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February 2017, 37(2): 685-723. doi: 10.3934/dcds.2017029

## A non-local bistable reaction-diffusion equation with a gap

 1 École des hautes études en sciences sociales, PSL Research University, and CNRS, CAMS, 190-198 avenue de France, F-75244 Paris Cedex 13, France 2 UNC Chapel Hill, Department of Mathematics, Phillips Hall, CB#3250, Chapel Hill, NC 27599-3250, USA

Received  August 2015 Revised  November 2016 Published  November 2016

Non-local reaction-diffusion equations arise naturally to account for diffusions involving jumps rather than local diffusions related to Brownian motion. In ecology, long distance dispersal require such frameworks. In this work we study a one-dimensional non-local reaction-diffusion equation with bistable type reaction. The heterogeneity here is due to a gap, some finite region where there is decay. Outside this gap region the equation is a classical homogeneous (space independent) non-local reaction-diffusion equation. This type of problem is motivated by applications in ecology, sociology, and physiology. We first establish the existence of a generalized traveling front that approaches a traveling wave solution as t-∞, propagating in a heterogeneous environment. We then study the problem of obstruction of solutions. In particular, we study the propagation properties of the generalized traveling front with significant use of the work of Bates, Fife, Ren and Wang in [5]. As in the local diffusion case, we prove that obstruction is possible if the gap is sufficiently large. An interesting difference between the local dispersal and the non-local dispersal is that in the latter the obstructing steady states are discontinuous. We characterize these jump discontinuities and discuss the scaling between the range of the dispersal and the critical length of the gap observed numerically. We further explore other differences between the local and the non-local dispersal cases. In this paper, we illustrate these properties by numerical simulations and we state a series of open problems.

Citation: Henri Berestycki, Nancy Rodríguez. A non-local bistable reaction-diffusion equation with a gap. Discrete & Continuous Dynamical Systems - A, 2017, 37 (2) : 685-723. doi: 10.3934/dcds.2017029
##### References:
 [1] G. Alberti and G. Bellettini, A non-local anisotropic model for phase transitions: Asymptotic behavior of rescaled energies [extended version], European Journal of Applied Mathematics, 9 (1998), 261-284. doi: 10.1017/S0956792598003453. [2] D. G. Aronson and H. F. Weinberger, Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation, in Partial differential equations and related topics, Springer, Berlin, Lecture Notes in Math. , 446 (1975), 5-49. [3] D. G. Aronson and H. F. Weinberger, Multidimentional non-linear diffusion arising in population genetics, Advances in Mathematics Math., 30 (1978), 33-76. doi: 10.1016/0001-8708(78)90130-5. [4] P. W. Bates and A. Chmaj, An integrodifferential model for phase transitions: Stationary solutions in higher space dimensions, Journal of statistical physics, 95 (1999), 1119-1139. doi: 10.1023/A:1004514803625. [5] P. W. Bates, P. Fife, X. Ren and X. Wang, Traveling waves in a convolution model for phase transitions, Archive for Rational Mechanics and Analysis, 138 (1997), 105-136. doi: 10.1007/s002050050037. [6] H. Berestycki and F. Hamel, Fronts and invasions in general domains, C. R. Math. Acad. Sci. Paris, 343 (2006), 711-716. doi: 10.1016/j.crma.2006.09.036. [7] H. Berestycki and F. Hamel, Generalized travelling waves for reaction-diffusion