Spatiotemporal Patterns of a Reaction–Diffusion Substrate–Inhibition Seelig Model

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.

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):

$$\begin{aligned} \displaystyle \frac{du}{dt}=\beta _1-u-\displaystyle \frac{\gamma _1uv}{1+u+v+Ku^2}=:f(u,v),\;\displaystyle \frac{dv}{dt}=\beta _2-\displaystyle \frac{\gamma _2uv}{1+u+v+Ku^2}=:g(u,v). \end{aligned}$$

(6.1)

System (6.1) has a positive equilibrium $(u_*,v_*)$, with

$$\begin{aligned} u_*:=\beta _1-\displaystyle \frac{\gamma _1}{\gamma _2}\beta _2,\;v_*:=\displaystyle \frac{\beta _2(1+u_*+Ku_*^2)}{\gamma _2u_*-\beta _2}, \end{aligned}$$

(6.2)

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

$$\begin{aligned} J(u_*,v_*):= \left( \begin{array}{cc} -1-\gamma _1j_0 &{}-\gamma _1k_0\\ -\gamma _2j_0, &{}-\gamma _2k_0 \end{array}\right) , \end{aligned}$$

(6.3)

where

$$\begin{aligned} j_0:=\displaystyle \frac{v_*(1+v_*-Ku_*^2)}{(1+u_*+v_*+Ku_*^2)^2},\;k_0:=\displaystyle \frac{u_*(1+u_*+Ku_*^2)}{(1+u_*+v_*+Ku_*^2)^2}. \end{aligned}$$

(6.4)

Then, the characteristic equation of (6.3) is given by

$$\begin{aligned} \mu ^2+(1+\gamma _1j_0+\gamma _2k_0)\mu +\gamma _2k_0=0. \end{aligned}$$

(6.5)

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

$$\begin{aligned} j_0>-\displaystyle \frac{1+\gamma _2k_0}{\gamma _1} \end{aligned}$$

(6.6)

holds, then $(u_*,v_*)$ is locally asymptotically stable in system (6.1). However, if

$$\begin{aligned} j_0<-\displaystyle \frac{1+\gamma _2k_0}{\gamma _1} \end{aligned}$$

(6.7)

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

$$\begin{aligned} K\in \bigg (0,\displaystyle \frac{\gamma _2+2}{2\beta _1}\bigg ]\cup \bigg [\displaystyle \frac{\gamma _2+2}{2\beta _1}+\displaystyle \frac{\epsilon _-}{2\beta _1^2},\displaystyle \frac{\gamma _2+2}{2\beta _1}+\displaystyle \frac{\epsilon _+}{2\beta _1^2}\bigg ], \end{aligned}$$

(6.8)

where

$$\begin{aligned} \epsilon _{\pm }:=\sqrt{9(1-\beta _1-\beta _2)^2+6\beta _1(\gamma _2+2)(1-\beta _1-\beta _2)}\pm 3(1-\beta _1-\beta _2)>0. \end{aligned}$$

(6.9)

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

$$\begin{aligned} \displaystyle \frac{\partial (fb)}{\partial u}+\displaystyle \frac{\partial (gb)}{\partial v}=\mathcal {W}(u)-(\gamma _1+1)v, \end{aligned}$$

(6.10)

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,

$$\begin{aligned} \Delta _{\mathcal {W}}=(2\beta _1K-2-\gamma _2)^2+12K(\beta _1+\beta _2-1). \end{aligned}$$

(6.11)

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. $\square$