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Existence of solutions for a class of abstract neutral differential equations
Traveling waves and entire solutions for an epidemic model with asymmetric dispersal
School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu 730000, China |
This paper is concerned with traveling waves and entire solutions of one epidemic model with asymmetric dispersal kernel function arising from the spread of an epidemic by oral-faecal transmission. The asymmetry of the kernel function will have an influence on two aspects: (ⅰ) the minimal wave speed of traveling wave fronts may be nonpositive, but we give a new restrictive condition on the kernel function to guarantee it is positive; (ⅱ) the two traveling wave solutions with the same speed spreading from right and left of $x$-axis may be different in shape, which further makes that the entire solutions with five or four parameters may be asymmetric and the entire solutions with three parameters increasing in $x$ may be different from those decreasing in $x$ in shape. As for traveling wave solutions, we get the existence, asymptotic behavior and uniqueness of the two traveling wave solutions spreading from right and left of $x$-axis, respectively. We further construct three new entire solutions with five, four or three parameters. Two comparison principles also be established.
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A mathematical model for the 1973 cholera epidemic in the European Mediterranean region, Rev. d'Epidemiol. Santé Publique, 27 (1979), 32-121.
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J. Carr and A. Chmaj,
Uniqueness of travelling waves for nonlocal monostable equations, Proc. Amer. Math. Soc., 132 (2004), 2433-2439.
doi: 10.1090/S0002-9939-04-07432-5. |
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X. Chen and J. S. Guo,
Existence and uniqueness of entire solutions for a reaction-diffusion equation, J. Differential Equations, 212 (2005), 62-84.
doi: 10.1016/j.jde.2004.10.028. |
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X. Chen, J. S. Guo and H. Ninomiya,
Entire solutions of reaction-diffusion equations with balanced bistable nonlinearities, Proc. Roy. Soc. Edinburgh Sect. A, 136 (2006), 1207-1237.
doi: 10.1017/S0308210500004959. |
[5] |
J. Coville, J. Dávila and S. Martínez,
Nonlocal anisotropic dispersal with monostable nonlinearity, J. Differential Equations, 244 (2008), 3080-3118.
doi: 10.1016/j.jde.2007.11.002. |
[6] |
J. Coville, Travelling fronts in asymmetric nonlocal reaction diffusion equations: the bistable and ignition cases Prépublication du CMM Hal-00696208. |
[7] |
J. Coville and L. Dupaigne,
On a non-local equation arising in population dynamics., Proc. Roy. Soc. Edinburgh. Sect. A, 137 (2007), 727-755.
doi: 10.1017/S0308210504000721. |
[8] |
W. Ellison and F. Ellison, Prime Numbers, A Wiley-Interscience Publication, John Wiley & Sons, New York; Hermann, Paris, 1985.
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Y. Fukao, Y. Morita and H. Ninomiya,
Some entire solutions of the Allen-Cahn equation, Taiwanese J. Math., 8 (2004), 15-32.
|
[10] |
J. S. Guo and Y. Morita,
Entire solutions of reaction-diffusion equations and an application to discrete diffusive equations, Discrete Contin. Dyn. Syst., 12 (2005), 193-212.
doi: 10.3934/dcds.2005.12.193. |
[11] |
J. S. Guo and C. H. Wu,
Entire solutions for a two-component competition system in a lattice, Tohoku. Math. J., 62 (2010), 17-28.
doi: 10.2748/tmj/1270041024. |
[12] |
F. Hamel and N. Nadirashvili,
Entire solutions of the KPP equation, Comm. Pure Appl. Math., 52 (1999), 1255-1276.
doi: 10.1002/(SICI)1097-0312(199910)52:10<1255::AID-CPA4>3.0.CO;2-W. |
[13] |
F. Hamel and N. Nadirashvili,
Travelling fronts and entire solutions of the Fisher-KPP equation in $\mathbb{R}^{N}$, Arch. Ration. Mech. Anal., 157 (2001), 91-163.
doi: 10.1007/PL00004238. |
[14] |
C. H. Hsu and T. S. Yang,
Existence, uniqueness, monotonicity and asymptotic behaviour of travelling waves for epidemic models, Nonlinearity, 26 (2013), 121-139.
doi: 10.1088/0951-7715/26/10/2925. |
[15] |
W. T. Li, Y. J Sun and Z. C. Wang,
Entire solutions in the Fisher-KPP equation with nonlocal
dispersal, Nonlinear Anal. Real World Appl., 11 (2010), 2302-2313.
doi: 10.1016/j.nonrwa.2009.07.005. |
[16] |
W. T. Li, Z. C. Wang and J. Wu,
Entire solutions in monostable reaction-diffusion equations with delayed nonlinearity, J. Differential Equations, 245 (2008), 102-129.
