# American Institute of Mathematical Sciences

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 and Continuous Dynamical Systems, 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), 73-109. doi: 10.1007/s00208-011-0722-8. [2] M. Alfaro and J. Coville, Rapid traveling waves in the nonlocal Fisher equation connect two unstable states, Appl. Math. Lett., 25 (2012), 2095-2099. doi: 10.1016/j.aml.2012.05.006. [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), 103-122. doi: 10.1007/s00033-002-8145-8. [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), to appear. [5] H. Berestycki, G. Nadin, B. Perthame and L. Ryzhik, The non-local Fisher-KPP equation: Traveling waves and steady states, Nonlinearity, 22 (2009), 2813-2844. doi: 10.1088/0951-7715/22/12/002. [6] H. Berestycki and L. Nirenberg, Traveling waves in cylinders, Ann. Inst. H. Poincare Anal. Non. Lineaire, 9 (1992), 497-572. [7] O. Bonnefon, J. Garnier, F. Hamel and L. Roques, Inside dynamics of delayed traveling waves, Math. Mod. Nat. Phen., 8 (2013), 42-59. doi: 10.1051/mmnp/20138305. [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), 3440-3470. doi: 10.1016/j.jde.2012.08.031. [9] A. Ducrot and G. Nadin, Asymptotic behaviour of travelling waves for the delayed Fisher-KPP equation, J. Differential Equations, 256 (2014), 3115-3140. doi: 10.1016/j.jde.2014.01.033. [10] A. Ivanov, C. Gomez and S. Trofimchuk, A note on the existence of non-monotone non-oscillating wavefronts,, preprint, (). [11] J. Fang and J. Wu, Monotone traveling waves for delayed Lotka-Volterra competition systems, Discrete Contin. Dynam. Systems, 32 (2012), 3043-3058. doi: 10.3934/dcds.2012.32.3043. [12] J. Fang and X.-Q. Zhao, Monotone wavefronts of the nonlocal Fisher-KPP equation, Nonlinearity, 24 (2011), 3043-3054. doi: 10.1088/0951-7715/24/11/002. [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), 229-261. doi: 10.1098/rspa.2005.1554. [14] T. Faria and S. Trofimchuk, Positive traveling fronts for reaction-diffusion systems with distributed delay, Nonlinearity, 23 (2010), 2457-2481. doi: 10.1088/0951-7715/23/10/006. [15] A. Gomez and S. Trofimchuk, Global continuation of monotone wavefronts, J. London Math. Soc., 89 (2014), 47-68. doi: 10.1112/jlms/jdt050. [16] A. Gomez and S. Trofimchuk, Monotone traveling wavefronts of the KPP-Fisher delayed equation, J. Differential Equations, 250 (2011), 1767-1787. doi: 10.1016/j.jde.2010.11.011. [17] J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations, Applied Mathematical Sciences, Springer-Verlag, 1993. [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), to appear. [19] T. Krisztin, Global dynamics of delay differential equations, Period. Math. Hungar., 56 (2008), 83-95. doi: 10.1007/s10998-008-5083-x. [20] M. Kwong and C. Ou, Existence and nonexistence of monotone traveling waves for the delayed Fisher equation, J. Differential Equations, 249 (2010), 728-745. doi: 10.1016/j.jde.2010.04.017. [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), 309-321. doi: 10.3934/dcds.2003.9.309. [22] E. Liz, V. Tkachenko and S. Trofimchuk, A global stability criterion for scalar functional differential equations, SIAM J. Math. Anal., 35 (2003), 596-622. doi: 10.1137/S0036141001399222. [23] S. Ma, Traveling wavefronts for delayed reaction-diffusion systems via a fixed point theorem, J. Differential Equations, 171 (2001), 294-314. doi: 10.1006/jdeq.2000.3846. [24] S. Ma, Traveling waves for non-local delayed diffusion equations via auxiliary equations, J. Differential Equations, 237 (2007), 259-277. doi: 10.1016/j.jde.2007.03.014. [25] J. Mallet-Paret, The Fredholm alternative for functional differential equations of mixed type, J. Dynam. Differential Equations, 11 (1999), 1-48. doi: 10.1023/A:1021889401235. [26] J. Mallet-Paret and G. Sell, Systems of delay differential equations I: Floquet multipliers and discrete Lyapunov functions, J. Differential Equations, 125 (1996), 385-440. doi: 10.1006/jdeq.1996.0036. [27] J. Mallet-Paret and G. Sell, The Poincare-Bendixson theorem for monotone cyclic feedback systems with delay, J. Differential Equations, 125 (1996), 441-489. doi: 10.1006/jdeq.1996.0037. [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, Ser. I, 349 (2011), 553-557. doi: 10.1016/j.crma.2011.03.008. [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), 33-41. doi: 10.1051/mmnp/20138304. [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), 1459-1491. doi: 10.3934/dcdsb.2013.18.1459. [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), 2307-2332. doi: 10.1016/j.jde.2008.06.023. [32] J. Wu and X. Zou, Traveling wave fronts of reaction-diffusion systems with delay, J. Dynam. Differential Equations, 13 (2001), 651-687. doi: 10.1023/A:1016690424892.

