# American Institute of Mathematical Sciences

April  2019, 39(4): 2077-2100. doi: 10.3934/dcds.2019087

## Pattern formation in the doubly-nonlocal Fisher-KPP equation

 1 Technical University of Munich, Faculty of Mathematics, Boltzmannstr. 3, 85748 Garching bei München, Germany 2 Gran Sasso Science Institute, Viale Francesco Crispi, 7, 67100 L'Aquila, Italy

Received  May 2018 Published  January 2019

Fund Project: CK would like to thank the VolkswagenStiftung for support via a Lichtenberg Professorship. PT wishes to express his gratitude to the "Bielefeld Young Researchers" Fund for the support through the Funding Line Postdocs: "Career Bridge Doctorate – Postdoc".

We study the existence, bifurcations, and stability of stationary solutions for the doubly-nonlocal Fisher-KPP equation. We prove using Lyapunov-Schmidt reduction that under suitable conditions on the parameters, a bifurcation from the non-trivial homogeneous state can occur. The kernel of the linearized operator at the bifurcation is two-dimensional and periodic stationary patterns are generated. Then we prove that these patterns are, again under suitable conditions, locally asymptotically stable. We also compare our results to previous work on the nonlocal Fisher-KPP equation containing a local diffusion term and a nonlocal reaction term. If the diffusion is approximated by a nonlocal kernel, we show that our results are consistent and reduce to the local ones in the local singular diffusion limit. Furthermore, we prove that there are parameter regimes, where no bifurcations can occur for the doubly-nonlocal Fisher-KPP equation. The results demonstrate that intricate different parameter regimes are possible. In summary, our results provide a very detailed classification of the multi-parameter dependence of the stationary solutions for the doubly-nonlocal Fisher-KPP equation.

