American Institute of Mathematical Sciences

Advanced
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

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 and 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, J. Differential Equations, 232 (2007), 104-133. doi: 10.1016/j.jde.2006.08.015.

[2]

M. Alfaro and J. Coville, Rapid traveling waves in the nonlocal Fisher equation connect two unstable states, Appl. Math. Lett., 25 (2012), 2095-2099. doi: 10.1016/j.aml.2012.05.006.

[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, Comm. Partial Differential Equations, 38 (2013), 2126-2154. doi: 10.1080/03605302.2013.828069.

[4]

M. Alfaro, J. Coville and G. Raoul, Bistable traveling waves for nonlocal reaction diffusion equations, Discrete Contin. Dyn. Syst., 34 (2014), 1775-1791. doi: 10.3934/dcds.2014.34.1775.

[5]

N. Apreutesei, A. Ducrot and V. Volpert, Traveling waves for integro-differential equations in population dynamics, Discrete Contin. Dyn. Syst. Ser. B, 11 (2009), 541-561. doi: 10.3934/dcdsb.2009.11.541.

[6]

D. G. Aronson and H. F. Weinberger, Multidimensional nonlinear diffusion arising in population genetics, Adv. in Math., 30 (1978), 33-76. doi: 10.1016/0001-8708(78)90130-5.

[7]

P. Ashwin, M. V. Bartuccelli, T. J. Bridges and S. A. Gourley, Traveling fronts for the KPP equation with spatio-temporal delay, Z. Angew. Math. Phys., 53 (2002), 103-122. doi: 10.1007/s00033-002-8145-8.

[8]

H. Berestycki, G. Nadin, B. Perthame and L. Ryzhik, The non-local Fisher-KPP equation: Travelling waves and steady states, Nonlinearity, 22 (2009), 2813-2844. doi: 10.1088/0951-7715/22/12/002.

[9]

J. Billingham, Dynamics of a strongly nonlocal reaction-diffusion population model, Nonlinearity, 17 (2004), 313-346. doi: 10.1088/0951-7715/17/1/018.

[10]

N. F. Britton, Aggregation and the competitive exclusion principle, J. Theoret. Biol., 136 (1989), 57-66. doi: 10.1016/S0022-5193(89)80189-4.

[11]

N. F. Britton, Spatial structures and periodic traveling waves in an integro-differential reaction-diffusion population model, SIAM J. Appl. Math., 50 (1990), 1663-1688. doi: 10.1137/0150099.

[12]

I. Demin and V. Volpert, Existence of waves for a nonlocal reaction-diffusion equation, Math. Model. Nat. Phenom., 5 (2010), 80-101. doi: 10.1051/mmnp/20105506.

[13]

K. Deng, On a nonlocal reaction-diffusion population model, Discrete Contin. Dyn. Syst. Ser. B, 9 (2008), 65-73. doi: 10.3934/dcdsb.2008.9.65.

[14]

G. Faye and M. Holzer, Modulated traveling fronts for a nonlocal Fisher-KPP equation: a dynamical systems approach, J. Differential Equations, 258 (2015), 2257-2289. doi: 10.1016/j.jde.2014.12.006.

[15]

J. Fang and X.-Q. Zhao, Monotone wave fronts of the nonlocal Fisher-KPP equation, Nonlinearity, 24 (2011), 3043-3054. doi: 10.1088/0951-7715/24/11/002.

[16]

S. Genieys and B. Perthame, Concentration in the nonlocal Fisher equation: the Hamilton-Jacobi limit, Math. Model. Nat. Phenom., 2 (2007), 135-151. doi: 10.1051/mmnp:2008029.

[17]

S. Genieys, V. Volpert and P. Auger, Pattern and waves for a model in population dynamics with nonlocal consumption of resources, Math. Model. Nat. Phenom., 1 (2006), 65-82. doi: 10.1051/mmnp:2006004.

[18]

A. Gomez and S. Trofimchuk, Monotone traveling wavefronts of the KPP-Fisher delayed equation, J. Differential Equations, 250 (2011), 1767-1787. doi: 10.1016/j.jde.2010.11.011.