doi: 10.1016/j.jde.2008.03.023. |
[17] |
W. T. Li, L. Zhang and G. B. Zhang,
Invasion entire solutions in a competition system with nonlocal dispersal, Discrete Contin. Dyn. Syst., 35 (2015), 1531-1560.
doi: 10.3934/dcds.2015.35.1531. |
[18] |
G. Lv,
Asymptotic behavior of traveling fronts and entire solutions for a nonlocal monostable equation, Nonlinear Anal., 72 (2010), 3659-3668.
doi: 10.1016/j.na.2009.12.047. |
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R. H. Martin and H. L. Smith,
Abstract functional differential equations and reaction-diffusion systems, Trans. Amer. Math. Soc., 321 (1990), 1-44.
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[20] |
Y. Morita and H. Ninomiya,
Entire solutions with merging fronts to reaction-diffusion equations, J. Dynam. Differential Equations, 18 (2006), 841-861.
doi: 10.1007/s10884-006-9046-x. |
[21] |
Y. Morita and K. Tachibana,
An entire solution to the Lotka-Volterra competition-diffusion equations, SIAM J. Math. Anal., 40 (2009), 2217-2240.
doi: 10.1137/080723715. |
[22] |
J. D. Murray, Mathematical Biology, Springer, Berlin-Heidelberg-New York, 1993.
![]() ![]() |
[23] |
S. Pan, W. T. Li and G. Lin,
Travelling wave fronts in nonlocal delayed reaction-diffusion systems and applications, Z. Angew. Math. Phys., 60 (2009), 377-392.
doi: 10.1007/s00033-007-7005-y. |
[24] |
Y. J. Sun, L. Zhang, W. T. Li and Z. C. Wang,
Entire solutions in nonlocal monostable equations: asymmetric case
(2015), submitted. |
[25] |
Y. J. Sun, W. T. Li and Z. C. Wang,
Traveling waves for a nonlocal anisotropic dispersal equation with monostable nonlinearity, Nonlinear Anal., 74 (2011), 814-826.
doi: 10.1016/j.na.2010.09.032. |
[26] |
A. I. Volpert, V. A. Volpert and V. A. Volpert,
Traveling Wave Solutions of Parabolic Systems, Translations of Mathematical Monographs, Vol. 140, Amer. Math. Soc. , Providence Rhode Island, 1994. |
[27] |
M. Wang and G. Lv,
Entire solutions of a diffusive and competitive Lotka-Volterra type system with nonlocal delay, Nonlinearity, 23 (2010), 1609-1630.
doi: 10.1088/0951-7715/23/7/005. |
[28] |
Z. C. Wang, W. T. Li and S. Ruan,
Entire solutions in bistable reaction-diffusion equations with nonlocal delayed nonlinearity, Trans. Amer. Math. Soc., 361 (2009), 2047-2084.
doi: 10.1090/S0002-9947-08-04694-1. |
[29] |
Z. C. Wang, W. T. Li and J. Wu,
Entire solutions in delayed lattice differential equations with monostable nonlinearity, SIAM J. Math. Anal., 40 (2009), 2392-2420.
doi: 10.1137/080727312. |
[30] |
S. L. Wu and C. H. Hsu,
Entire solutions with merging fronts to a bistable periodic lattice dynamical system, Discrete Contin. Dyn. Syst., 36 (2016), 2329-2346.
doi: 10.3934/dcds.2016.36.2329. |
[31] |
S. L. Wu and C. H. Hsu,
Existence of entire solutions for delayed monostable epidemic models, Trans. Amer. Math. Soc., 368 (2016), 6033-6062.
doi: 10.1090/tran/6526. |
[32] |
S. L. Wu and H. Wang,
Front-like entire solutions for monostable reaction-diffusion systems, J. Dynam. Differential Equations, 25 (2013), 505-533.
doi: 10.1007/s10884-013-9293-6. |
[33] |
D. Xu and X. Q. Zhao,
Erratum to "Bistable waves in an epidemic model", J. Dynam. Differential Equations, 17 (2005), 219-247.
doi: 10.1007/s10884-005-6294-0. |
[34] |
H. Yagisita,
Backward global solutions characterizing annihilation dynamics of travelling fronts, Publ. Res. Inst. Math. Sci., 39 (2003), 117-164.
doi: 10.2977/prims/1145476150. |
[35] |
L. Zhang, W. T. Li, Z. C. Wang and Y. J. Sun,
Entire solutions in nonlocal bistable equations: asymmetric case
(2016), submitted. |
[36] |
L. Zhang, W. T. Li and S. L. Wu,
Multi-type entire solutions in a nonlocal dispersal epidemic model, J. Dynam. Differential Equations, 28 (2016), 189-224.