Citation: Christian Kuehn, Pasha Tkachov. Pattern formation in the doubly-nonlocal Fisher-KPP equation. Discrete and Continuous Dynamical Systems, 2019, 39 (4) : 2077-2100. doi: 10.3934/dcds.2019087
##### References:
 [1] F. Achleitner and C. Kuehn, On Bounded Positive Stationary Solutions for a Nonlocal Fisher-KPP Equation, Nonlinear Anal., 112 (2015), 15-29.  doi: 10.1016/j.na.2014.09.004. [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] N. Apreutesei, N. Bessonov, V. Volpert and V. Vougalter, Spatial structures and generalized travelling waves for an integro-differential equation, Discrete Continuous Dynam. Systems - B, 13 (2010), 537-557. doi: 10.3934/dcdsb.2010.13.537. [4] P. Auger, S. Genieys and V. Volpert, Pattern and Waves for a Model in Population Dynamics with Nonlocal Consumption of Resources, Mathematical Modelling of Natural Phenomena, 1 (2006), 65-82.  doi: 10.1051/mmnp:2006004. [5] O. Aydogmus, Phase transitions in a logistic metapopulation model with nonlocal interactions, Bull Math Biol, 80 (2018), 228-253.  doi: 10.1007/s11538-017-0373-3. [6] F. Barbosa, A. Penna, R. Ferreira, K. Novais, J. da Cunha and F. Oliveira, Pattern transitions and complexity for a nonlocal logistic map, Physica A, 473 (2017), 301-312.  doi: 10.1016/j.physa.2016.12.082. [7] 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. [8] B. Bolker and S. W. Pacala, Using moment equations to understand stochastically driven spatial pattern formation in ecological systems, Theor. Popul. Biol., 52 (1997), 179-197.  doi: 10.1006/tpbi.1997.1331. [9] E. Bouin, J. Garnier, C. Henderson and F. Patout, Thin front limit of an integro–differential Fisher–KPP equation with fat–tailed kernels, SIAM J. Math. Anal., 50 (2018), 3365-3394.  doi: 10.1137/17M1132501. [10] H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer, New York, 2011. [11] N. Britton, Spatial structures and periodic travelling waves in an integro-differential reaction-diffusion population model, SIAM J. Appl. Math., 50 (1990), 1663-1688.  doi: 10.1137/0150099. [12] N. Britton, Aggregation and the competitive exclusion principle, Journal of Theoretical Biology, 136 (1989), 57-66.  doi: 10.1016/S0022-5193(89)80189-4. [13] J. Coville and L. Dupaigne, On a non-local equation arising in population dynamics, Proc. R. Soc. A, 137 (2007), 725-755.  doi: 10.1017/S0308210504000721. [14] J. L. Daleckii and M. G. Krein, Stability of Solutions of Differential Equations in Banach Space, Transl. Math. Monographs (AMS, Providence), 1974. [15] R. Durrett, Crabgrass, measles and gypsy moths: An introduction to modern probability, Bulletin (New Series) of the American Mathematical Society, 18 (1988), 117-143.  doi: 10.1090/S0273-0979-1988-15625-X. [16] G. Faye and M. Holzer, Modulated traveling fronts for a nonlocal Fisher–KPP equation: A dynamical systems approach, Journal of Differential Equations, 258 (2015), 2257-2289.  doi: 10.1016/j.jde.2014.12.006. [17] D. Finkelshtein, Y. Kondratiev and O. Kutoviy, Semigroup approach to birth-and-death stochastic dynamics in continuum, J. Funct. Anal., 262 (2012), 1274-1308.  doi: 10.1016/j.jfa.2011.11.005. [18] D. Finkelshtein, Y. Kondratiev and P. Tkachov, Traveling waves and long-time behavior in a doubly nonlocal Fisher–KPP equation, preprint, arXiv: 1508.02215. [19] D. Finkelshtein, Y. Kondratiev and P. Tkachov, Accelerated front propagation for monostable equations with nonlocal diffusion, preprint, arXiv: 1611.09329. [20] D. Finkelshtein and P. Tkachov, The hair-trigger effect for a class of nonlocal nonlinear equations, Nonlinearity, 31 (2018), 2442-2479.  doi: 10.1088/1361-6544/aab1cb. [21] D. Finkelshtein and P. Tkachov, Accelerated nonlocal nonsymmetric dispersion for monostable equations on the real line, Applicable Analysis, (2017), 1-25. doi: 10.1080/00036811.2017.1400537. [22] N. Fournier and S. Méléard, A microscopic probabilistic description of a locally regulated population and macroscopic approximations, The Annals of Applied Probability, 14 (2004), 1880-1919.  doi: 10.1214/105051604000000882. [23] M. Fuentes, M. Kuperman and V. Kenkre, Nonlocal Interaction Effects on Pattern Formation in Population Dynamics, Phys. Rev. Lett., 91 (2003), 158104. doi: 10.1103/PhysRevLett.91.158104. [24] J. Garnier, Accelerating solutions in integro-differential equations, SIAM J. Math. Anal., 43 (2011), 1955-1974.  doi: 10.1137/10080693X. [25] S. Gourley, Travelling front solutions of a nonlocal Fisher equation, J. Math. Biol., 41 (2000), 272-284.  doi: 10.1007/s002850000047. [26] S. Gourley, M. Chaplain and F. Davidson, Spatio-temporal pattern formation in a nonlocal reaction-diffusion equation, Dynamical Systems, 16 (2001), 173-192.  doi: 10.1080/14689360116914. [27] 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. [28] H. Kielhöfer, Bifurcation Theory, Applied Mathematical Sciences, 156, 2012. [29] M. Krukowski, A functional analysis point of view on compactness theorems in function spaces, preprint, arXiv: 1801.01898. [30] D. Mollison, Possible velocities for a simple epidemic, Advances in Appl. Probability, 4 (1972), 233-257.  doi: 10.2307/1425997. [31] D. Mollison, The rate of spatial propagation of simple epidemics, In Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability (Univ. California, Berkeley, Calif., 1970/1971), Vol. III: Probability theory, pages 579-614. Univ. California Press, Berkeley, Calif., 1972. [32] J. M. Ortega and W. C. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, New York-London, 1970. [33] M. Reed and B. Simon, Methods of Modern Mathematical Physics, vol. I. Acad. Press, New York, 1978. [34] M. Reed and B. Simon, Methods of Modern Mathematical Physics, vol. IV, Acad. Press, New York, 1978.

