September  2014, 34(9): 3511-3533. doi: 10.3934/dcds.2014.34.3511

Slowly oscillating wavefronts of the KPP-Fisher delayed equation

1. 

Mathematical Institute, Silesian University, 746 01 Opava, Czech Republic

2. 

Instituto de Matemática y Física, Universidad de Talca, Casilla 747, Talca

Received  June 2013 Revised  November 2013 Published  March 2014

This paper concerns the semi-wavefronts (i.e. bounded solutions $u=\phi(x\cdot\nu +ct) >0,$ $ |\nu|=1, $ satisfying $\phi(-\infty)=0$) to the delayed KPP-Fisher equation $u_t(t,x) = \Delta u(t,x) + u(t,x)(1-u(t-\tau,x)), \ u \geq 0,\ x \in \mathbb{R}^m.$ First, we show that the profile $\phi$ of each semi-wavefront should be either monotone or eventually sine-like slowly oscillating around the positive equilibrium. Then a solution to the problem of existence of semi-wavefronts is provided. Next, we prove that the semi-wavefronts are in fact wavefronts (i.e. additionally $\phi(+\infty)=1$) if $c \geq 2$ and $\tau \leq 1$; our proof uses dynamical properties of an auxiliary one-dimensional map with the negative Schwarzian. However, we also show that, for $c \geq 2$ and $\tau \geq 1.87$, each semi-wavefront profile $\phi(t)$ should develop non-decaying oscillations around $1$ as $t \to +\infty$.
Citation: Karel Hasik, Sergei Trofimchuk. Slowly oscillating wavefronts of the KPP-Fisher delayed equation. Discrete & Continuous Dynamical Systems - A, 2014, 34 (9) : 3511-3533. doi: 10.3934/dcds.2014.34.3511
References:
[1]

M. Aguerrea, C. Gomez and S. Trofimchuk, On uniqueness of semi-wavefronts (Diekmann-Kaper theory of a nonlinear convolution equation re-visited),, Math. Ann., 354 (2012), 73.  doi: 10.1007/s00208-011-0722-8.  Google Scholar

[2]

M. Alfaro and J. Coville, Rapid traveling waves in the nonlocal Fisher equation connect two unstable states,, Appl. Math. Lett., 25 (2012), 2095.  doi: 10.1016/j.aml.2012.05.006.  Google Scholar

[3]

P. Ashwin, M. Bartuccelli, T. Bridges and S. Gourley, Traveling fronts for the KPP equation with spatio-temporal delay,, Z. Angew. Math. Phys., 53 (2002), 103.  doi: 10.1007/s00033-002-8145-8.  Google Scholar

[4]

B. Bánhelyi, T. Csendes, T. Krisztin and A. Neumaier, Global attractivity of the zero solution for Wright's equation,, SIAM J. Appl. Dynam. Syst., (2014).   Google Scholar

[5]

H. Berestycki, G. Nadin, B. Perthame and L. Ryzhik, The non-local Fisher-KPP equation: Traveling waves and steady states,, Nonlinearity, 22 (2009), 2813.  doi: 10.1088/0951-7715/22/12/002.  Google Scholar

[6]

H. Berestycki and L. Nirenberg, Traveling waves in cylinders,, Ann. Inst. H. Poincare Anal. Non. Lineaire, 9 (1992), 497.   Google Scholar

[7]

O. Bonnefon, J. Garnier, F. Hamel and L. Roques, Inside dynamics of delayed traveling waves,, Math. Mod. Nat. Phen., 8 (2013), 42.  doi: 10.1051/mmnp/20138305.  Google Scholar

[8]

S. Chen and J. Shi, Stability and Hopf bifurcation in a diffusive logistic population model with nonlocal delay effect,, J. Differential Equations, 253 (2012), 3440.  doi: 10.1016/j.jde.2012.08.031.  Google Scholar

[9]

A. Ducrot and G. Nadin, Asymptotic behaviour of travelling waves for the delayed Fisher-KPP equation,, J. Differential Equations, 256 (2014), 3115.  doi: 10.1016/j.jde.2014.01.033.  Google Scholar

[10]

A. Ivanov, C. Gomez and S. Trofimchuk, A note on the existence of non-monotone non-oscillating wavefronts,, preprint, ().   Google Scholar

[11]