equations, Contemporary Mathematics, 446 (2007), 101-123. doi: 10.1090/conm/446/08627. [8] H. Berestycki, B. Larrouturou and P. L. Lions, Multi-dimensional travelling-wave solutions of a flame propagation model, Archive for Rational Mechanics and Analysis, 111 (1990), 33-49. doi: 10.1007/BF00375699. [9] H. Berestycki, H. Matano and F. Hamel, Bistable traveling waves around an obstacle, Communications on Pure and Applied Mathematics, 62 (2009), 729-788. doi: 10.1002/cpa.20275. [10] H. Berestycki, B. Nicolaenko and B. Scheurer, Traveling wave solutions to combustion models and their singular limits, SIAM Journal on Mathematical Analysis, 16 (1985), 1207-1242. doi: 10.1137/0516088. [11] H. Berestycki and L. Nirenberg, On the method of moving planes and the sliding method, Boletim da Sociedade Brasileira de Matematica, 22 (1991), 1-37. doi: 10.1007/BF01244896. [12] H. Berestycki, N. Rodríguez and L. Ryzhik, Traveling wave solutions in a reaction-diffusion model for criminal activity, Multiscale Modeling & Simulation, 11 (2013), 1097-1126. [13] H. Berestycki, J.-M. Roquejoffre and L. Rossi, The periodic patch model for population dynamics with fractional diffusion, Discrete Contin. Dyn. Syst. Ser. S, 4 (2011), 1-13. doi: 10.3934/dcdss.2011.4.1. [14] E. Chasseigne, M. Chaves and J. D. Rossi, Asymptotic behavior for nonlocal diffusion equations, Journal de Mathématiques Pures et Appliquées, 86 (2006), 271-291. doi: 10.1016/j.matpur.2006.04.005. [15] F. Chen, Almost periodic traveling waves of nonlocal evolution equations, Nonlinear Analysis, Theory, Methods and Applications, 50 (2002), 807-838. doi: 10.1016/S0362-546X(01)00787-8. [16] X. Chen, Existence, uniqueness, and asymptotic stability of traveling waves in nonlocal evolution equations, Adv. Differential Equations, 2 (1997), 125-160. [17] C. Cortazar, M. Elgueta and J. D. Rossi, Nonlocal diffusion problems that approximate the heat equation with Dirichlet boundary conditions, Israel Journal of Mathematics, 170 (2009), 53-60. [18] J. Coville, Travelling fronts in asymmetric nonlocal reaction diffusion equations: The bistable and ignition cases, Prépublication du CMM, Hal-006962 (2007), 1-43. [19] J. Coville, J. Dávila and S. Martínez, Nonlocal anisotropic dispersal with monostable nonlinearity, Journal of Differential Equations, 244 (2008), 3080-3118. doi: 10.1016/j.jde.2007.11.002. [20] J. Coville and L. Dupaigne, On a non-local equation arising in population dynamics, Proc. Roy. Soc. Edinburgh Sect. A., 137 (2007), 727-755. [21] G. B. Ermentrout and J. B. McLeod, Existence and uniqueness of travelling waves for a neural network, Proceedings of the Royal Society of Edinburgh, 123 (1993), 461-478. [22] P. C. Fife, An integrodifferential analog of semilinear parabolic PDEs, Partial differential equations and applications, Lecture Notes in Pure and Appl. Math., 177 (1996), 137-145. doi: 10.3109/14659899609084991. [23] P. C. Fife and J. B. McLeod, The approach of solutions of nonlinear diffusion equations to travelling front solutions, Arch. Ration. Mech. Anal., 65 (1977), 335-361. [24] P. C. Fife and X. Wang, A convolution model for interfacial motion: The generation and propagation of internal layers in higher space dimensions, Adv. Differential Equations, 3 (1998), 85-110. [25] R. A. Fisher, The wave of advance of advantageous genes, Ann. Eugenics, 7 (1937), 353-369. [26] J. Garnier, Accelerating solutions in integro-differential equations, SIAM J. Appl. Math., 43 (2011), 1955-1974. doi: 10.1137/10080693X. [27] V. Hutson and S. Martinez, The evolution of dispersal, Journal of Math. Bio, 47 (2003), 