show all references

##### References:
 [1] F. Achleitner and C. Kuehn, On Bounded Positive Stationary Solutions for a Nonlocal Fisher-KPP Equation, Nonlinear Anal., 112 (2015), 15-29.  doi: 10.1016/j.na.2014.09.004. [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] N. Apreutesei, N. Bessonov, V. Volpert and V. Vougalter, Spatial structures and generalized travelling waves for an integro-differential equation, Discrete Continuous Dynam. Systems - B, 13 (2010), 537-557. doi: 10.3934/dcdsb.2010.13.537. [4] P. Auger, S. Genieys and V. Volpert, Pattern and Waves for a Model in Population Dynamics with Nonlocal Consumption of Resources, Mathematical Modelling of Natural Phenomena, 1 (2006), 65-82.  doi: 10.1051/mmnp:2006004. [5] O. Aydogmus, Phase transitions in a logistic metapopulation model with nonlocal interactions, Bull Math Biol, 80 (2018), 228-253.  doi: 10.1007/s11538-017-0373-3. [6] F. Barbosa, A. Penna, R. Ferreira, K. Novais, J. da Cunha and F. Oliveira, Pattern transitions and complexity for a nonlocal logistic map, Physica A, 473 (2017), 301-312.  doi: 10.1016/j.physa.2016.12.082. [7] 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. [8] B. Bolker and S. W. Pacala, Using moment equations to understand stochastically driven spatial pattern formation in ecological systems, Theor. Popul. Biol., 52 (1997), 179-197.  doi: 10.1006/tpbi.1997.1331. [9] E. Bouin, J. Garnier, C. Henderson and F. Patout, Thin front limit of an integro–differential Fisher–KPP equation with fat–tailed kernels, SIAM J. Math. Anal., 50 (2018), 3365-3394.  doi: 10.1137/17M1132501. [10] H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer, New York, 2011. [11] N. Britton, Spatial structures and periodic travelling waves in an integro-differential reaction-diffusion population model, SIAM J. Appl. Math., 50 (1990), 1663-1688.  doi: 10.1137/0150099. [12] N. Britton, Aggregation and the competitive exclusion principle, Journal of Theoretical Biology, 136 (1989), 57-66.  doi: 10.1016/S0022-5193(89)80189-4. [13] J. Coville and L. Dupaigne, On a non-local equation arising in population dynamics, Proc. R. Soc. A, 137 (2007), 725-755.  doi: 10.1017/S0308210504000721. [14] J. L. Daleckii and M. G. Krein, Stability of Solutions of Differential Equations in Banach Space, Transl. Math. Monographs (AMS, Providence), 1974. [15] R. Durrett, Crabgrass, measles and gypsy moths: An introduction to modern probability, Bulletin (New Series) of the American Mathematical Society, 18 (1988), 117-143.  doi: 10.1090/S0273-0979-1988-15625-X. [16] G. Faye and M. Holzer, Modulated traveling fronts for a nonlocal Fisher–KPP equation: A dynamical systems approach, Journal of Differential Equations, 258 (2015), 2257-2289.  doi: 10.1016/j.jde.2014.12.006. [17] D. Finkelshtein, Y. Kondratiev and O. Kutoviy, Semigroup approach to birth-and-death stochastic dynamics in continuum, J. Funct. Anal., 262 (2012), 1274-1308.  doi: 10.1016/j.jfa.2011.11.005. [18] D. Finkelshtein, Y. Kondratiev and P. Tkachov, Traveling waves and long-time behavior in a doubly nonlocal Fisher–KPP equation, preprint, arXiv: 1508.02215. [19] D. Finkelshtein, Y. Kondratiev and P. Tkachov, Accelerated front propagation for monostable equations with nonlocal diffusion, preprint, arXiv: 1611.09329. [20] D. Finkelshtein and P. Tkachov, The hair-trigger effect for a class of nonlocal nonlinear equations, Nonlinearity, 31 (2018), 2442-2479.  