doi: 10.1007/s10884-014-9416-8. |
[37] |
X. Q. Zhao and W. Wang,
Fisher waves in an epidemic model, Discrete Contin. Dyn. Syst. B, 4 (2004), 1117-1128.
doi: 10.3934/dcdsb.2004.4.1117. |
show all references
References:
[1] |
V. Capasso and S. L. Paveri-Fontana,
A mathematical model for the 1973 cholera epidemic in the European Mediterranean region, Rev. d'Epidemiol. Santé Publique, 27 (1979), 32-121.
|
[2] |
J. Carr and A. Chmaj,
Uniqueness of travelling waves for nonlocal monostable equations, Proc. Amer. Math. Soc., 132 (2004), 2433-2439.
doi: 10.1090/S0002-9939-04-07432-5. |
[3] |
X. Chen and J. S. Guo,
Existence and uniqueness of entire solutions for a reaction-diffusion equation, J. Differential Equations, 212 (2005), 62-84.
doi: 10.1016/j.jde.2004.10.028. |
[4] |
X. Chen, J. S. Guo and H. Ninomiya,
Entire solutions of reaction-diffusion equations with balanced bistable nonlinearities, Proc. Roy. Soc. Edinburgh Sect. A, 136 (2006), 1207-1237.
doi: 10.1017/S0308210500004959. |
[5] |
J. Coville, J. Dávila and S. Martínez,
Nonlocal anisotropic dispersal with monostable nonlinearity, J. Differential Equations, 244 (2008), 3080-3118.
doi: 10.1016/j.jde.2007.11.002. |
[6] |
J. Coville, Travelling fronts in asymmetric nonlocal reaction diffusion equations: the bistable and ignition cases Prépublication du CMM Hal-00696208. |
[7] |
J. Coville and L. Dupaigne,
On a non-local equation arising in population dynamics., Proc. Roy. Soc. Edinburgh. Sect. A, 137 (2007), 727-755.
doi: 10.1017/S0308210504000721. |
[8] |
W. Ellison and F. Ellison, Prime Numbers, A Wiley-Interscience Publication, John Wiley & Sons, New York; Hermann, Paris, 1985.
![]() ![]() |
[9] |
Y. Fukao, Y. Morita and H. Ninomiya,
Some entire solutions of the Allen-Cahn equation, Taiwanese J. Math., 8 (2004), 15-32.
|
[10] |
J. S. Guo and Y. Morita,
Entire solutions of reaction-diffusion equations and an application to discrete diffusive equations, Discrete Contin. Dyn. Syst., 12 (2005), 193-212.
doi: 10.3934/dcds.2005.12.193. |
[11] |
J. S. Guo and C. H. Wu,
Entire solutions for a two-component competition system in a lattice, Tohoku. Math. J., 62 (2010), 17-28.
doi: 10.2748/tmj/1270041024. |
[12] |
F. Hamel and N. Nadirashvili,
Entire solutions of the KPP equation, Comm. Pure Appl. Math., 52 (1999), 1255-1276.
doi: 10.1002/(SICI)1097-0312(199910)52:10<1255::AID-CPA4>3.0.CO;2-W. |
[13] |
F. Hamel and N. Nadirashvili,
Travelling fronts and entire solutions of the Fisher-KPP equation in $\mathbb{R}^{N}$, Arch. Ration. Mech. Anal., 157 (2001), 91-163.
doi: 10.1007/PL00004238. |
[14] |
C. H. Hsu and T. S. Yang,
Existence, uniqueness, monotonicity and asymptotic behaviour of travelling waves for epidemic models, Nonlinearity, 26 (2013), 121-139.
doi: 10.1088/0951-7715/26/10/2925. |
[15] |
W. T. Li, Y. J Sun and Z. C. Wang,
Entire solutions in the Fisher-KPP equation with nonlocal
dispersal, Nonlinear Anal. Real World Appl., 11 (2010), 2302-2313.
doi: 10.1016/j.nonrwa.2009.07.005. |
[16] |
W. T. Li, Z. C. Wang and J. Wu,
Entire solutions in monostable reaction-diffusion equations with delayed nonlinearity, J. Differential Equations, 245 (2008), 102-129.
doi: 10.1016/j.jde.2008.03.023. |
[17] |
W. T. Li, L. Zhang and G. B. Zhang,
Invasion entire solutions in a competition system with nonlocal dispersal, Discrete Contin. Dyn. Syst., 35 (2015), 1531-1560.
