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 and 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.

[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.

[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.

[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.

[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.

[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.

[7]

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

[8]

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

[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.

[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.

[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.

[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.

[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.

[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.

[15]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[23]

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

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.

[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.

[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.

[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.

[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.

[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.

[7]

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

[8]

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

[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.

[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.

[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.

[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.

[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.

[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.

[15]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[23]

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

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).
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