Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Turing instability and Turing-Hopf bifurcation in diffusive Schnakenberg
  systems with gene expression time delay   For delayed reaction-diffusion Schnakenberg systems with Neumann boundary
conditions, critical conditions for Turing instability are derived, which are
necessary and sufficient. And existence conditions for Turing, Hopf and
Turing-Hopf bifurcations are established. Normal forms truncated to order 3 at
Turing-Hopf singularity of codimension 2, are derived. By investigating
Turing-Hopf bifurcation, the parameter regions for the stability of a periodic
solution, a pair of spatially inhomogeneous steady states and a pair of
spatially inhomogeneous periodic solutions, are derived in $(\tau,\varepsilon)$
parameter plane ($\tau$ for time delay, $\varepsilon$ for diffusion rate). It
is revealed that joint effects of diffusion and delay can lead to the
occurrence of mixed spatial and temporal patterns. Moreover, it is also
demonstrated that various spatially inhomogeneous patterns with different
spatial frequencies can be achieved via changing the diffusion rate. And, the
phenomenon that time delay may induce a failure of Turing instability observed
by Gaffney and Monk (2006) are theoretically explained.

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The categories of this paper are: acc-phys
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The categories of this paper are: astro-ph
1
The categories of this paper are: cs
8
The categories of this paper are: hep-ex
4
The categories of this paper are: cond-mat
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The categories of this paper are: patt-sol
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The categories of this paper are: chem-ph
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The categories of this paper are: eess
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The categories of this paper are: dg-ga
9
The categories of this paper are: nlin
5
The categories of this paper are: gr-qc
11
The categories of this paper are: hep-lat
0
The categories of this paper are: hep-ph
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The categories of this paper are: nucl-th
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The categories of this paper are: quant-ph
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The categories of this paper are: math-ph