waves in the case of p/d > e(2), which to the best of our knowledge was open. Finally, some numerical simulations are carried out. When e < p/d <= e(2), we demonstrate numerically that after a long time the solution behaves like a monotone traveling wave for a small time-delay, and behaves like an oscillatory traveling wave for a big time-delay. When p/d > e(2), if the time-delay is small, then the solution numerically behaves like a monotone/nonmonotone traveling wave, but if the time-delay is big, then the solution is numerically demonstrated to be chaotically oscillatory but not an oscillatory traveling wave. These either confirm and support our theoretical results or open up some new phenomena for future research.
EXPONENTIAL STABILITY OF NONMONOTONE TRAVELING WAVES FOR NICHOLSON\'S BLOWFLIES EQUATION
(2014-01-01)
Lin, Chi-Kun
;
Lin, Chi-Tien
;
Lin, Yanping
;
Mei, Ming
;
應用數學系
This paper is concerned with Nicholson\'s blowflies equation, a kind of time-delayed reaction-diffusion equation. It is known that when the ratio of birth rate coefficient and death rate coefficient satisfies 1 < p/d <= e, the equation is monotone and possesses monotone traveling wavefronts, which have been intensively studied in previous research. However, when p/d > e, the equation losses its monotonicity, and its traveling waves are oscillatory when the time-delay r or the wave speed c is large, which causes the study of stability of these nonmonotone traveling waves to be challenging. In this paper, we use the technical weighted energy method to prove that when e < p/d <= e(2), all noncritical traveling waves phi(x + ct) with c > c(*) > 0 are exponentially stable, where c(*) > 0 is the minimum wave speed. Here, we allow the traveling wave to be either monotone or nonmonotone with any speed c > c(*) and any size of the time-delay r > 0; however, when p/d > e(2) with a small time-delay r < [pi-arctan root ln p/d(ln p/d -2)]/d root ln p/d(ln p/d -2), all noncritical traveling waves phi(x + ct) with c > c(*) > 0 are exponentially stable, too. As a corollary, we also prove the uniqueness of traveling waves in the case of p/d > e(2), which to the best of our knowledge was open. Finally, some numerical simulations are carried out. When e < p/d <= e(2), we demonstrate numerically that after a long time the solution behaves like a monotone traveling wave for a small time-delay, and behaves like an oscillatory traveling wave for a big time-delay. When p/d > e(2), if the time-delay is small, then the solution numerically behaves like a monotone/nonmonotone traveling wave, but if the time-delay is big, then the solution is numerically demonstrated to be chaotically oscillatory but not an oscillatory traveling wave. These either confirm and support our theoretical results or open up some new phenomena for future research.
Traveling wavefronts for time-delayed reaction-diffusion equation: (I) Local nonlinearity
(2009-07-15)
Mei, Ming
;
Lin, Chi-Kun
;
Lin, Chi-Tien
;
So, Joseph W. -H.
;
應用數學系
In this paper, we study a class of time-delayed reaction-diffusion equation with local nonlinearity for the birth rate. For all wave-fronts with the speed c > c(*), where c(*) > 0 is the critical wave speed, we prove that these wavefronts are asymptotically stable, when the initial perturbation around the traveling waves decays exponentially as x -> -infinity, but the initial perturbation can be arbitrarily large in other locations. This essentially improves the stability results obtained by Mei. So. Li and Shen [M. Mei, J.W.-H. So, M.Y. Li, S.S.R Shen, Asymptotic stability of traveling waves for the Nicholson's blowflies equation with diffusion, Proc. Roy. Soc. Edinburgh Sect. A 134 (2004) 579-594] for the speed c > 2 root D(m) (epsilon p - d(m)) with small initial perturbation and by Lin and Mei [C.-K. Lin, M. Mei, On travelling wavefronts of the Nicholson's blowflies equations with diffusion, submitted for publication] for c > c(*) with sufficiently small delay time r approximate to 0. The approach adopted in this paper is the technical weighted energy method used in [M. Mei, J.W.-H. So, M.Y. Li, S.S.R Shen, Asymptotic stability of traveling waves for the Nicholson's blowflies equation with diffusion, Proc. Roy. Soc. Edinburgh Sect. A 134 (2004) 579-594], but inspired by Gourley [S.A. Gourley, Linear stability of travelling fronts in an age-structured reaction-diffusion Population model, Quart. J. Mech. Appl. Math. 58 (2005) 257-268] and based on the property of the critical wavefronts, the weight function is carefully selected and it plays a key role in proving the stability for any c > c(*) and for an arbitrary time-delay r > 0. (C) 2009 Elsevier Inc. All rights reserved.
Traveling wavefronts for time-delayed reaction-diffusion equation: (II) Nonlocal nonlinearity
(2009-07-15)
Mei, Ming
;
Lin, Chi-Kun
;
Lin, Chi-Tien
;
So, Joseph W. -H.
;
應用數學系
This is the second part of a series of study oil the stability of traveling wavefronts of reaction-diffusion equations with time delays. In this paper we will consider a nonlocal time-delayed reaction-diffusion equation. When the initial perturbation around the traveling wave decays exponentially as x -> -infinity (but the initial perturbation call be arbitrarily large in other locations), we prove the asymptotic stability of all traveling waves for the reaction-diffusion equation, including even the slower waves whose speed are close to the critical speed. This essentially improves the previous stability results by Mei and So [M. Mei, J.W.-H. So, Stability of strong traveling waves for a nonlocal time-delayed reaction-diffusion equation, Proc. Roy. Soc. Edinburgh Sect. A 138 (2008) 551-568] for the speed c > 2 root D(m)(3 epsilon p-2d(m)) with a small initial perturbation. The approach we use here is the weighted energy method, but the weight function is more tricky to construct due to the property of the critical wavefront, and the difficulty arising from the nonlocal nonlinearity is also overcome. Finally, by using the Crank-Nicholson scheme, we present some numerical results which confirm Our theoretical Study. (C) 2009 Elsevier Inc. All rights reserved.