December  2021, 41(12): 5979-6000. doi: 10.3934/dcds.2021103

On pushed wavefronts of monostable equation with unimodal delayed reaction

1. 

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

2. 

Instituto de Matemática, Universidad de Talca, Casilla 747, Talca, Chile

* Corresponding author: Sergei Trofimchuk

Received  October 2020 Published  December 2021 Early access  June 2021

We study the Mackey-Glass type monostable delayed reaction-diffusion equation with a unimodal birth function $ g(u) $. This model, designed to describe evolution of single species populations, is considered here in the presence of the weak Allee effect ($ g(u_0)>g'(0)u_0 $ for some $ u_0>0 $). We focus our attention on the existence of slow monotonic traveling fronts to the equation: under given assumptions, this problem seems to be rather difficult since the usual positivity and monotonicity arguments are not effective. First, we solve the front existence problem for small delays, $ h \in [0,h_p] $, where $ h_p $, given by an explicit formula, is optimal in a certain sense. Then we take a representative piece-wise linear unimodal birth function which makes possible explicit computation of traveling fronts. In this case, we find out that a) increase of delay can destroy asymptotically stable pushed fronts; b) the set of all admissible wavefront speeds has usual structure of a semi-infinite interval $ [c_*, +\infty) $; c) for each $ h\geq 0 $, the pushed wavefront is unique (if it exists); d) pushed wave can oscillate slowly around the positive equilibrium for sufficiently large delays.

Citation: Karel Hasík, Jana Kopfová, Petra Nábělková, Sergei Trofimchuk. On pushed wavefronts of monostable equation with unimodal delayed reaction. Discrete & Continuous Dynamical Systems, 2021, 41 (12) : 5979-6000. doi: 10.3934/dcds.2021103
References:
[1]

M. AguerreaC. 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.  Google Scholar

[2]

S. Ai, Traveling wave fronts for generalized Fisher equations with spatio-temporal delays, J. Differential Equations, 232 (2007), 104-133.  doi: 10.1016/j.jde.2006.08.015.  Google Scholar

[3]

M. AlfaroA. Ducrot and T. Giletti, Travelling waves for a non- monotone bistable equation with delay: Existence and oscillations, Proc. Lond. Math. Soc., 116 (2018), 729-759.  doi: 10.1112/plms.12092.  Google Scholar

[4]

M. Bani-YaghoubG. YaoM. Fujiwara and D. E. Amundsen, Understanding the interplay between density dependent birth function and maturation time delay using a reaction-diffusion population model, Ecological Complexity, 21 (2015), 14-26.  doi: 10.1016/j.ecocom.2014.10.007.  Google Scholar

[5]

R. D. Benguria and M. C. Depassier, Variational characterization of the speed of propagation of fronts for the nonlinear diffusion equation,, Comm. Math. Phys., 175 (1996), 221-227.  doi: 10.1007/BF02101631.  Google Scholar

[6]

P. Erm and B. L. Phillips, Evolution transforms pushed waves into pulled waves, The American Naturalist, 195 (2020), E87–E99. doi: 10.1086/707324.  Google Scholar

[7]

B. H. Gilding and R. Kersner, Travelling Waves in Nonlinear Diffusion-Convection Reaction, Birkhäuser, 2004. doi: 10.1007/978-3-0348-7964-4.  Google Scholar

[8]

A. Gomez and S. Trofimchuk, Global continuation of monotone wavefronts, J. London Math. Soc., 89 (2014), 47-68.  doi: 10.1112/jlms/jdt050.  Google Scholar

[9]

K. P. Hadeler, Topics in Mathematical Biology, Lecture Notes on Mathematical Modelling in the Life Sciences, Springer, 2017. doi: 10.1007/978-3-319-65621-2.  Google Scholar

[10]

J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations, Applied Mathematical Sciences, Springer-Verlag, 1993. doi: 10.1007/978-1-4612-4342-7.  Google Scholar

[11]

X. Liang and X.-Q. Zhao, Spreading speeds and traveling waves for abstract monostable evolution systems,, J. Functional Anal., 259 (2010), 857-903.  doi: 10.1016/j.jfa.2010.04.018.  Google Scholar

[12]

J. Mallet-Paret,, The Fredholm alternative for functional differential equations of mixed type, J. Dynam. Diff. Eqns. 11 (1999), 1–47. doi: 10.1023/A:1021889401235.  Google Scholar

[13]

G. NadinL. RossiL. Ryzhik and B. Perthame, Wave-like solutions for nonlocal reaction-diffusion equations: A toy model, Math. Model. Nat. Phenom., 8 (2013), 33-41.  doi: 10.1051/mmnp/20138304.  Google Scholar