J. Fang and J. Wu, Monotone traveling waves for delayed Lotka-Volterra competition systems,, Discrete Contin. Dynam. Systems, 32 (2012), 3043.  doi: 10.3934/dcds.2012.32.3043.  Google Scholar

[12]

J. Fang and X.-Q. Zhao, Monotone wavefronts of the nonlocal Fisher-KPP equation,, Nonlinearity, 24 (2011), 3043.  doi: 10.1088/0951-7715/24/11/002.  Google Scholar

[13]

T. Faria, W. Huang and J. Wu, Traveling waves for delayed reaction-diffusion equations with global response,, Proc. Roy. Soc. London Ser. A, 462 (2006), 229.  doi: 10.1098/rspa.2005.1554.  Google Scholar

[14]

T. Faria and S. Trofimchuk, Positive traveling fronts for reaction-diffusion systems with distributed delay,, Nonlinearity, 23 (2010), 2457.  doi: 10.1088/0951-7715/23/10/006.  Google Scholar

[15]

A. Gomez and S. Trofimchuk, Global continuation of monotone wavefronts,, J. London Math. Soc., 89 (2014), 47.  doi: 10.1112/jlms/jdt050.  Google Scholar

[16]

A. Gomez and S. Trofimchuk, Monotone traveling wavefronts of the KPP-Fisher delayed equation,, J. Differential Equations, 250 (2011), 1767.  doi: 10.1016/j.jde.2010.11.011.  Google Scholar

[17]

J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations,, Applied Mathematical Sciences, (1993).   Google Scholar

[18]

K. Hasik and S. Trofimchuk, An extension of the Wright's 3/2-theorem for the KPP-Fisher delayed equation,, Proc. Amer. Math. Soc., (2014).   Google Scholar

[19]

T. Krisztin, Global dynamics of delay differential equations,, Period. Math. Hungar., 56 (2008), 83.  doi: 10.1007/s10998-008-5083-x.  Google Scholar

[20]

M. Kwong and C. Ou, Existence and nonexistence of monotone traveling waves for the delayed Fisher equation,, J. Differential Equations, 249 (2010), 728.  doi: 10.1016/j.jde.2010.04.017.  Google Scholar

[21]

E. Liz, M. Pinto, G. Robledo, V. Tkachenko and S. Trofimchuk, Wright type delay differential equations with negative Schwarzian,, Discrete Contin. Dynam. Systems, 9 (2003), 309.  doi: 10.3934/dcds.2003.9.309.  Google Scholar

[22]

E. Liz, V. Tkachenko and S. Trofimchuk, A global stability criterion for scalar functional differential equations,, SIAM J. Math. Anal., 35 (2003), 596.  doi: 10.1137/S0036141001399222.  Google Scholar

[23]

S. Ma, Traveling wavefronts for delayed reaction-diffusion systems via a fixed point theorem,, J. Differential Equations, 171 (2001), 294.  doi: 10.1006/jdeq.2000.3846.  Google Scholar

[24]

S. Ma, Traveling waves for non-local delayed diffusion equations via auxiliary equations,, J. Differential Equations, 237 (2007), 259.  doi: 10.1016/j.jde.2007.03.014.  Google Scholar

[25]

J. Mallet-Paret, The Fredholm alternative for functional differential equations of mixed type,, J. Dynam. Differential Equations, 11 (1999), 1.  doi: 10.1023/A:1021889401235.  Google Scholar

[26]

J. Mallet-Paret and G. Sell, Systems of delay differential equations I: Floquet multipliers and discrete Lyapunov functions,, J. Differential Equations, 125 (1996), 385.  doi: 10.1006/jdeq.1996.0036.  Google Scholar

[27]

J. Mallet-Paret and G. Sell, The Poincare-Bendixson theorem for monotone cyclic feedback systems with delay,, J. Differential Equations, 125 (1996), 441.  doi: 10.1006/jdeq.1996.0037.  Google Scholar

[28]

G. Nadin, B. Perthame and M. Tang, Can a traveling wave connect two unstable states? The case of the nonlocal Fisher equation,, C. R. Acad. Sci. Paris, 349 (2011), 553.  doi: 10.1016/j.crma.2011.03.008.  Google Scholar

[29]

G. Nadin, L. Rossi, L. Ryzhik and B. Perthame, Wave-like solutions for nonlocal reaction-diffusion equations: A toy model,, Math. Mod. Nat. Phen., 8 (2013), 33.  doi: 10.1051/mmnp/20138304.  Google Scholar

