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

On pushed wavefronts of monostable equation with unimodal delayed reaction

Karel Hasík 1, , Jana Kopfová 1, , Petra Nábělková 1, and  Sergei Trofimchuk 2,,

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.

Keywords: Traveling front, pushed wave, minimal speed.
Mathematics Subject Classification: Primary: 34K10, 35K57; Secondary: 92D25.
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. 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.  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. Alfaro, A. 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-Yaghoub, G. Yao, M. 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

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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

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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. Nadin, L. Rossi, L. 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. 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.  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. Wu, T.-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. 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.  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. Alfaro, A. 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-Yaghoub, G. Yao, M. 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. Nadin, L. Rossi, L. 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. 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.  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. Wu, T.-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 $
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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).
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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).
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