483-517. [28] J. Jacobsen, Y. Jin and M. A. Lewis, Integrodifference models for persistence in temporally varying river environments, Journal of Mathematical Biology, 70 (2015), 549-590. doi: 10.1007/s00285-014-0774-y. [29] Y. Kanel', Certain problems of burning-theory equations, Soviet Mathematics-Doklady, 2 (1961), 48-51. [30] T. Lewis and J. Keener, Wave-block in excitable media due to regions of depressed excitability, SIAM J. Appl. Math., 61 (2000), 293-316. doi: 10.1137/S0036139998349298. [31] T. A. Lim and A. Zlatos, Transition fronts for inhomogeneous Fisher-KPP reactions and non-local diffusion, Trans. Amer. Math. Soc., 368 (2016), 8615-8631. doi: 10.1090/tran/6602. [32] H. Matano, Traveling waves in spatially random media, RIMS Kokyuroku, 1337 (2003), 1-9. [33] R. Meaney, Commuters and Marauders: An examination of the spatial behavior of serial criminals, Journal of Investigative Psychology and Offender Profiling, 1 (2004), 121-137. doi: 10.1002/jip.12. [34] B. Perthame and P. E. Souganidis, Front propagation for a jump process model arising in spatial ecology, Discrete Contin. Dyn. Syst., 13 (2005), 1235-1246. doi: 10.3934/dcds.2005.13.1235. [35] J. Riviera, Traveling wave solutions for a nonlocal reaction-diffusion model of influenza A Drift, DCDS-B, 13 (2010), 157-174. doi: 10.3934/dcdsb.2010.13.157. [36] N. Rodríguez, On an integro-differential model for pest control in a heterogeneous environment, Journal of Mathematical Biology, 70 (2014), 1177-1206. [37] K. Schumacher, Travelling-front solutions for integro-differential equations. Ⅰ, J. Reine Angew. Math., 316 (1980), 54-70. doi: 10.1515/crll.1980.316.54. [38] Y.-J. Sun, W.-T. Li and Z.-C. Wang, Entire solutions in nonlocal dispersal equations with bistable nonlinearity, Journal of Differential Equations, 251 (2011), 551-581. [39] H. Yagisita, Existence and nonexistence of traveling waves for a nonlocal monostable equation, Publications of the Research Institute for Mathematical Sciences, 45 (2009), 925-953. doi: 10.2977/prims/1260476648. [40] H. Yagisita, Existence of traveling wave solutions for a nonlocal bistable equation: An abstract approach, Publ. RIMS, Kyoto Univ., 45 (2009), 955-979.

show all references

##### References:
 [1] G. Alberti and G. Bellettini, A non-local anisotropic model for phase transitions: Asymptotic behavior of rescaled energies [extended version], European Journal of Applied Mathematics, 9 (1998), 261-284. doi: 10.1017/S0956792598003453. [2] D. G. Aronson and H. F. Weinberger, Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation, in Partial differential equations and related topics, Springer, Berlin, Lecture Notes in Math. , 446 (1975), 5-49. [3] D. G. Aronson and H. F. Weinberger, Multidimentional non-linear diffusion arising in population genetics, Advances in Mathematics Math., 30 (1978), 33-76. doi: 10.1016/0001-8708(78)90130-5. [4] P. W. Bates and A. Chmaj, An integrodifferential model for phase transitions: Stationary solutions in higher space dimensions, Journal of statistical physics, 95 (1999), 1119-1139. doi: 10.1023/A:1004514803625. [5] P. W. Bates, P. Fife, X. Ren and X. Wang, Traveling waves in a convolution model for phase transitions, Archive for Rational Mechanics and Analysis, 138 (1997), 105-136. doi: 10.1007/s002050050037. [6] H. Berestycki and F. Hamel, Fronts and invasions in general domains, C. R. Math. Acad. Sci. Paris, 343 (2006), 711-716. doi: 10.1016/j.crma.2006.09.036. [7] H. Berestycki and F. Hamel, Generalized travelling waves for reaction-diffusion equations, Contemporary Mathematics, 446 (2007), 101-123. doi: 