Traveling wave solutions in a nonlocal reaction-diffusion population model

Previous Article
Bogdanov-Takens bifurcation of codimension 3 in a predator-prey model with constant-yield predator harvesting
CPAA Home
This Issue
Next Article

May 2016, 15(3): 1057-1076. doi: 10.3934/cpaa.2016.15.1057

Traveling wave solutions in a nonlocal reaction-diffusion population model

Bang-Sheng Han ^1, and Zhi-Cheng Wang ^1,

1.	School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu 730000, China, China

Received October 2015 Revised December 2015 Published February 2016

Abstract
Related Papers

This paper is concerned with a nonlocal reaction-diffusion equation with the form \begin{eqnarray} \frac{\partial u}{\partial t}=\frac{\partial^{2}u}{\partial x^{2}}+u\left\{ 1+\alpha u-\beta u^{2}-(1+\alpha-\beta)(\phi\ast u) \right\}, \quad (t,x)\in (0,\infty) \times \mathbb{R}, \end{eqnarray} where $\alpha $ and $\beta$ are positive constants, $0<\beta<1+\alpha$. We prove that there exists a number $c^*\geq 2$ such that the equation admits a positive traveling wave solution connecting the zero equilibrium to an unknown positive steady state for each speed $c>c^*$. At the same time, we show that there is no such traveling wave solutions for speed $c<2$. For sufficiently large speed $c>c^*$, we further show that the steady state is the unique positive equilibrium. Using the lower and upper solutions method, we also establish the existence of monotone traveling wave fronts connecting the zero equilibrium and the positive equilibrium. Finally, for a specific kernel function $\phi(x):=\frac{1}{2\sigma}e^{-\frac{|x|}{\sigma}}$ ($\sigma>0$), by numerical simulations we show that the traveling wave solutions may connects the zero equilibrium to a periodic steady state as $\sigma$ is increased. Furthermore, by the stability analysis we explain why and when a periodic steady state can appear.

Keywords: monostable, Nonlocal reaction-diffusion equation, linear stability analysis., traveling waves, lower and upper solutions, numerical simulation.

Mathematics Subject Classification: 35C07, 35B40, 35K57, 92D2.

Citation: Bang-Sheng Han, Zhi-Cheng Wang. Traveling wave solutions in a nonlocal reaction-diffusion population model. Communications on Pure & Applied Analysis, 2016, 15 (3) : 1057-1076. doi: 10.3934/cpaa.2016.15.1057

References:

[1]	S. Ai, Traveling wave fronts for generalized Fisher equations with spatio-temporal delays,, \emph{J. Differential Equations}, 232 (2007), 104. doi: 10.1016/j.jde.2006.08.015. Google Scholar
[2]	M. Alfaro and J. Coville, Rapid traveling waves in the nonlocal Fisher equation connect two unstable states,, \emph{Appl. Math. Lett.}, 25 (2012), 2095. doi: 10.1016/j.aml.2012.05.006. Google Scholar
[3]	M. Alfaro, J. Coville and G. Raoul, Traveling waves in a nonlocal reaction-diffusion equation as a model for a population structured by a space variable and a phenotypic trait,, \emph{Comm. Partial Differential Equations}, 38 (2013), 2126. doi: 10.1080/03605302.2013.828069. Google Scholar
[4]	M. Alfaro, J. Coville and G. Raoul, Bistable traveling waves for nonlocal reaction diffusion equations,, \emph{Discrete Contin. Dyn. Syst.}, 34 (2014), 1775. doi: 10.3934/dcds.2014.34.1775. Google Scholar
[5]	N. Apreutesei, A. Ducrot and V. Volpert, Traveling waves for integro-differential equations in population dynamics,, \emph{Discrete Contin. Dyn. Syst. Ser. B}, 11 (2009), 541. doi: 10.3934/dcdsb.2009.11.541. Google Scholar
[6]	D. G. Aronson and H. F. Weinberger, Multidimensional nonlinear diffusion arising in population genetics,, \emph{Adv. in Math.}, 30 (1978), 33. doi: 10.1016/0001-8708(78)90130-5. Google Scholar
[7]	P. Ashwin, M. V. Bartuccelli, T. J. Bridges and S. A. Gourley, Traveling fronts for the KPP equation with spatio-temporal delay,, \emph{Z. Angew. Math. Phys.}, 53 (2002), 103. doi: 10.1007/s00033-002-8145-8. Google Scholar
[8]	H. Berestycki, G. Nadin, B. Perthame and L. Ryzhik, The non-local Fisher-KPP equation: Travelling waves and steady states,, \emph{Nonlinearity}, 22 (2009), 2813. doi: 10.1088/0951-7715/22/12/002. Google Scholar
[9]	J. Billingham, Dynamics of a strongly nonlocal reaction-diffusion population model,, \emph{Nonlinearity}, 17 (2004), 313. doi: 10.1088/0951-7715/17/1/018. Google Scholar
[10]	N. F. Britton, Aggregation and the competitive exclusion principle,, \emph{J. Theoret. Biol.}, 136 (1989), 57. doi: 10.1016/S0022-5193(89)80189-4. Google Scholar
[11]	N. F. Britton, Spatial structures and periodic traveling waves in an integro-differential reaction-diffusion population model,, \emph{SIAM J. Appl. Math.}, 50 (1990), 1663. doi: 10.1137/0150099. Google Scholar
[12]	I. Demin and V. Volpert, Existence of waves for a nonlocal reaction-diffusion equation,, \emph{Math. Model. Nat. Phenom.}, 5 (2010), 80. doi: 10.1051/mmnp/20105506. Google Scholar
[13]	K. Deng, On a nonlocal reaction-diffusion population model,, \emph{Discrete Contin. Dyn. Syst. Ser. B}, 9 (2008), 65. doi: 10.3934/dcdsb.2008.9.65. Google Scholar
[14]	G. Faye and M. Holzer, Modulated traveling fronts for a nonlocal Fisher-KPP equation: a dynamical systems approach,, \emph{J. Differential Equations}, 258 (2015), 2257. doi: 10.1016/j.jde.2014.12.006. Google Scholar
[15]	J. Fang and X.-Q. Zhao, Monotone wave fronts of the nonlocal Fisher-KPP equation,, \emph{Nonlinearity}, 24 (2011), 3043. doi: 10.1088/0951-7715/24/11/002. Google Scholar
[16]	S. Genieys and B. Perthame, Concentration in the nonlocal Fisher equation: the Hamilton-Jacobi limit,, \emph{Math. Model. Nat. Phenom.}, 2 (2007), 135. doi: 10.1051/mmnp:2008029. Google Scholar
[17]	S. Genieys, V. Volpert and P. Auger, Pattern and waves for a model in population dynamics with nonlocal consumption of resources,, \emph{Math. Model. Nat. Phenom.}, 1 (2006), 65. doi: 10.1051/mmnp:2006004. Google Scholar
[18]	A. Gomez and S. Trofimchuk, Monotone traveling wavefronts of the KPP-Fisher delayed equation,, \emph{J. Differential Equations}, 250 (2011), 1767. doi: 10.1016/j.jde.2010.11.011. Google Scholar
[19]	G. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order,, Classics in Mathematics, (2001). Google Scholar
