# American Institute of Mathematical Sciences

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

## Traveling wave solutions in a nonlocal reaction-diffusion population model

 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.
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

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##### 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
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