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Pattern formation in the doubly-nonlocal Fisher-KPP equation

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"

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  • 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.

    Mathematics Subject Classification: Primary: 35K57, 47G20, 37G99; Secondary: 92D15.


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  • 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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