[20]	S. A. Gourley, Traveling front solutions of a nonlocal Fisher equation,, \emph{J. Math. Biol.}, 41 (2000), 272. doi: 10.1007/s002850000047. Google Scholar
[21]	S. A. Gourley and N. F. Britton, Instability of traveling wave solutions of a population model with nonlocal effects,, \emph{IMA J. Appl. Math.}, 51 (1993), 299. doi: 10.1093/imamat/51.3.299. Google Scholar
[22]	S. A. Gourley and N. F. Britton, On a modified Volterra population equation with diffusion,, \emph{Nonlinear Anal.}, 21 (1993), 389. doi: 10.1016/0362-546X(93)90082-4. Google Scholar
[23]	S. A. Gourley and N. F. Britton, A predator-prey reaction-diffusion system with nonlocal effects,, \emph{J. Math. Biol.}, 34 (1996), 297. doi: 10.1007/BF00160498. Google Scholar
[24]	S. A. Gourley, M. A. J. Chaplain and F. A. Davidson, Spatio-temporal pattern formation in a nonlocal reaction-diffusion equation,, \emph{Dyn. Syst.}, 16 (2001), 173. doi: 10.1080/14689360116914. Google Scholar
[25]	F. Hamel and L. Ryzhik, On the nonlocal Fisher-KPP equation: steady states, spreading speed and global bounds,, \emph{Nonlinearity}, 27 (2014), 2735. doi: 10.1088/0951-7715/27/11/2735. Google Scholar
[26]	G. Nadin, B. Perthame, L. Rossi and L. Ryzhik, Wave-like solutions for nonlocal reaction-diffusion equations: A toy model,, \emph{Math. Model. Nat. Phenom.}, 8 (2013), 33. doi: 10.1051/mmnp/20138304. Google Scholar
[27]	G. Nadin, B. Perthame and M. Tang, Can a traveling wave connect two unstable states? The case of the nonlocal Fisher equation,, \emph{C. R. Math. Acad. Sci. Paris}, 349 (2011), 553. doi: 10.1016/j.crma.2011.03.008. Google Scholar
[28]	D. H. Sattinger, On the stability of waves of nonlinear parabolic systems,, \emph{Adv. Math.}, 22 (1976), 312. doi: 10.1016/0001-8708(76)90098-0. Google Scholar
[29]	C. Ou and J. Wu, Traveling wavefronts in a delayed food-limited population model,, \emph{SIAM J. Math. Anal.}, 39 (2007), 103. doi: 10.1137/050638011. Google Scholar
[30]	A. Volpert, V. Volpert and V. Volpert, Traveling Wave Solutions of Parabolic Systems,, Translated from the Russian manuscript by James F. Heyda. Translations of Mathematical Monographs, (1994). Google Scholar
[31]	Z.-C. Wang and W.-T. Li, Monotone travelling fronts of a food-limited population model with nonlocal delay,, \emph{Nonlinear Anal. Real World Appl.}, 8 (2007), 699. doi: 10.1016/j.nonrwa.2006.03.001. Google Scholar
[32]	Z.-C. Wang and W.-T. Li, Traveling fronts in diffusive and cooperative Lotka-Volterra system with nonlocal delays,, \emph{Z. Angew. Math. Phys.}, 58 (2007), 571. doi: 10.1007/s00033-006-5125-4. Google Scholar
[33]	Z.-C. Wang, W.-T. Li and S. Ruan, Traveling wave fronts in reaction-diffusion systems with spatio-temporal delays,, \emph{J. Differential Equations}, 222 (2006), 185. doi: 10.1016/j.jde.2005.08.010. Google Scholar
[34]	Z.-C. Wang, W.-T. Li and S. Ruan, Existence and stability of traveling wave fronts in reaction advection diffusion equations with nonlocal delay,, \emph{J. Differential Equations}, 238 (2007), 153. doi: 10.1016/j.jde.2007.03.025. Google Scholar
[35]	Z.-C. Wang and J. Wu, Traveling waves in a bio-reactor model with stage-structure,, \emph{J. Math. Anal. Appl.}, 385 (2012), 683. doi: 10.1016/j.jmaa.2011.06.084. Google Scholar
[36]	Z.-C. Wang, J. Wu and R. Liu, Traveling waves of the spread of avian influenza,, \emph{Proc. Amer. Math. Soc.}, 140 (2012), 3931. doi: 10.1090/S0002-9939-2012-11246-8. Google Scholar
[37]	G. X. Yang and J. Xu, Analysis of spatiotemporal patterns in a single species reaction-diffusion model with spatiotemporal delay,, \emph{Nonlinear Anal. Real World Appl.}, 22 (2015), 54. doi: 10.1016/j.nonrwa.2014.07.013. Google Scholar
[38]	Q. Ye, Z. Li, M. Wang and Y. Wu, Introduction of Reaction-Diffusion Equations,, Science Publish, (2011). Google Scholar