[30]

W. Sun and M. Tang, Relaxation method for one dimensional traveling waves of singular and nonlocal equations,, Discrete Contin. Dynam. Systems B, 18 (2013), 1459.  doi: 10.3934/dcdsb.2013.18.1459.  Google Scholar

[31]

E. Trofimchuk, V. Tkachenko and S. Trofimchuk, Slowly oscillating wave solutions of a single species reaction-diffusion equation with delay,, J. Differential Equations, 245 (2008), 2307.  doi: 10.1016/j.jde.2008.06.023.  Google Scholar

[32]

J. Wu and X. Zou, Traveling wave fronts of reaction-diffusion systems with delay,, J. Dynam. Differential Equations, 13 (2001), 651.  doi: 10.1023/A:1016690424892.  Google Scholar

show all references

References:
[1]

M. Aguerrea, C. Gomez and S. Trofimchuk, On uniqueness of semi-wavefronts (Diekmann-Kaper theory of a nonlinear convolution equation re-visited),, Math. Ann., 354 (2012), 73.  doi: 10.1007/s00208-011-0722-8.  Google Scholar

[2]

M. Alfaro and J. Coville, Rapid traveling waves in the nonlocal Fisher equation connect two unstable states,, Appl. Math. Lett., 25 (2012), 2095.  doi: 10.1016/j.aml.2012.05.006.  Google Scholar

[3]

P. Ashwin, M. Bartuccelli, T. Bridges and S. Gourley, Traveling fronts for the KPP equation with spatio-temporal delay,, Z. Angew. Math. Phys., 53 (2002), 103.  doi: 10.1007/s00033-002-8145-8.  Google Scholar

[4]

B. Bánhelyi, T. Csendes, T. Krisztin and A. Neumaier, Global attractivity of the zero solution for Wright's equation,, SIAM J. Appl. Dynam. Syst., (2014).   Google Scholar

[5]

H. Berestycki, G. Nadin, B. Perthame and L. Ryzhik, The non-local Fisher-KPP equation: Traveling waves and steady states,, Nonlinearity, 22 (2009), 2813.  doi: 10.1088/0951-7715/22/12/002.  Google Scholar

[6]

H. Berestycki and L. Nirenberg, Traveling waves in cylinders,, Ann. Inst. H. Poincare Anal. Non. Lineaire, 9 (1992), 497.   Google Scholar

[7]

O. Bonnefon, J. Garnier, F. Hamel and L. Roques, Inside dynamics of delayed traveling waves,, Math. Mod. Nat. Phen., 8 (2013), 42.  doi: 10.1051/mmnp/20138305.  Google Scholar

[8]

S. Chen and J. Shi, Stability and Hopf bifurcation in a diffusive logistic population model with nonlocal delay effect,, J. Differential Equations, 253 (2012), 3440.  doi: 10.1016/j.jde.2012.08.031.  Google Scholar

[9]

A. Ducrot and G. Nadin, Asymptotic behaviour of travelling waves for the delayed Fisher-KPP equation,, J. Differential Equations, 256 (2014), 3115.  doi: 10.1016/j.jde.2014.01.033.  Google Scholar

[10]

A. Ivanov, C. Gomez and S. Trofimchuk, A note on the existence of non-monotone non-oscillating wavefronts,, preprint, ().   Google Scholar

[11]

J. Fang and J. Wu, Monotone traveling waves for delayed Lotka-Volterra competition systems,, Discrete Contin. Dynam. Systems, 32 (2012), 3043.  doi: 10.3934/dcds.2012.32.3043.  Google Scholar

[12]

J. Fang and X.-Q. Zhao, Monotone wavefronts of the nonlocal Fisher-KPP equation,, Nonlinearity, 24 (2011), 3043.  doi: 10.1088/0951-7715/24/11/002.  Google Scholar

[13]

T. Faria, W. Huang and J. Wu, Traveling waves for delayed reaction-diffusion equations with global response,, Proc. Roy. Soc. London Ser. A, 462 (2006), 229.  doi: 10.1098/rspa.2005.1554.  Google Scholar

[14]

T. Faria and S. Trofimchuk, Positive traveling fronts for reaction-diffusion systems with distributed delay,, Nonlinearity, 23 (2010), 2457.  doi: 10.1088/0951-7715/23/10/006.  Google Scholar

