Journal
JOURNAL OF DYNAMICS AND DIFFERENTIAL EQUATIONS
Volume 29, Issue 1, Pages 219-241Publisher
SPRINGER
DOI: 10.1007/s10884-015-9444-z
Keywords
Seelig reaction-diffusion chemical model; Invariant rectangle; Lumped parameter assumption; Global bifurcation analysis; Turing patterns
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Funding
- National Natural Science Foundation of China [11371108]
- Program for New Century Excellent Talents in University from Ministry of Education [NECT-13-0755]
- Scientific Research Foundation for the Returned Overseas Chinese Scholars of Heilongjiang Province [LC2012C36, 2013RFLXJ025]
- NSF [DMS-1220342]
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In this paper, the spatiotemporal patterns of a reaction-diffusion substrate-inhibition chemical Seelig model are considered. We first prove that this parabolic Seelig model has an invariant rectangle in the phase plane which attracts all the solutions of the model regardless of the initial values. Then, we consider the long time behaviors of the solutions in the invariant rectangle. In particular, we prove that, under suitable lumped parameter assumption conditions, these solutions either converge exponentially to the unique positive constant steady states or to the spatially homogeneous periodic solutions. Finally, we study the existence and non-existence of Turing patterns. To find parameter ranges where system does not exhibit Turing patterns, we use the properties of non-constant steady states, including obtaining several useful estimates. To seek the parameter ranges where system possesses Turing patterns, we use the techniques of global bifurcation theory. These two different parameter ranges are distinguished in a delicate bifurcation diagram. Moreover, numerical experiments are also presented to support and strengthen our analytical analysis.
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