[19]

G. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Classics in Mathematics, Springer-Verlag, Berlin, 2001.

[20]

S. A. Gourley, Traveling front solutions of a nonlocal Fisher equation, J. Math. Biol., 41 (2000), 272-284. doi: 10.1007/s002850000047.

[21]

S. A. Gourley and N. F. Britton, Instability of traveling wave solutions of a population model with nonlocal effects, IMA J. Appl. Math., 51 (1993), 299-310. doi: 10.1093/imamat/51.3.299.

[22]

S. A. Gourley and N. F. Britton, On a modified Volterra population equation with diffusion, Nonlinear Anal., 21 (1993), 389-395. doi: 10.1016/0362-546X(93)90082-4.

[23]

S. A. Gourley and N. F. Britton, A predator-prey reaction-diffusion system with nonlocal effects, J. Math. Biol., 34 (1996), 297-333. doi: 10.1007/BF00160498.

[24]

S. A. Gourley, M. A. J. Chaplain and F. A. Davidson, Spatio-temporal pattern formation in a nonlocal reaction-diffusion equation, Dyn. Syst., 16 (2001), 173-192. doi: 10.1080/14689360116914.

[25]

F. Hamel and L. Ryzhik, On the nonlocal Fisher-KPP equation: steady states, spreading speed and global bounds, Nonlinearity, 27 (2014), 2735-2753. doi: 10.1088/0951-7715/27/11/2735.

[26]

G. Nadin, B. Perthame, L. Rossi and L. Ryzhik, Wave-like solutions for nonlocal reaction-diffusion equations: A toy model, Math. Model. Nat. Phenom., 8 (2013), 33-41. doi: 10.1051/mmnp/20138304.

[27]

G. Nadin, B. Perthame and M. Tang, Can a traveling wave connect two unstable states? The case of the nonlocal Fisher equation, C. R. Math. Acad. Sci. Paris, 349 (2011), 553-557. doi: 10.1016/j.crma.2011.03.008.

[28]

D. H. Sattinger, On the stability of waves of nonlinear parabolic systems, Adv. Math., 22 (1976), 312-355. doi: 10.1016/0001-8708(76)90098-0.

[29]

C. Ou and J. Wu, Traveling wavefronts in a delayed food-limited population model, SIAM J. Math. Anal., 39 (2007), 103-125. doi: 10.1137/050638011.

[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, 140. American Mathematical Society, Providence, RI, 1994.

[31]

Z.-C. Wang and W.-T. Li, Monotone travelling fronts of a food-limited population model with nonlocal delay, Nonlinear Anal. Real World Appl., 8 (2007), 699-712. doi: 10.1016/j.nonrwa.2006.03.001.

[32]

Z.-C. Wang and W.-T. Li, Traveling fronts in diffusive and cooperative Lotka-Volterra system with nonlocal delays, Z. Angew. Math. Phys., 58 (2007), 571-591. doi: 10.1007/s00033-006-5125-4.

[33]

Z.-C. Wang, W.-T. Li and S. Ruan, Traveling wave fronts in reaction-diffusion systems with spatio-temporal delays, J. Differential Equations, 222 (2006), 185-232. doi: 10.1016/j.jde.2005.08.010.

[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, J. Differential Equations, 238 (2007), 153-200. doi: 10.1016/j.jde.2007.03.025.

[35]

Z.-C. Wang and J. Wu, Traveling waves in a bio-reactor model with stage-structure, J. Math. Anal. Appl., 385 (2012), 683-692. doi: 10.1016/j.jmaa.2011.06.084.

[36]

Z.-C. Wang, J. Wu and R. Liu, Traveling waves of the spread of avian influenza, Proc. Amer. Math. Soc., 140 (2012), 3931-3946. doi: 10.1090/S0002-9939-2012-11246-8.

[37]

G. X. Yang and J. Xu, Analysis of spatiotemporal patterns in a single species reaction-diffusion model with spatiotemporal delay, Nonlinear Anal. Real World Appl., 22 (2015), 54-65. doi: 10.1016/j.nonrwa.2014.07.013.

