Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Pattern formation in the doubly-nonlocal Fisher-KPP equation   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.

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The categories of this paper are: atom-ph
17
The categories of this paper are: q-fin
14
The categories of this paper are: math-ph
22
The categories of this paper are: chao-dyn
5
The categories of this paper are: gr-qc
9
The categories of this paper are: nlin
12
The categories of this paper are: nucl-th
11
The categories of this paper are: hep-lat
28
The categories of this paper are: alg-geom
18
The categories of this paper are: econ
16
The categories of this paper are: stat
31
The categories of this paper are: ao-sci
3
The categories of this paper are: physics
37
The categories of this paper are: supr-con
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The categories of this paper are: patt-sol
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The categories of this paper are: funct-an