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##### 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), 73-109. doi: 10.1007/s00208-011-0722-8. [2] M. Alfaro and J. Coville, Rapid traveling waves in the nonlocal Fisher equation connect two unstable states, Appl. Math. Lett., 25 (2012), 2095-2099. doi: 10.1016/j.aml.2012.05.006. [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), 103-122. doi: 10.1007/s00033-002-8145-8. [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), to appear. [5] H. Berestycki, G. Nadin, B. Perthame and L. Ryzhik, The non-local Fisher-KPP equation: Traveling waves and steady states, Nonlinearity, 22 (2009), 2813-2844. doi: 10.1088/0951-7715/22/12/002. [6] H. Berestycki and L. Nirenberg, Traveling waves in cylinders, Ann. Inst. H. Poincare Anal. Non. Lineaire, 9 (1992), 497-572. [7] O. Bonnefon, J. Garnier, F. Hamel and L. Roques, Inside dynamics of delayed traveling waves, Math. Mod. Nat. Phen., 8 (2013), 42-59. doi: 10.1051/mmnp/20138305. [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), 3440-3470. doi: 10.1016/j.jde.2012.08.031. [9] A. Ducrot and G. Nadin, Asymptotic behaviour of travelling waves for the delayed Fisher-KPP equation, J. Differential Equations, 256 (2014), 3115-3140. doi: 10.1016/j.jde.2014.01.033. [10] A. Ivanov, C. Gomez and S. Trofimchuk, A note on the existence of non-monotone non-oscillating wavefronts,, preprint, (). [11] J. Fang and J. Wu, Monotone traveling waves for delayed Lotka-Volterra competition systems, Discrete Contin. Dynam. Systems, 32 (2012), 3043-3058. doi: 10.3934/dcds.2012.32.3043. [12] J. Fang and X.-Q. Zhao, Monotone wavefronts of the nonlocal Fisher-KPP equation, Nonlinearity, 24 (2011), 3043-3054. doi: 10.1088/0951-7715/24/11/002. [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), 229-261. doi: 10.1098/rspa.2005.1554. [14] T. Faria and S. Trofimchuk, Positive traveling fronts for reaction-diffusion systems with distributed delay, Nonlinearity, 23 (2010), 2457-2481. doi: 10.1088/0951-7715/23/10/006. [15] A. Gomez and S. Trofimchuk, Global continuation of monotone wavefronts, J. London Math. Soc., 89 (2014), 47-68. doi: 10.1112/jlms/jdt050. [16] A. Gomez and S. Trofimchuk, Monotone traveling wavefronts of the KPP-Fisher delayed equation, J. Differential Equations, 250 (2011), 1767-1787. doi: 10.1016/j.jde.2010.11.011. [17] J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations, Applied Mathematical Sciences, Springer-Verlag, 1993. [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), to appear. [19] T. Krisztin, Global dynamics of delay differential equations, Period. Math. Hungar., 56 (2008), 83-95. doi: 10.1007/s10998-008-5083-x. [20] M. Kwong and C. Ou, Existence and nonexistence of monotone traveling waves for the delayed Fisher equation, J. Differential Equations, 249 (2010), 728-745. doi: 10.1016/j.jde.2010.04.017. [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), 309-321. doi: 10.3934/dcds.2003.9.309. [22] E. Liz, V. Tkachenko and S. Trofimchuk, A global stability criterion for scalar functional differential equations, SIAM J. Math. Anal., 35 (2003), 596-622. doi: 10.1137/S0036141001399222. [23] S. Ma, Traveling wavefronts for delayed reaction-diffusion systems via a fixed point theorem, J. Differential Equations, 171 (2001), 294-314. doi: 10.1006/jdeq.2000.3846. [24] S. Ma, Traveling waves for non-local delayed diffusion equations via auxiliary equations, J. Differential Equations, 237 (2007), 259-277. doi: 10.1016/j.jde.2007.03.014. [25] J. Mallet-Paret, The Fredholm alternative for functional differential equations of mixed type, J. Dynam. Differential Equations, 11 (1999), 1-48. doi: 10.1023/A:1021889401235. [26] J. Mallet-Paret and G. Sell, Systems of delay differential equations I: Floquet multipliers and discrete Lyapunov functions, J. Differential Equations, 125 (1996), 385-440. doi: 10.1006/jdeq.1996.0036. [27] J. Mallet-Paret and G. Sell, The Poincare-Bendixson theorem for monotone cyclic feedback systems with delay, J. Differential Equations, 125 (1996), 441-489. doi: 10.1006/jdeq.1996.0037. [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, Ser. I, 349 (2011), 553-557. doi: 10.1016/j.crma.2011.03.008. [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), 33-41. doi: 10.1051/mmnp/20138304. [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), 1459-1491. doi: 10.3934/dcdsb.2013.18.1459. [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), 2307-2332. doi: 10.1016/j.jde.2008.06.023. [32] J. Wu and X. Zou, Traveling wave fronts of reaction-diffusion systems with delay, J. Dynam. Differential Equations, 13 (2001), 651-687. doi: 10.1023/A:1016690424892.
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