[38]

Q. Ye, Z. Li, M. Wang and Y. Wu, Introduction of Reaction-Diffusion Equations, Science Publish, Beijing, 2011.

show all references

References:
[1]

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.

[2]

M. Alfaro and J. Coville, Rapid traveling waves in the nonlocal Fisher equation connect two unstable states, Appl. Math. Lett., 25 (2012), 2095-2099. doi: 10.1016/j.aml.2012.05.006.

[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, Comm. Partial Differential Equations, 38 (2013), 2126-2154. doi: 10.1080/03605302.2013.828069.

[4]

M. Alfaro, J. Coville and G. Raoul, Bistable traveling waves for nonlocal reaction diffusion equations, Discrete Contin. Dyn. Syst., 34 (2014), 1775-1791. doi: 10.3934/dcds.2014.34.1775.

[5]

N. Apreutesei, A. Ducrot and V. Volpert, Traveling waves for integro-differential equations in population dynamics, Discrete Contin. Dyn. Syst. Ser. B, 11 (2009), 541-561. doi: 10.3934/dcdsb.2009.11.541.

[6]

D. G. Aronson and H. F. Weinberger, Multidimensional nonlinear diffusion arising in population genetics, Adv. in Math., 30 (1978), 33-76. doi: 10.1016/0001-8708(78)90130-5.

[7]

P. Ashwin, M. V. Bartuccelli, T. J. Bridges and S. A. Gourley, Traveling fronts for the KPP equation with spatio-temporal delay, Z. Angew. Math. Phys., 53 (2002), 103-122. doi: 10.1007/s00033-002-8145-8.

[8]

H. Berestycki, G. Nadin, B. Perthame and L. Ryzhik, The non-local Fisher-KPP equation: Travelling waves and steady states, Nonlinearity, 22 (2009), 2813-2844. doi: 10.1088/0951-7715/22/12/002.

[9]

J. Billingham, Dynamics of a strongly nonlocal reaction-diffusion population model, Nonlinearity, 17 (2004), 313-346. doi: 10.1088/0951-7715/17/1/018.

[10]

N. F. Britton, Aggregation and the competitive exclusion principle, J. Theoret. Biol., 136 (1989), 57-66. doi: 10.1016/S0022-5193(89)80189-4.

[11]

N. F. Britton, Spatial structures and periodic traveling waves in an integro-differential reaction-diffusion population model, SIAM J. Appl. Math., 50 (1990), 1663-1688. doi: 10.1137/0150099.

[12]

I. Demin and V. Volpert, Existence of waves for a nonlocal reaction-diffusion equation, Math. Model. Nat. Phenom., 5 (2010), 80-101. doi: 10.1051/mmnp/20105506.

[13]

K. Deng, On a nonlocal reaction-diffusion population model, Discrete Contin. Dyn. Syst. Ser. B, 9 (2008), 65-73. doi: 10.3934/dcdsb.2008.9.65.

[14]

G. Faye and M. Holzer, Modulated traveling fronts for a nonlocal Fisher-KPP equation: a dynamical systems approach, J. Differential Equations, 258 (2015), 2257-2289. doi: 10.1016/j.jde.2014.12.006.

[15]

J. Fang and X.-Q. Zhao, Monotone wave fronts of the nonlocal Fisher-KPP equation, Nonlinearity, 24 (2011), 3043-3054. doi: 10.1088/0951-7715/24/11/002.

[16]

S. Genieys and B. Perthame, Concentration in the nonlocal Fisher equation: the Hamilton-Jacobi limit, Math. Model. Nat. Phenom., 2 (2007), 135-151. doi: 10.1051/mmnp:2008029.

[17]

S. Genieys, V. Volpert and P. Auger, Pattern and waves for a model in population dynamics with nonlocal consumption of resources, Math. Model. Nat. Phenom., 1 (2006), 65-82. doi: 10.1051/mmnp:2006004.

[18]

A. Gomez and S. Trofimchuk, Monotone traveling wavefronts of the KPP-Fisher delayed equation, J. Differential Equations, 250 (2011), 1767-1787. doi: 10.1016/j.jde.2010.11.011.

[19]

G. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Classics in Mathematics, Springer-Verlag, Berlin, 2001.