[14]

C. Ou and J. Wu, Persistence of wavefronts in delayed nonlocal reaction-diffusion equations, J. Differential Equations, 235 (2007), 219–261. doi: 10.1016/j.jde.2006.12.010.  Google Scholar

[15]

W. van Saarloos, Front propagation into unstable states, Physics Reports, 386 (2003), 29–222. Google Scholar

[16]

E. Trofimchuk, M. Pinto and S. Trofimchuk, Monotone waves for non-monotone and non-local monostable reaction-diffusion equations,, J. Differential Equations, 261 (2016), 1203–1236. doi: 10.1016/j.jde.2016.03.039.  Google Scholar

[17]

E. Trofimchuk, M. Pinto and S. Trofimchuk, Pushed traveling fronts in monostable equations with monotone delayed reaction,, Discrete Contin. Dyn. Syst., 33 (2013) 2169–2187. doi: 10.3934/dcds.2013.33.2169.  Google Scholar

[18]

E. Trofimchuk and S. Trofimchuk, Admissible wavefronts speeds for a single species reaction-diffusion equation with delay,, Discrete Contin. Dyn. Syst., 20 (2008), 407-423.  doi: 10.3934/dcds.2008.20.407.  Google Scholar

[19]

E. TrofimchukV. 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.  Google Scholar

[20]

S. Trofimchuk and V. Volpert, Global continuation of monotone waves for bistable delayed equations with unimodal nonlinearities, Nonlinearity, 32 (2019), 2593-2632.  doi: 10.1088/1361-6544/ab0e23.  Google Scholar

[21]

S.-L. WuT.-C. Niu and C.-H. Hsu, Global asymptotic stability of pushed traveling fronts for monostable delayed reaction-diffusion equations, Discrete Contin. Dyn. Syst., 37 (2017), 3467-3486.  doi: 10.3934/dcds.2017147.  Google Scholar

[22]

J. Wu and X. Zou, Traveling wave fronts of reaction-diffusion systems with delay,, J. Dynam. Diff. Eqns., 13 (2001), 651-687.  doi: 10.1023/A:1016690424892.  Google Scholar

[23]

J. Xin, Front propagation in heterogeneous media,, SIAM Review, 42 (2000), 161-230.  doi: 10.1137/S0036144599364296.  Google Scholar

show all references

References:
[1]

M. AguerreaC. 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.  Google Scholar

[2]

S. Ai, Traveling wave fronts for generalized Fisher equations with spatio-temporal delays, J. Differential Equations, 232 (2007), 104-133.  doi: 10.1016/j.jde.2006.08.015.  Google Scholar

[3]

M. AlfaroA. Ducrot and T. Giletti, Travelling waves for a non- monotone bistable equation with delay: Existence and oscillations, Proc. Lond. Math. Soc., 116 (2018), 729-759.  doi: 10.1112/plms.12092.  Google Scholar

[4]

M. Bani-YaghoubG. YaoM. Fujiwara and D. E. Amundsen, Understanding the interplay between density dependent birth function and maturation time delay using a reaction-diffusion population model, Ecological Complexity, 21 (2015), 14-26.  doi: 10.1016/j.ecocom.2014.10.007.  Google Scholar

[5]

R. D. Benguria and M. C. Depassier, Variational characterization of the speed of propagation of fronts for the nonlinear diffusion equation,, Comm. Math. Phys., 175 (1996), 221-227.  doi: 10.1007/BF02101631.  Google Scholar

[6]

P. Erm and B. L. Phillips, Evolution transforms pushed waves into pulled waves, The American Naturalist, 195 (2020), E87–E99. doi: 10.1086/707324.  Google Scholar

[7]

B. H. Gilding and R. Kersner, Travelling Waves in Nonlinear Diffusion-Convection Reaction, Birkhäuser, 2004. doi: 10.1007/978-3-0348-7964-4.  Google Scholar

[8]

A. Gomez and S. Trofimchuk, Global continuation of monotone wavefronts, J. London Math. Soc., 89 (2014), 47-68.  doi: 10.1112/jlms/jdt050.  Google Scholar

[9]

K. P. Hadeler, Topics in Mathematical Biology, Lecture Notes on Mathematical Modelling in the Life Sciences, Springer, 2017. doi: 10.1007/978-3-319-65621-2.  Google Scholar

[10]

J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations, Applied Mathematical Sciences, Springer-Verlag, 1993. doi: 10.1007/978-1-4612-4342-7.  Google Scholar

[11]