doi: 10.1088/1361-6544/aab1cb. [21] D. Finkelshtein and P. Tkachov, Accelerated nonlocal nonsymmetric dispersion for monostable equations on the real line, Applicable Analysis, (2017), 1-25. doi: 10.1080/00036811.2017.1400537. [22] N. Fournier and S. Méléard, A microscopic probabilistic description of a locally regulated population and macroscopic approximations, The Annals of Applied Probability, 14 (2004), 1880-1919.  doi: 10.1214/105051604000000882. [23] M. Fuentes, M. Kuperman and V. Kenkre, Nonlocal Interaction Effects on Pattern Formation in Population Dynamics, Phys. Rev. Lett., 91 (2003), 158104. doi: 10.1103/PhysRevLett.91.158104. [24] J. Garnier, Accelerating solutions in integro-differential equations, SIAM J. Math. Anal., 43 (2011), 1955-1974.  doi: 10.1137/10080693X. [25] S. Gourley, Travelling front solutions of a nonlocal Fisher equation, J. Math. Biol., 41 (2000), 272-284.  doi: 10.1007/s002850000047. [26] S. Gourley, M. Chaplain and F. Davidson, Spatio-temporal pattern formation in a nonlocal reaction-diffusion equation, Dynamical Systems, 16 (2001), 173-192.  doi: 10.1080/14689360116914. [27] 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. [28] H. Kielhöfer, Bifurcation Theory, Applied Mathematical Sciences, 156, 2012. [29] M. Krukowski, A functional analysis point of view on compactness theorems in function spaces, preprint, arXiv: 1801.01898. [30] D. Mollison, Possible velocities for a simple epidemic, Advances in Appl. Probability, 4 (1972), 233-257.  doi: 10.2307/1425997. [31] D. Mollison, The rate of spatial propagation of simple epidemics, In Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability (Univ. California, Berkeley, Calif., 1970/1971), Vol. III: Probability theory, pages 579-614. Univ. California Press, Berkeley, Calif., 1972. [32] J. M. Ortega and W. C. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, New York-London, 1970. [33] M. Reed and B. Simon, Methods of Modern Mathematical Physics, vol. I. Acad. Press, New York, 1978. [34] M. Reed and B. Simon, Methods of Modern Mathematical Physics, vol. IV, Acad. Press, New York, 1978.
Sketch of the existence of classes of periodic solutions for small $\varepsilon$ and $\delta$
Computation of ${\text{sgn}} \left( \alpha \left( \varepsilon ,p \right) \right)$, where −1 is shown in white and +1 in black. This is shown only for illustration purposes and conditions on $\alpha$ can be checked analytically
Computation of ${\text{sgn}} \left( \omega \left( \varepsilon ,p \right) \right)$. Again we show −1 in white and +1 in black
Computation of ${\text{sgn}} \left( \alpha \left( \varepsilon ,p \right) \right)$; same conventions as for plots above
Computation of ${\text{sgn}} \left( \omega \left( \varepsilon ,p \right) \right)$; same conventions as for plots above
 [1] Jianping Gao, Shangjiang Guo, Wenxian Shen. Persistence and time periodic positive solutions of doubly nonlocal Fisher-KPP equations in time periodic and space heterogeneous media. Discrete and Continuous Dynamical Systems - B, 2021, 26 (5) : 2645-2676. doi: 10.3934/dcdsb.2020199 [2] Christian Pötzsche. Nonautonomous bifurcation of bounded solutions I: A Lyapunov-Schmidt approach. Discrete and Continuous Dynamical Systems - B, 2010, 14 (2) : 739-776. doi: 10.3934/dcdsb.2010.14.739 [3] Wenxian Shen, Zhongwei Shen. Transition fronts in nonlocal Fisher-KPP equations in time heterogeneous media. Communications on Pure and Applied Analysis, 2016, 15 (4) : 1193-1213. doi: 10.3934/cpaa.2016.15.1193 [4] Heinz Schättler, Urszula Ledzewicz. Lyapunov-Schmidt reduction for optimal control problems. Discrete and Continuous Dynamical Systems - B, 2012, 17 (6) : 