[20]

S. A. Gourley, Traveling front solutions of a nonlocal Fisher equation, J. Math. Biol., 41 (2000), 272-284. doi: 10.1007/s002850000047.

[21]

S. A. Gourley and N. F. Britton, Instability of traveling wave solutions of a population model with nonlocal effects, IMA J. Appl. Math., 51 (1993), 299-310. doi: 10.1093/imamat/51.3.299.

[22]

S. A. Gourley and N. F. Britton, On a modified Volterra population equation with diffusion, Nonlinear Anal., 21 (1993), 389-395. doi: 10.1016/0362-546X(93)90082-4.

[23]

S. A. Gourley and N. F. Britton, A predator-prey reaction-diffusion system with nonlocal effects, J. Math. Biol., 34 (1996), 297-333. doi: 10.1007/BF00160498.

[24]

S. A. Gourley, M. A. J. Chaplain and F. A. Davidson, Spatio-temporal pattern formation in a nonlocal reaction-diffusion equation, Dyn. Syst., 16 (2001), 173-192. doi: 10.1080/14689360116914.

[25]

F. Hamel and L. Ryzhik, On the nonlocal Fisher-KPP equation: steady states, spreading speed and global bounds, Nonlinearity, 27 (2014), 2735-2753. doi: 10.1088/0951-7715/27/11/2735.

[26]

G. Nadin, B. Perthame, L. Rossi and L. Ryzhik, Wave-like solutions for nonlocal reaction-diffusion equations: A toy model, Math. Model. Nat. Phenom., 8 (2013), 33-41. doi: 10.1051/mmnp/20138304.

[27]

G. Nadin, B. Perthame and M. Tang, Can a traveling wave connect two unstable states? The case of the nonlocal Fisher equation, C. R. Math. Acad. Sci. Paris, 349 (2011), 553-557. doi: 10.1016/j.crma.2011.03.008.

[28]

D. H. Sattinger, On the stability of waves of nonlinear parabolic systems, Adv. Math., 22 (1976), 312-355. doi: 10.1016/0001-8708(76)90098-0.

[29]

C. Ou and J. Wu, Traveling wavefronts in a delayed food-limited population model, SIAM J. Math. Anal., 39 (2007), 103-125. doi: 10.1137/050638011.

[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, 140. American Mathematical Society, Providence, RI, 1994.

[31]

Z.-C. Wang and W.-T. Li, Monotone travelling fronts of a food-limited population model with nonlocal delay, Nonlinear Anal. Real World Appl., 8 (2007), 699-712. doi: 10.1016/j.nonrwa.2006.03.001.

[32]

Z.-C. Wang and W.-T. Li, Traveling fronts in diffusive and cooperative Lotka-Volterra system with nonlocal delays, Z. Angew. Math. Phys., 58 (2007), 571-591. doi: 10.1007/s00033-006-5125-4.

[33]

Z.-C. Wang, W.-T. Li and S. Ruan, Traveling wave fronts in reaction-diffusion systems with spatio-temporal delays, J. Differential Equations, 222 (2006), 185-232. doi: 10.1016/j.jde.2005.08.010.

[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, J. Differential Equations, 238 (2007), 153-200. doi: 10.1016/j.jde.2007.03.025.

[35]

Z.-C. Wang and J. Wu, Traveling waves in a bio-reactor model with stage-structure, J. Math. Anal. Appl., 385 (2012), 683-692. doi: 10.1016/j.jmaa.2011.06.084.

[36]

Z.-C. Wang, J. Wu and R. Liu, Traveling waves of the spread of avian influenza, Proc. Amer. Math. Soc., 140 (2012), 3931-3946. doi: 10.1090/S0002-9939-2012-11246-8.

[37]

G. X. Yang and J. Xu, Analysis of spatiotemporal patterns in a single species reaction-diffusion model with spatiotemporal delay, Nonlinear Anal. Real World Appl., 22 (2015), 54-65. doi: 10.1016/j.nonrwa.2014.07.013.

[38]

Q. Ye, Z. Li, M. Wang and Y. Wu, Introduction of Reaction-Diffusion Equations, Science Publish, Beijing, 2011.

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