show all references

References:

[1]	S. Ai, Traveling wave fronts for generalized Fisher equations with spatio-temporal delays,, \emph{J. Differential Equations}, 232 (2007), 104. doi: 10.1016/j.jde.2006.08.015. Google Scholar
[2]	M. Alfaro and J. Coville, Rapid traveling waves in the nonlocal Fisher equation connect two unstable states,, \emph{Appl. Math. Lett.}, 25 (2012), 2095. doi: 10.1016/j.aml.2012.05.006. Google Scholar
[3]	M. Alfaro, J. Coville and G. Raoul, Traveling waves in a nonlocal reaction-diffusion equation as a model for a population structured by a space variable and a phenotypic trait,, \emph{Comm. Partial Differential Equations}, 38 (2013), 2126. doi: 10.1080/03605302.2013.828069. Google Scholar
[4]	M. Alfaro, J. Coville and G. Raoul, Bistable traveling waves for nonlocal reaction diffusion equations,, \emph{Discrete Contin. Dyn. Syst.}, 34 (2014), 1775. doi: 10.3934/dcds.2014.34.1775. Google Scholar
[5]	N. Apreutesei, A. Ducrot and V. Volpert, Traveling waves for integro-differential equations in population dynamics,, \emph{Discrete Contin. Dyn. Syst. Ser. B}, 11 (2009), 541. doi: 10.3934/dcdsb.2009.11.541. Google Scholar
[6]	D. G. Aronson and H. F. Weinberger, Multidimensional nonlinear diffusion arising in population genetics,, \emph{Adv. in Math.}, 30 (1978), 33. doi: 10.1016/0001-8708(78)90130-5. Google Scholar
[7]	P. Ashwin, M. V. Bartuccelli, T. J. Bridges and S. A. Gourley, Traveling fronts for the KPP equation with spatio-temporal delay,, \emph{Z. Angew. Math. Phys.}, 53 (2002), 103. doi: 10.1007/s00033-002-8145-8. Google Scholar
[8]	H. Berestycki, G. Nadin, B. Perthame and L. Ryzhik, The non-local Fisher-KPP equation: Travelling waves and steady states,, \emph{Nonlinearity}, 22 (2009), 2813. doi: 10.1088/0951-7715/22/12/002. Google Scholar
[9]	J. Billingham, Dynamics of a strongly nonlocal reaction-diffusion population model,, \emph{Nonlinearity}, 17 (2004), 313. doi: 10.1088/0951-7715/17/1/018. Google Scholar
[10]	N. F. Britton, Aggregation and the competitive exclusion principle,, \emph{J. Theoret. Biol.}, 136 (1989), 57. doi: 10.1016/S0022-5193(89)80189-4. Google Scholar
[11]	N. F. Britton, Spatial structures and periodic traveling waves in an integro-differential reaction-diffusion population model,, \emph{SIAM J. Appl. Math.}, 50 (1990), 1663. doi: 10.1137/0150099. Google Scholar
[12]	I. Demin and V. Volpert, Existence of waves for a nonlocal reaction-diffusion equation,, \emph{Math. Model. Nat. Phenom.}, 5 (2010), 80. doi: 10.1051/mmnp/20105506. Google Scholar