10.1090/conm/446/08627. [8] H. Berestycki, B. Larrouturou and P. L. Lions, Multi-dimensional travelling-wave solutions of a flame propagation model, Archive for Rational Mechanics and Analysis, 111 (1990), 33-49. doi: 10.1007/BF00375699. [9] H. Berestycki, H. Matano and F. Hamel, Bistable traveling waves around an obstacle, Communications on Pure and Applied Mathematics, 62 (2009), 729-788. doi: 10.1002/cpa.20275. [10] H. Berestycki, B. Nicolaenko and B. Scheurer, Traveling wave solutions to combustion models and their singular limits, SIAM Journal on Mathematical Analysis, 16 (1985), 1207-1242. doi: 10.1137/0516088. [11] H. Berestycki and L. Nirenberg, On the method of moving planes and the sliding method, Boletim da Sociedade Brasileira de Matematica, 22 (1991), 1-37. doi: 10.1007/BF01244896. [12] H. Berestycki, N. Rodríguez and L. Ryzhik, Traveling wave solutions in a reaction-diffusion model for criminal activity, Multiscale Modeling & Simulation, 11 (2013), 1097-1126. [13] H. Berestycki, J.-M. Roquejoffre and L. Rossi, The periodic patch model for population dynamics with fractional diffusion, Discrete Contin. Dyn. Syst. Ser. S, 4 (2011), 1-13. doi: 10.3934/dcdss.2011.4.1. [14] E. Chasseigne, M. Chaves and J. D. Rossi, Asymptotic behavior for nonlocal diffusion equations, Journal de Mathématiques Pures et Appliquées, 86 (2006), 271-291. doi: 10.1016/j.matpur.2006.04.005. [15] F. Chen, Almost periodic traveling waves of nonlocal evolution equations, Nonlinear Analysis, Theory, Methods and Applications, 50 (2002), 807-838. doi: 10.1016/S0362-546X(01)00787-8. [16] X. Chen, Existence, uniqueness, and asymptotic stability of traveling waves in nonlocal evolution equations, Adv. Differential Equations, 2 (1997), 125-160. [17] C. Cortazar, M. Elgueta and J. D. Rossi, Nonlocal diffusion problems that approximate the heat equation with Dirichlet boundary conditions, Israel Journal of Mathematics, 170 (2009), 53-60. [18] J. Coville, Travelling fronts in asymmetric nonlocal reaction diffusion equations: The bistable and ignition cases, Prépublication du CMM, Hal-006962 (2007), 1-43. [19] J. Coville, J. Dávila and S. Martínez, Nonlocal anisotropic dispersal with monostable nonlinearity, Journal of Differential Equations, 244 (2008), 3080-3118. doi: 10.1016/j.jde.2007.11.002. [20] J. Coville and L. Dupaigne, On a non-local equation arising in population dynamics, Proc. Roy. Soc. Edinburgh Sect. A., 137 (2007), 727-755. [21] G. B. Ermentrout and J. B. McLeod, Existence and uniqueness of travelling waves for a neural network, Proceedings of the Royal Society of Edinburgh, 123 (1993), 461-478. [22] P. C. Fife, An integrodifferential analog of semilinear parabolic PDEs, Partial differential equations and applications, Lecture Notes in Pure and Appl. Math., 177 (1996), 137-145. doi: 10.3109/14659899609084991. [23] P. C. Fife and J. B. McLeod, The approach of solutions of nonlinear diffusion equations to travelling front solutions, Arch. Ration. Mech. Anal., 65 (1977), 335-361. [24] P. C. Fife and X. Wang, A convolution model for interfacial motion: The generation and propagation of internal layers in higher space dimensions, Adv. Differential Equations, 3 (1998), 85-110. [25] R. A. Fisher, The wave of advance of advantageous genes, Ann. Eugenics, 7 (1937), 353-369. [26] J. Garnier, Accelerating solutions in integro-differential equations, SIAM J. Appl. Math., 43 (2011), 1955-1974. doi: 10.1137/10080693X. [27] V. Hutson and S. Martinez, The evolution of dispersal, Journal of Math. Bio, 47 (2003), 483-517. [28] J. Jacobsen, Y. Jin and M. A. Lewis, Integrodifference models