X. Liang and X.-Q. Zhao, Spreading speeds and traveling waves for abstract monostable evolution systems,, J. Functional Anal., 259 (2010), 857-903.  doi: 10.1016/j.jfa.2010.04.018.  Google Scholar

[12]

J. Mallet-Paret,, The Fredholm alternative for functional differential equations of mixed type, J. Dynam. Diff. Eqns. 11 (1999), 1–47. doi: 10.1023/A:1021889401235.  Google Scholar

[13]

G. NadinL. RossiL. Ryzhik and B. Perthame, Wave-like solutions for nonlocal reaction-diffusion equations: A toy model, Math. Model. Nat. Phenom., 8 (2013), 33-41.  doi: 10.1051/mmnp/20138304.  Google Scholar

[14]

C. Ou and J. Wu, Persistence of wavefronts in delayed nonlocal reaction-diffusion equations, J. Differential Equations, 235 (2007), 219–261. doi: 10.1016/j.jde.2006.12.010.  Google Scholar

[15]

W. van Saarloos, Front propagation into unstable states, Physics Reports, 386 (2003), 29–222. Google Scholar

[16]

E. Trofimchuk, M. Pinto and S. Trofimchuk, Monotone waves for non-monotone and non-local monostable reaction-diffusion equations,, J. Differential Equations, 261 (2016), 1203–1236. doi: 10.1016/j.jde.2016.03.039.  Google Scholar

[17]

E. Trofimchuk, M. Pinto and S. Trofimchuk, Pushed traveling fronts in monostable equations with monotone delayed reaction,, Discrete Contin. Dyn. Syst., 33 (2013) 2169–2187. doi: 10.3934/dcds.2013.33.2169.  Google Scholar

[18]

E. Trofimchuk and S. Trofimchuk, Admissible wavefronts speeds for a single species reaction-diffusion equation with delay,, Discrete Contin. Dyn. Syst., 20 (2008), 407-423.  doi: 10.3934/dcds.2008.20.407.  Google Scholar

[19]

E. TrofimchukV. 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.  Google Scholar

[20]

S. Trofimchuk and V. Volpert, Global continuation of monotone waves for bistable delayed equations with unimodal nonlinearities, Nonlinearity, 32 (2019), 2593-2632.  doi: 10.1088/1361-6544/ab0e23.  Google Scholar

[21]

S.-L. WuT.-C. Niu and C.-H. Hsu, Global asymptotic stability of pushed traveling fronts for monostable delayed reaction-diffusion equations, Discrete Contin. Dyn. Syst., 37 (2017), 3467-3486.  doi: 10.3934/dcds.2017147.  Google Scholar

[22]

J. Wu and X. Zou, Traveling wave fronts of reaction-diffusion systems with delay,, J. Dynam. Diff. Eqns., 13 (2001), 651-687.  doi: 10.1023/A:1016690424892.  Google Scholar

[23]

J. Xin, Front propagation in heterogeneous media,, SIAM Review, 42 (2000), 161-230.  doi: 10.1137/S0036144599364296.  Google Scholar

Figure 1.  Toy model: piece-wise linear birth function $ g $
Figure 2.  Vertical asymptote $ h = h_* $ followed (in the counter-clockwise direction) by the graphs of $ c(h), c_\#(h), c_*(h),c_\kappa(h) $. The cases $ k = 1.5 $ (left) and $ k = 1.2 $ (right).
Figure 3.  Snapshots of solution $ u(t,x) $ to the Cauchy problem (25), $ k = 1.2 $, converging to the pushed wavefront, at the indicated sequence of times $ t = t_j $. The cases $ h = 0.5 $ (left), $ h = 6 $ (right).
[1]

Elena Trofimchuk, Manuel Pinto, Sergei Trofimchuk. On the minimal speed of front propagation in a model of the Belousov-Zhabotinsky reaction. Discrete & Continuous Dynamical Systems - B, 2014, 19 (6) : 1769-1781. doi: 10.3934/dcdsb.2014.19.1769

[2]

Jong-Shenq Guo, Ken-Ichi Nakamura, Toshiko Ogiwara, Chang-Hong Wu. The sign of traveling wave speed in bistable dynamics. Discrete & Continuous Dynamical Systems, 2020, 40 (6) : 3451-3466. doi: 10.3934/dcds.2020047

[3]

Linghai Zhang. Wave speed analysis of traveling wave fronts in delayed synaptically coupled neuronal networks. Discrete & Continuous Dynamical Systems, 2014, 34 (5) : 2405-2450. doi: 10.3934/dcds.2014.34.2405

[4]