[15]

A. Gomez and S. Trofimchuk, Global continuation of monotone wavefronts,, J. London Math. Soc., 89 (2014), 47.  doi: 10.1112/jlms/jdt050.  Google Scholar

[16]

A. Gomez and S. Trofimchuk, Monotone traveling wavefronts of the KPP-Fisher delayed equation,, J. Differential Equations, 250 (2011), 1767.  doi: 10.1016/j.jde.2010.11.011.  Google Scholar

[17]

J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations,, Applied Mathematical Sciences, (1993).   Google Scholar

[18]

K. Hasik and S. Trofimchuk, An extension of the Wright's 3/2-theorem for the KPP-Fisher delayed equation,, Proc. Amer. Math. Soc., (2014).   Google Scholar

[19]

T. Krisztin, Global dynamics of delay differential equations,, Period. Math. Hungar., 56 (2008), 83.  doi: 10.1007/s10998-008-5083-x.  Google Scholar

[20]

M. Kwong and C. Ou, Existence and nonexistence of monotone traveling waves for the delayed Fisher equation,, J. Differential Equations, 249 (2010), 728.  doi: 10.1016/j.jde.2010.04.017.  Google Scholar

[21]

E. Liz, M. Pinto, G. Robledo, V. Tkachenko and S. Trofimchuk, Wright type delay differential equations with negative Schwarzian,, Discrete Contin. Dynam. Systems, 9 (2003), 309.  doi: 10.3934/dcds.2003.9.309.  Google Scholar

[22]

E. Liz, V. Tkachenko and S. Trofimchuk, A global stability criterion for scalar functional differential equations,, SIAM J. Math. Anal., 35 (2003), 596.  doi: 10.1137/S0036141001399222.  Google Scholar

[23]

S. Ma, Traveling wavefronts for delayed reaction-diffusion systems via a fixed point theorem,, J. Differential Equations, 171 (2001), 294.  doi: 10.1006/jdeq.2000.3846.  Google Scholar

[24]

S. Ma, Traveling waves for non-local delayed diffusion equations via auxiliary equations,, J. Differential Equations, 237 (2007), 259.  doi: 10.1016/j.jde.2007.03.014.  Google Scholar

[25]

J. Mallet-Paret, The Fredholm alternative for functional differential equations of mixed type,, J. Dynam. Differential Equations, 11 (1999), 1.  doi: 10.1023/A:1021889401235.  Google Scholar

[26]

J. Mallet-Paret and G. Sell, Systems of delay differential equations I: Floquet multipliers and discrete Lyapunov functions,, J. Differential Equations, 125 (1996), 385.  doi: 10.1006/jdeq.1996.0036.  Google Scholar

[27]

J. Mallet-Paret and G. Sell, The Poincare-Bendixson theorem for monotone cyclic feedback systems with delay,, J. Differential Equations, 125 (1996), 441.  doi: 10.1006/jdeq.1996.0037.  Google Scholar

[28]

G. Nadin, B. Perthame and M. Tang, Can a traveling wave connect two unstable states? The case of the nonlocal Fisher equation,, C. R. Acad. Sci. Paris, 349 (2011), 553.  doi: 10.1016/j.crma.2011.03.008.  Google Scholar

[29]

G. Nadin, L. Rossi, L. Ryzhik and B. Perthame, Wave-like solutions for nonlocal reaction-diffusion equations: A toy model,, Math. Mod. Nat. Phen., 8 (2013), 33.  doi: 10.1051/mmnp/20138304.  Google Scholar

[30]

W. Sun and M. Tang, Relaxation method for one dimensional traveling waves of singular and nonlocal equations,, Discrete Contin. Dynam. Systems B, 18 (2013), 1459.  doi: 10.3934/dcdsb.2013.18.1459.  Google Scholar

[31]

E. Trofimchuk, V. Tkachenko and S. Trofimchuk, Slowly oscillating wave solutions of a single species reaction-diffusion equation with delay,, J. Differential Equations, 245 (2008), 2307.  doi: 10.1016/j.jde.2008.06.023.  Google Scholar

[32]

J. Wu and X. Zou, Traveling wave fronts of reaction-diffusion systems with delay,, J. Dynam. Differential Equations, 13 (2001), 651.  doi: 10.1023/A:1016690424892.  Google Scholar

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