doi: 10.3934/dcds.2015.35.1531. |
[18] |
G. Lv,
Asymptotic behavior of traveling fronts and entire solutions for a nonlocal monostable equation, Nonlinear Anal., 72 (2010), 3659-3668.
doi: 10.1016/j.na.2009.12.047. |
[19] |
R. H. Martin and H. L. Smith,
Abstract functional differential equations and reaction-diffusion systems, Trans. Amer. Math. Soc., 321 (1990), 1-44.
doi: 10.2307/2001590. |
[20] |
Y. Morita and H. Ninomiya,
Entire solutions with merging fronts to reaction-diffusion equations, J. Dynam. Differential Equations, 18 (2006), 841-861.
doi: 10.1007/s10884-006-9046-x. |
[21] |
Y. Morita and K. Tachibana,
An entire solution to the Lotka-Volterra competition-diffusion equations, SIAM J. Math. Anal., 40 (2009), 2217-2240.
doi: 10.1137/080723715. |
[22] |
J. D. Murray, Mathematical Biology, Springer, Berlin-Heidelberg-New York, 1993.
![]() ![]() |
[23] |
S. Pan, W. T. Li and G. Lin,
Travelling wave fronts in nonlocal delayed reaction-diffusion systems and applications, Z. Angew. Math. Phys., 60 (2009), 377-392.
doi: 10.1007/s00033-007-7005-y. |
[24] |
Y. J. Sun, L. Zhang, W. T. Li and Z. C. Wang,
Entire solutions in nonlocal monostable equations: asymmetric case
(2015), submitted. |
[25] |
Y. J. Sun, W. T. Li and Z. C. Wang,
Traveling waves for a nonlocal anisotropic dispersal equation with monostable nonlinearity, Nonlinear Anal., 74 (2011), 814-826.
doi: 10.1016/j.na.2010.09.032. |
[26] |
A. I. Volpert, V. A. Volpert and V. A. Volpert,
Traveling Wave Solutions of Parabolic Systems, Translations of Mathematical Monographs, Vol. 140, Amer. Math. Soc. , Providence Rhode Island, 1994. |
[27] |
M. Wang and G. Lv,
Entire solutions of a diffusive and competitive Lotka-Volterra type system with nonlocal delay, Nonlinearity, 23 (2010), 1609-1630.
doi: 10.1088/0951-7715/23/7/005. |
[28] |
Z. C. Wang, W. T. Li and S. Ruan,
Entire solutions in bistable reaction-diffusion equations with nonlocal delayed nonlinearity, Trans. Amer. Math. Soc., 361 (2009), 2047-2084.
doi: 10.1090/S0002-9947-08-04694-1. |
[29] |
Z. C. Wang, W. T. Li and J. Wu,
Entire solutions in delayed lattice differential equations with monostable nonlinearity, SIAM J. Math. Anal., 40 (2009), 2392-2420.
doi: 10.1137/080727312. |
[30] |
S. L. Wu and C. H. Hsu,
Entire solutions with merging fronts to a bistable periodic lattice dynamical system, Discrete Contin. Dyn. Syst., 36 (2016), 2329-2346.
doi: 10.3934/dcds.2016.36.2329. |
[31] |
S. L. Wu and C. H. Hsu,
Existence of entire solutions for delayed monostable epidemic models, Trans. Amer. Math. Soc., 368 (2016), 6033-6062.
doi: 10.1090/tran/6526. |
[32] |
S. L. Wu and H. Wang,
Front-like entire solutions for monostable reaction-diffusion systems, J. Dynam. Differential Equations, 25 (2013), 505-533.
doi: 10.1007/s10884-013-9293-6. |
[33] |
D. Xu and X. Q. Zhao,
Erratum to "Bistable waves in an epidemic model", J. Dynam. Differential Equations, 17 (2005), 219-247.
doi: 10.1007/s10884-005-6294-0. |
[34] |
H. Yagisita,
Backward global solutions characterizing annihilation dynamics of travelling fronts, Publ. Res. Inst. Math. Sci., 39 (2003), 117-164.
doi: 10.2977/prims/1145476150. |
[35] |
L. Zhang, W. T. Li, Z. C. Wang and Y. J. Sun,
Entire solutions in nonlocal bistable equations: asymmetric case
(2016), submitted. |
[36] |
L. Zhang, W. T. Li and S. L. Wu,
Multi-type entire solutions in a nonlocal dispersal epidemic model, J. Dynam. Differential Equations, 28 (2016), 189-224.
doi: 10.1007/s10884-014-9416-8. |
[37] |
X. Q. Zhao and W. Wang,
Fisher waves in an epidemic model, Discrete Contin. Dyn. Syst. B, 4 (2004), 1117-1128.
doi: 10.3934/dcdsb.2004.4.1117. |
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