Jiamin Cao, Peixuan Weng. Single spreading speed and traveling wave solutions of a diffusive pioneer-climax model without cooperative property. Communications on Pure & Applied Analysis, 2017, 16 (4) : 1405-1426. doi: 10.3934/cpaa.2017067

[5]

Elena Trofimchuk, Manuel Pinto, Sergei Trofimchuk. Pushed traveling fronts in monostable equations with monotone delayed reaction. Discrete & Continuous Dynamical Systems, 2013, 33 (5) : 2169-2187. doi: 10.3934/dcds.2013.33.2169

[6]

Yuri Latushkin, Roland Schnaubelt, Xinyao Yang. Stable foliations near a traveling front for reaction diffusion systems. Discrete & Continuous Dynamical Systems - B, 2017, 22 (8) : 3145-3165. doi: 10.3934/dcdsb.2017168

[7]

Jong-Shenq Guo, Hirokazu Ninomiya, Chin-Chin Wu. Existence of a rotating wave pattern in a disk for a wave front interaction model. Communications on Pure & Applied Analysis, 2013, 12 (2) : 1049-1063. doi: 10.3934/cpaa.2013.12.1049

[8]

Shi-Liang Wu, Tong-Chang Niu, Cheng-Hsiung Hsu. Global asymptotic stability of pushed traveling fronts for monostable delayed reaction-diffusion equations. Discrete & Continuous Dynamical Systems, 2017, 37 (6) : 3467-3486. doi: 10.3934/dcds.2017147

[9]

Chufen Wu, Peixuan Weng. Asymptotic speed of propagation and traveling wavefronts for a SIR epidemic model. Discrete & Continuous Dynamical Systems - B, 2011, 15 (3) : 867-892. doi: 10.3934/dcdsb.2011.15.867

[10]

Dashun Xu, Xiao-Qiang Zhao. Asymptotic speed of spread and traveling waves for a nonlocal epidemic model. Discrete & Continuous Dynamical Systems - B, 2005, 5 (4) : 1043-1056. doi: 10.3934/dcdsb.2005.5.1043

[11]

Yang Wang, Xiong Li. Uniqueness of traveling front solutions for the Lotka-Volterra system in the weak competition case. Discrete & Continuous Dynamical Systems - B, 2019, 24 (7) : 3067-3075. doi: 10.3934/dcdsb.2018300

[12]

Monica Conti, Elsa M. Marchini, Vittorino Pata. Semilinear wave equations of viscoelasticity in the minimal state framework. Discrete & Continuous Dynamical Systems, 2010, 27 (4) : 1535-1552. doi: 10.3934/dcds.2010.27.1535

[13]

Guo Lin, Wan-Tong Li. Traveling wave solutions of a competitive recursion. Discrete & Continuous Dynamical Systems - B, 2012, 17 (1) : 173-189. doi: 10.3934/dcdsb.2012.17.173

[14]

Sebastian Acosta. A control approach to recover the wave speed (conformal factor) from one measurement. Inverse Problems & Imaging, 2015, 9 (2) : 301-315. doi: 10.3934/ipi.2015.9.301

[15]

Chase Mathison. Thermoacoustic Tomography with circular integrating detectors and variable wave speed. Inverse Problems & Imaging, 2020, 14 (4) : 665-682. doi: 10.3934/ipi.2020030

[16]

Jong-Shenq Guo, Ying-Chih Lin. The sign of the wave speed for the Lotka-Volterra competition-diffusion system. Communications on Pure & Applied Analysis, 2013, 12 (5) : 2083-2090. doi: 10.3934/cpaa.2013.12.2083

[17]

Marcelo Disconzi, Daniel Toundykov, Justin T. Webster. Front matter. Evolution Equations & Control Theory, 2016, 5 (4) : i-iii. doi: 10.3934/eect.201604i

[18]

Zhenguo Bai, Tingting Zhao. Spreading speed and traveling waves for a non-local delayed reaction-diffusion system without quasi-monotonicity. Discrete & Continuous Dynamical Systems - B, 2018, 23 (10) : 4063-4085. doi: 10.3934/dcdsb.2018126

[19]

Peixuan Weng. Spreading speed and traveling wavefront of an age-structured population diffusing in a 2D lattice strip. Discrete & Continuous Dynamical Systems - B, 2009, 12 (4) : 883-904. doi: 10.3934/dcdsb.2009.12.883

[20]

Chang-Hong Wu. Spreading speed and traveling waves for a two-species weak competition system with free boundary. Discrete & Continuous Dynamical Systems - B, 2013, 18 (9) : 2441-2455. doi: 10.3934/dcdsb.2013.18.2441

2020 Impact Factor: 1.392

Article outline

Figures and Tables

[Back to Top]