Abstract
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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Acknowledgments
F. Yi was partially supported by 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) and (2013RFLXJ025). N. Tuncer was partially supported by NSF Grant DMS-1220342.
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Appendix: The Dynamics of ODEs
Appendix: The Dynamics of ODEs
In this section, we consider the local/global asymptotic stability of \((u_*,v_*)\), as well as the occurrence of stable periodic solutions of the following Ordinary Differential Equations (ODEs):
System (6.1) has a positive equilibrium \((u_*,v_*)\), with
if and only if \(\displaystyle \frac{\beta _1}{1+\gamma _1}>\frac{\beta _2}{\gamma _2}\) holds.
The linearized operator of system (6.1) evaluated at \((u_*,v_*)\) is given by
where
Then, the characteristic equation of (6.3) is given by
Lemma 9
Suppose that \(\displaystyle \frac{\beta _1}{1+\gamma _1}>\frac{\beta _2}{\gamma _2}\) is satisfied so that \((u_*,v_*)\) is the unique positive equilibrium of (6.1). If
holds, then \((u_*,v_*)\) is locally asymptotically stable in system (6.1). However, if
holds, then \((u_*,v_*)\) is unstable in system (6.1), and the system (6.1) has a locally orbitally stable periodic orbit, denoted by \((p(t),q(t))\).
Proof
Suppose that (6.6) holds. Then, all the eigenvalues of (6.5) has strictly negative real parts, thus \((u_*,v_*)\) is locally asymptotically stable; While if (6.7) holds, then (6.5) has one eigenvalue with positive real parts, thus \((u_*,v_*)\) is unstable. According to Theorem 2, the solutions is bounded, then from Poincare–Bendixson theorem, we conclude the existence of a locally orbitally stable periodic orbit, denoted by \((p(t),q(t))\). \(\square \)
The next result is on the global asymptotic stability of the positive equilibrium \((u_*,v_*)\) in (6.1):
Lemma 10
Suppose that \(\displaystyle \frac{\beta _1}{1+\gamma _1}>\frac{\beta _2}{\gamma _2}\) is satisfied so that \((u_*,v_*)\) is the unique positive equilibrium of (6.1). Assume also that \(0<\beta _1+\beta _2\le 1\) holds. Then, \((u_*,v_*)\) is globally asymptotically stable in system (6.1), if
where
Proof
We first use the Dulac criteria to exclude the existence of periodic orbits in the first quadrant. Define \(b(u,v)=1+u+v+Ku^2\), then, we have
where \(\mathcal {W}(u):=-3Ku^2-(\gamma _2+2-2\beta _1K)u+\beta _1+\beta _2-1\).
Let \(u_\mathcal {W}\) be the symmetry axis of the function \(\mathcal {W}(u)\). Then, \(u_\mathcal {W}=\frac{1}{3}\beta _1K-\frac{1}{6}(2+\gamma _2)\). If \( K\in \big (0,\displaystyle \frac{\gamma _2+2}{2\beta _1}\big ]\) holds, we have \(u_\mathcal {W}\le 0\). Thus, \(\mathcal {W}(u)\le 0\), which indicates that under \(\partial (fb)/\partial u+\partial (gb)/\partial v<0\) in the first quadrant.
On the other hand, let \(\Delta _{\mathcal {W}}\) be the discriminant of the function \(\mathcal {W}(u)\). Then,
Suppose that \( K\in \big [\displaystyle \frac{\gamma _2+2}{2\beta _1}+\displaystyle \frac{\epsilon _-}{2\beta _1^2},\displaystyle \frac{\gamma _2+2}{2\beta _1}+\displaystyle \frac{\epsilon _+}{2\beta _1^2}\big ]\) holds. Then \(\Delta _{\mathcal {W}}\le 0\). Again, we can conclude that \(\mathcal {W}(u)\le 0\), which indicates that under \(\partial (fb)/\partial u+\partial (gb)/\partial v<0\) in the first quadrant.
So far, under (6.8) and \(0<\beta _1+\beta _2\le 1\), by Dulac criteria, system (6.1) does not have closed orbits in the first quadrant. By Theorem 2, it follows that the solution is bounded. Thus, by Poincare–Bendixson theorem, we know that \((u_*,v_*)\) is globally asymptotically stable in ODEs. \(\square \)
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Yi, F., Liu, S. & Tuncer, N. Spatiotemporal Patterns of a Reaction–Diffusion Substrate–Inhibition Seelig Model. J Dyn Diff Equat 29, 219–241 (2017). https://doi.org/10.1007/s10884-015-9444-z
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DOI: https://doi.org/10.1007/s10884-015-9444-z