[13]	K. Deng, On a nonlocal reaction-diffusion population model,, \emph{Discrete Contin. Dyn. Syst. Ser. B}, 9 (2008), 65. doi: 10.3934/dcdsb.2008.9.65. Google Scholar
[14]	G. Faye and M. Holzer, Modulated traveling fronts for a nonlocal Fisher-KPP equation: a dynamical systems approach,, \emph{J. Differential Equations}, 258 (2015), 2257. doi: 10.1016/j.jde.2014.12.006. Google Scholar
[15]	J. Fang and X.-Q. Zhao, Monotone wave fronts of the nonlocal Fisher-KPP equation,, \emph{Nonlinearity}, 24 (2011), 3043. doi: 10.1088/0951-7715/24/11/002. Google Scholar
[16]	S. Genieys and B. Perthame, Concentration in the nonlocal Fisher equation: the Hamilton-Jacobi limit,, \emph{Math. Model. Nat. Phenom.}, 2 (2007), 135. doi: 10.1051/mmnp:2008029. Google Scholar
[17]	S. Genieys, V. Volpert and P. Auger, Pattern and waves for a model in population dynamics with nonlocal consumption of resources,, \emph{Math. Model. Nat. Phenom.}, 1 (2006), 65. doi: 10.1051/mmnp:2006004. Google Scholar
[18]	A. Gomez and S. Trofimchuk, Monotone traveling wavefronts of the KPP-Fisher delayed equation,, \emph{J. Differential Equations}, 250 (2011), 1767. doi: 10.1016/j.jde.2010.11.011. Google Scholar
[19]	G. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order,, Classics in Mathematics, (2001). Google Scholar
[20]	S. A. Gourley, Traveling front solutions of a nonlocal Fisher equation,, \emph{J. Math. Biol.}, 41 (2000), 272. doi: 10.1007/s002850000047. Google Scholar
[21]	S. A. Gourley and N. F. Britton, Instability of traveling wave solutions of a population model with nonlocal effects,, \emph{IMA J. Appl. Math.}, 51 (1993), 299. doi: 10.1093/imamat/51.3.299. Google Scholar
[22]	S. A. Gourley and N. F. Britton, On a modified Volterra population equation with diffusion,, \emph{Nonlinear Anal.}, 21 (1993), 389. doi: 10.1016/0362-546X(93)90082-4. Google Scholar
[23]	S. A. Gourley and N. F. Britton, A predator-prey reaction-diffusion system with nonlocal effects,, \emph{J. Math. Biol.}, 34 (1996), 297. doi: 10.1007/BF00160498. Google Scholar
[24]	S. A. Gourley, M. A. J. Chaplain and F. A. Davidson, Spatio-temporal pattern formation in a nonlocal reaction-diffusion equation,, \emph{Dyn. Syst.}, 16 (2001), 173. doi: 10.1080/14689360116914. Google Scholar
[25]	F. Hamel and L. Ryzhik, On the nonlocal Fisher-KPP equation: steady states, spreading speed and global bounds,, \emph{Nonlinearity}, 27 (2014), 2735. doi: 10.1088/0951-7715/27/11/2735. Google Scholar
