Figure 1.
Toy model: piece-wise linear birth function $ g $
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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
| 1. | Mathematical Institute, Silesian University, 746 01 Opava, Czech Republic |
| 2. | Instituto de Matemática, Universidad de Talca, Casilla 747, Talca, Chile |
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.
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 and Continuous Dynamical Systems,
2021, 41
(12)
: 5979-6000.
doi: 10.3934/dcds.2021103
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References:
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doi: 10.1007/s00208-011-0722-8. |
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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. |
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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. |
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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. |
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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. |
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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. |
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B. H. Gilding and R. Kersner, Travelling Waves in Nonlinear Diffusion-Convection Reaction, Birkhäuser, 2004.
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Global continuation of monotone wavefronts, J. London Math. Soc., 89 (2014), 47-68.
doi: 10.1112/jlms/jdt050. |
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K. P. Hadeler, Topics in Mathematical Biology, Lecture Notes on Mathematical Modelling in the Life Sciences, Springer, 2017.
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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. |
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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. 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. |
| [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. 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. |
| [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. 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. |
| [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. 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] |
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. 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. |
| [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. |
| [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. 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. |
| [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. 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. |
| [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. 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. |
| [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 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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