2201-2223. doi: 10.3934/dcdsb.2012.17.2201 [5] Patrick Martinez, Judith Vancostenoble. Lipschitz stability for the growth rate coefficients in a nonlinear Fisher-KPP equation. Discrete and Continuous Dynamical Systems - S, 2021, 14 (2) : 695-721. doi: 10.3934/dcdss.2020362 [6] Lina Wang, Xueli Bai, Yang Cao. Exponential stability of the traveling fronts for a viscous Fisher-KPP equation. Discrete and Continuous Dynamical Systems - B, 2014, 19 (3) : 801-815. doi: 10.3934/dcdsb.2014.19.801 [7] Aijun Zhang. Traveling wave solutions of periodic nonlocal Fisher-KPP equations with non-compact asymmetric kernel. Discrete and Continuous Dynamical Systems - S, 2022  doi: 10.3934/dcdss.2022061 [8] Matthieu Alfaro, Arnaud Ducrot. Sharp interface limit of the Fisher-KPP equation. Communications on Pure and Applied Analysis, 2012, 11 (1) : 1-18. doi: 10.3934/cpaa.2012.11.1 [9] Denghui Wu, Zhen-Hui Bu. Multidimensional stability of pyramidal traveling fronts in degenerate Fisher-KPP monostable and combustion equations. Electronic Research Archive, 2021, 29 (6) : 3721-3740. doi: 10.3934/era.2021058 [10] Benjamin Contri. Fisher-KPP equations and applications to a model in medical sciences. Networks and Heterogeneous Media, 2018, 13 (1) : 119-153. doi: 10.3934/nhm.2018006 [11] François Hamel, James Nolen, Jean-Michel Roquejoffre, Lenya Ryzhik. A short proof of the logarithmic Bramson correction in Fisher-KPP equations. Networks and Heterogeneous Media, 2013, 8 (1) : 275-289. doi: 10.3934/nhm.2013.8.275 [12] Hiroshi Matsuzawa. A free boundary problem for the Fisher-KPP equation with a given moving boundary. Communications on Pure and Applied Analysis, 2018, 17 (5) : 1821-1852. doi: 10.3934/cpaa.2018087 [13] Matt Holzer. A proof of anomalous invasion speeds in a system of coupled Fisher-KPP equations. Discrete and Continuous Dynamical Systems, 2016, 36 (4) : 2069-2084. doi: 10.3934/dcds.2016.36.2069 [14] Margarita Arias, Juan Campos, Cristina Marcelli. Fastness and continuous dependence in front propagation in Fisher-KPP equations. Discrete and Continuous Dynamical Systems - B, 2009, 11 (1) : 11-30. doi: 10.3934/dcdsb.2009.11.11 [15] Jean-Michel Roquejoffre, Luca Rossi, Violaine Roussier-Michon. Sharp large time behaviour in $N$-dimensional Fisher-KPP equations. Discrete and Continuous Dynamical Systems, 2019, 39 (12) : 7265-7290. doi: 10.3934/dcds.2019303 [16] Zhenzhen Wang, Tianshou Zhou. Asymptotic behaviors and stochastic traveling waves in stochastic Fisher-KPP equations. Discrete and Continuous Dynamical Systems - B, 2021, 26 (9) : 5023-5045. doi: 10.3934/dcdsb.2020323 [17] Aaron Hoffman, Matt Holzer. Invasion fronts on graphs: The Fisher-KPP equation on homogeneous trees and Erdős-Réyni graphs. Discrete and Continuous Dynamical Systems - B, 2019, 24 (2) : 671-694. doi: 10.3934/dcdsb.2018202 [18] Patrick Martinez, Jean-Michel Roquejoffre. The rate of attraction of super-critical waves in a Fisher-KPP type model with shear flow. Communications on Pure and Applied Analysis, 2012, 11 (6) : 2445-2472. doi: 10.3934/cpaa.2012.11.2445 [19] Gregoire Nadin. How does the spreading speed associated with the Fisher-KPP equation depend on random stationary diffusion and reaction terms?. Discrete and Continuous Dynamical Systems - B, 2015, 20 (6) : 1785-1803. doi: 10.3934/dcdsb.2015.20.1785 [20] Aijun Zhang. Traveling wave solutions with mixed dispersal for spatially periodic Fisher-KPP equations. Conference Publications, 2013, 2013 (special) : 815-824. doi: 10.3934/proc.2013.2013.815

2020 Impact Factor: 1.392