[26]	G. Nadin, B. Perthame, L. Rossi and L. Ryzhik, Wave-like solutions for nonlocal reaction-diffusion equations: A toy model,, \emph{Math. Model. Nat. Phenom.}, 8 (2013), 33. doi: 10.1051/mmnp/20138304. Google Scholar
[27]	G. Nadin, B. Perthame and M. Tang, Can a traveling wave connect two unstable states? The case of the nonlocal Fisher equation,, \emph{C. R. Math. Acad. Sci. Paris}, 349 (2011), 553. doi: 10.1016/j.crma.2011.03.008. Google Scholar
[28]	D. H. Sattinger, On the stability of waves of nonlinear parabolic systems,, \emph{Adv. Math.}, 22 (1976), 312. doi: 10.1016/0001-8708(76)90098-0. Google Scholar
[29]	C. Ou and J. Wu, Traveling wavefronts in a delayed food-limited population model,, \emph{SIAM J. Math. Anal.}, 39 (2007), 103. doi: 10.1137/050638011. Google Scholar
[30]	A. Volpert, V. Volpert and V. Volpert, Traveling Wave Solutions of Parabolic Systems,, Translated from the Russian manuscript by James F. Heyda. Translations of Mathematical Monographs, (1994). Google Scholar
[31]	Z.-C. Wang and W.-T. Li, Monotone travelling fronts of a food-limited population model with nonlocal delay,, \emph{Nonlinear Anal. Real World Appl.}, 8 (2007), 699. doi: 10.1016/j.nonrwa.2006.03.001. Google Scholar
[32]	Z.-C. Wang and W.-T. Li, Traveling fronts in diffusive and cooperative Lotka-Volterra system with nonlocal delays,, \emph{Z. Angew. Math. Phys.}, 58 (2007), 571. doi: 10.1007/s00033-006-5125-4. Google Scholar
[33]	Z.-C. Wang, W.-T. Li and S. Ruan, Traveling wave fronts in reaction-diffusion systems with spatio-temporal delays,, \emph{J. Differential Equations}, 222 (2006), 185. doi: 10.1016/j.jde.2005.08.010. Google Scholar
[34]	Z.-C. Wang, W.-T. Li and S. Ruan, Existence and stability of traveling wave fronts in reaction advection diffusion equations with nonlocal delay,, \emph{J. Differential Equations}, 238 (2007), 153. doi: 10.1016/j.jde.2007.03.025. Google Scholar
[35]	Z.-C. Wang and J. Wu, Traveling waves in a bio-reactor model with stage-structure,, \emph{J. Math. Anal. Appl.}, 385 (2012), 683. doi: 10.1016/j.jmaa.2011.06.084. Google Scholar
[36]	Z.-C. Wang, J. Wu and R. Liu, Traveling waves of the spread of avian influenza,, \emph{Proc. Amer. Math. Soc.}, 140 (2012), 3931. doi: 10.1090/S0002-9939-2012-11246-8. Google Scholar
[37]	G. X. Yang and J. Xu, Analysis of spatiotemporal patterns in a single species reaction-diffusion model with spatiotemporal delay,, \emph{Nonlinear Anal. Real World Appl.}, 22 (2015), 54. doi: 10.1016/j.nonrwa.2014.07.013. Google Scholar
[38]	Q. Ye, Z. Li, M. Wang and Y. Wu, Introduction of Reaction-Diffusion Equations,, Science Publish, (2011). Google Scholar

