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 & Continuous Dynamical Systems - A, 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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P. AugerS. 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.  Google Scholar

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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.  Google Scholar

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F. BarbosaA. PennaR. FerreiraK. NovaisJ. 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.  Google Scholar

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H. BerestyckiG. NadinB. 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.  Google Scholar

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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.  Google Scholar

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E. BouinJ. GarnierC. 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.  Google Scholar

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H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer, New York, 2011.  Google Scholar

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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.  Google Scholar

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N. Britton, Aggregation and the competitive exclusion principle, Journal of Theoretical Biology, 136 (1989), 57-66.  doi: 10.1016/S0022-5193(89)80189-4.  Google Scholar

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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.  Google Scholar

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J. L. Daleckii and M. G. Krein, Stability of Solutions of Differential Equations in Banach Space, Transl. Math. Monographs (AMS, Providence), 1974.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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D. FinkelshteinY. 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.  Google Scholar

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D. Finkelshtein, Y. Kondratiev and P. Tkachov, Traveling waves and long-time behavior in a doubly nonlocal Fisher–KPP equation, preprint, arXiv: 1508.02215. Google Scholar

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D. Finkelshtein, Y. Kondratiev and P. Tkachov, Accelerated front propagation for monostable equations with nonlocal diffusion, preprint, arXiv: 1611.09329. Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

J. Garnier, Accelerating solutions in integro-differential equations, SIAM J. Math. Anal., 43 (2011), 1955-1974.  doi: 10.1137/10080693X.  Google Scholar

[25]

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

[26]

S. GourleyM. Chaplain and F. Davidson, Spatio-temporal pattern formation in a nonlocal reaction-diffusion equation, Dynamical Systems, 16 (2001), 173-192.  doi: 10.1080/14689360116914.  Google Scholar

[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.  Google Scholar

[28]

H. Kielhöfer, Bifurcation Theory, Applied Mathematical Sciences, 156, 2012. Google Scholar

[29]

M. Krukowski, A functional analysis point of view on compactness theorems in function spaces, preprint, arXiv: 1801.01898. Google Scholar

[30]

D. Mollison, Possible velocities for a simple epidemic, Advances in Appl. Probability, 4 (1972), 233-257.  doi: 10.2307/1425997.  Google Scholar

[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.  Google Scholar

[32] J. M. Ortega and W. C. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, New York-London, 1970.   Google Scholar
[33]

M. Reed and B. Simon, Methods of Modern Mathematical Physics, vol. I. Acad. Press, New York, 1978. Google Scholar

[34]

M. Reed and B. Simon, Methods of Modern Mathematical Physics, vol. IV, Acad. Press, New York, 1978. Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

P. AugerS. 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.  Google Scholar

[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.  Google Scholar

[6]

F. BarbosaA. PennaR. FerreiraK. NovaisJ. 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.  Google Scholar

[7]

H. BerestyckiG. NadinB. 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.  Google Scholar

[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.  Google Scholar

[9]

E. BouinJ. GarnierC. 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.  Google Scholar

[10]

H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer, New York, 2011.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

J. L. Daleckii and M. G. Krein, Stability of Solutions of Differential Equations in Banach Space, Transl. Math. Monographs (AMS, Providence), 1974.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

D. FinkelshteinY. 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.  Google Scholar

[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. Google Scholar

[19]

D. Finkelshtein, Y. Kondratiev and P. Tkachov, Accelerated front propagation for monostable equations with nonlocal diffusion, preprint, arXiv: 1611.09329. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

J. Garnier, Accelerating solutions in integro-differential equations, SIAM J. Math. Anal., 43 (2011), 1955-1974.  doi: 10.1137/10080693X.  Google Scholar

[25]

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

[26]

S. GourleyM. Chaplain and F. Davidson, Spatio-temporal pattern formation in a nonlocal reaction-diffusion equation, Dynamical Systems, 16 (2001), 173-192.  doi: 10.1080/14689360116914.  Google Scholar

[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.  Google Scholar

[28]

H. Kielhöfer, Bifurcation Theory, Applied Mathematical Sciences, 156, 2012. Google Scholar

[29]

M. Krukowski, A functional analysis point of view on compactness theorems in function spaces, preprint, arXiv: 1801.01898. Google Scholar

[30]

D. Mollison, Possible velocities for a simple epidemic, Advances in Appl. Probability, 4 (1972), 233-257.  doi: 10.2307/1425997.  Google Scholar

[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.  Google Scholar

[32] J. M. Ortega and W. C. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, New York-London, 1970.   Google Scholar
[33]

M. Reed and B. Simon, Methods of Modern Mathematical Physics, vol. I. Acad. Press, New York, 1978. Google Scholar

[34]

M. Reed and B. Simon, Methods of Modern Mathematical Physics, vol. IV, Acad. Press, New York, 1978. Google Scholar

Figure 1.  Sketch of the existence of classes of periodic solutions for small $\varepsilon $ and $\delta $
Figure 2.  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
Figure 3.  Computation of ${\text{sgn}} \left( \omega \left( \varepsilon ,p \right) \right)$. Again we show −1 in white and +1 in black
Figure 4.  Computation of ${\text{sgn}} \left( \alpha \left( \varepsilon ,p \right) \right)$; same conventions as for plots above
Figure 5.  Computation of ${\text{sgn}} \left( \omega \left( \varepsilon ,p \right) \right)$; same conventions as for plots above
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