for persistence in temporally varying river environments, Journal of Mathematical Biology, 70 (2015), 549-590. doi: 10.1007/s00285-014-0774-y. [29] Y. Kanel', Certain problems of burning-theory equations, Soviet Mathematics-Doklady, 2 (1961), 48-51. [30] T. Lewis and J. Keener, Wave-block in excitable media due to regions of depressed excitability, SIAM J. Appl. Math., 61 (2000), 293-316. doi: 10.1137/S0036139998349298. [31] T. A. Lim and A. Zlatos, Transition fronts for inhomogeneous Fisher-KPP reactions and non-local diffusion, Trans. Amer. Math. Soc., 368 (2016), 8615-8631. doi: 10.1090/tran/6602. [32] H. Matano, Traveling waves in spatially random media, RIMS Kokyuroku, 1337 (2003), 1-9. [33] R. Meaney, Commuters and Marauders: An examination of the spatial behavior of serial criminals, Journal of Investigative Psychology and Offender Profiling, 1 (2004), 121-137. doi: 10.1002/jip.12. [34] B. Perthame and P. E. Souganidis, Front propagation for a jump process model arising in spatial ecology, Discrete Contin. Dyn. Syst., 13 (2005), 1235-1246. doi: 10.3934/dcds.2005.13.1235. [35] J. Riviera, Traveling wave solutions for a nonlocal reaction-diffusion model of influenza A Drift, DCDS-B, 13 (2010), 157-174. doi: 10.3934/dcdsb.2010.13.157. [36] N. Rodríguez, On an integro-differential model for pest control in a heterogeneous environment, Journal of Mathematical Biology, 70 (2014), 1177-1206. [37] K. Schumacher, Travelling-front solutions for integro-differential equations. Ⅰ, J. Reine Angew. Math., 316 (1980), 54-70. doi: 10.1515/crll.1980.316.54. [38] Y.-J. Sun, W.-T. Li and Z.-C. Wang, Entire solutions in nonlocal dispersal equations with bistable nonlinearity, Journal of Differential Equations, 251 (2011), 551-581. [39] H. Yagisita, Existence and nonexistence of traveling waves for a nonlocal monostable equation, Publications of the Research Institute for Mathematical Sciences, 45 (2009), 925-953. doi: 10.2977/prims/1260476648. [40] H. Yagisita, Existence of traveling wave solutions for a nonlocal bistable equation: An abstract approach, Publ. RIMS, Kyoto Univ., 45 (2009), 955-979.
Results from numerical simulations for the gap region problem with probability distribution satisfying $(H1)$ with $\sigma =3$ and $\sigma=5$. Figure (1a) displays a gap region of size 10. In Figure (1b) the length of the gap region is 15. Note that the gap has been shifted for the numerical computations.
Illustration of the change in the right-most value of the support of the subsolutions. The $x-axis$ provides the center of the support of the subsolutions. Figure 2a illustrates a case when $L=5.$ Here, one only observes the expected discontinuities, when $x=0$ (corresponding to $(i)$ in the figure) and when $x=L$ enter the domain (corresponding to $(ii)$ in the Figure). Figure 2b illustrates the case when $L=12$. On the contrary to the previous case, we see that there is an additional discontinuity (corresponding to $(iii)$)
Numerical solutions to the Cauchy problem (2.10) with a step function initial condition at time $t=200$. Figure 3a illustrates the case when the gap is too small to prevent propagation (L = 5). Figure 3a illustrates the case when the gap is sufficiently large to obstruct propagation (L = 12).
Family of subsolutions as the domain is shifted from left to right. Here, $L=12$ and the discontinuity can be seen clearly in sequence of subfigures 8-10.
Family of subsolutions as the domain is shifted from left to right. Here, L = 12 and the discontinuity can be seen clearly in sequence of subfigures 8-10.
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