[1]	Zhaosheng Feng. Traveling waves to a reaction-diffusion equation. Conference Publications, 2007, 2007 (Special) : 382-390. doi: 10.3934/proc.2007.2007.382
[2]	Yicheng Jiang, Kaijun Zhang. Stability of traveling waves for nonlocal time-delayed reaction-diffusion equations. Kinetic & Related Models, 2018, 11 (5) : 1235-1253. doi: 10.3934/krm.2018048
[3]	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 - A, 2017, 37 (6) : 3467-3486. doi: 10.3934/dcds.2017147
[4]	Shi-Liang Wu, Yu-Juan Sun, San-Yang Liu. Traveling fronts and entire solutions in partially degenerate reaction-diffusion systems with monostable nonlinearity. Discrete & Continuous Dynamical Systems - A, 2013, 33 (2) : 921-946. doi: 10.3934/dcds.2013.33.921
[5]	Manjun Ma, Xiao-Qiang Zhao. Monostable waves and spreading speed for a reaction-diffusion model with seasonal succession. Discrete & Continuous Dynamical Systems - B, 2016, 21 (2) : 591-606. doi: 10.3934/dcdsb.2016.21.591
[6]	Michaël Bages, Patrick Martinez. Existence of pulsating waves in a monostable reaction-diffusion system in solid combustion. Discrete & Continuous Dynamical Systems - B, 2010, 14 (3) : 817-869. doi: 10.3934/dcdsb.2010.14.817
[7]	Rui Huang, Ming Mei, Yong Wang. Planar traveling waves for nonlocal dispersion equation with monostable nonlinearity. Discrete & Continuous Dynamical Systems - A, 2012, 32 (10) : 3621-3649. doi: 10.3934/dcds.2012.32.3621
[8]	Samira Boussaïd, Danielle Hilhorst, Thanh Nam Nguyen. Convergence to steady state for the solutions of a nonlocal reaction-diffusion equation. Evolution Equations & Control Theory, 2015, 4 (1) : 39-59. doi: 10.3934/eect.2015.4.39
[9]	Zhao-Xing Yang, Guo-Bao Zhang, Ge Tian, Zhaosheng Feng. Stability of non-monotone non-critical traveling waves in discrete reaction-diffusion equations with time delay. Discrete & Continuous Dynamical Systems - S, 2017, 10 (3) : 581-603. doi: 10.3934/dcdss.2017029
[10]	Xiaojie Hou, Wei Feng. Traveling waves and their stability in a coupled reaction diffusion system. Communications on Pure & Applied Analysis, 2011, 10 (1) : 141-160. doi: 10.3934/cpaa.2011.10.141
[11]	Joaquin Riviera, Yi Li. Existence of traveling wave solutions for a nonlocal reaction-diffusion model of influenza a drift. Discrete & Continuous Dynamical Systems - B, 2010, 13 (1) : 157-174. doi: 10.3934/dcdsb.2010.13.157
[12]	Masaharu Taniguchi. Instability of planar traveling waves in bistable reaction-diffusion systems. Discrete & Continuous Dynamical Systems - B, 2003, 3 (1) : 21-44. doi: 10.3934/dcdsb.2003.3.21
[13]	Ming Mei, Yau Shu Wong. Novel stability results for traveling wavefronts in an age-structured reaction-diffusion equation. Mathematical Biosciences & Engineering, 2009, 6 (4) : 743-752. doi: 10.3934/mbe.2009.6.743
[14]	Xiaojie Hou, Yi Li. Local stability of traveling-wave solutions of nonlinear reaction-diffusion equations. Discrete & Continuous Dynamical Systems - A, 2006, 15 (2) : 681-701. doi: 10.3934/dcds.2006.15.681
[15]	Shi-Liang Wu, Wan-Tong Li, San-Yang Liu. Exponential stability of traveling fronts in monostable reaction-advection-diffusion equations with non-local delay. Discrete & Continuous Dynamical Systems - B, 2012, 17 (1) : 347-366. doi: 10.3934/dcdsb.2012.17.347
[16]	Ming Mei. Stability of traveling wavefronts for time-delayed reaction-diffusion equations. Conference Publications, 2009, 2009 (Special) : 526-535. doi: 10.3934/proc.2009.2009.526
[17]	Guo Lin, Haiyan Wang. Traveling wave solutions of a reaction-diffusion equation with state-dependent delay. Communications on Pure & Applied Analysis, 2016, 15 (2) : 319-334. doi: 10.3934/cpaa.2016.15.319
[18]	Costică Moroşanu. Stability and errors analysis of two iterative schemes of fractional steps type associated to a nonlinear reaction-diffusion equation. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 1-21. doi: 10.3934/dcdss.2020089
[19]	Lianzhang Bao, Zhengfang Zhou. Traveling wave solutions for a one dimensional model of cell-to-cell adhesion and diffusion with monostable reaction term. Discrete & Continuous Dynamical Systems - S, 2017, 10 (3) : 395-412. doi: 10.3934/dcdss.2017019
[20]	Grigori Chapiro, Lucas Furtado, Dan Marchesin, Stephen Schecter. Stability of interacting traveling waves in reaction-convection-diffusion systems. Conference Publications, 2015, 2015 (special) : 258-266. doi: 10.3934/proc.2015.0258

2018 Impact Factor: 0.925

American Institute of Mathematical Sciences