近年来, 越来越多的研究关注自催化模型的动力学行为, 通过对反应扩散系统的定性分析和数值模拟, 更清楚地展示反应物的振荡现象, 对化学过程的认知产生重要意义.文献[1]研究了一类自催化三分子生化反应扩散模型, 建立了正平衡点的稳定性和非常数稳态解的存在性.本文考虑具有可逆效应的自催化三分子生化反应扩散模型, 其化学反应机制为

$ P \stackrel{K_{1}}{\longrightarrow}U, \quad U+2V \mathop{\rightleftharpoons }\limits^{K_{2}}_{K_{-2}} 3V, \quad V\stackrel{K_{m}}{\longrightarrow}C, $

其中$ K_{1}, K_{2}, K_{-2}, K_{m} $表示反应速率, $ P, U $为反应物, $ V $不仅是反应物还是产物, $ C $为某种惰性产物.在上述化学反应过程中, $ V $不但没有被取代或消耗, 反而增加了, 因此$ V $为催化剂.碘酸-亚砷酸体系^[2]就是这一反应机制最好的近似, $ U $和$ V $可分别代表$ IO_{3}^{-} $和$ I^{-} $.假设$ P $为原始反应物且存在大量过剩, 它的浓度始终保持在初值$ a $不变, 则上述化学反应过程可简化为

(1.1) $ \begin{equation} \left\{ \begin{array}{ll} u_{t}-d_{1}\Delta u = a-uv^{2}+cv^{3}, &\quad x\in \Omega, t>0, \\ v_{t}-d_{2}\Delta v = uv^{2}-cv^{3}-bv, &\quad x\in \Omega, t>0, \\ \partial_{\nu}u = \partial_{\nu}v = 0, &\quad x\in \partial\Omega, t>0, \\ \end{array} \right. \end{equation} $

其中$ \Omega\subset {{\Bbb R}} ^{N}(N\geq1) $为有界开集且具有光滑边界$ \partial\Omega $, $ \nu $表示$ \partial\Omega $上单位外法向量. $ u, v $分别表示反应物$ U $和催化剂$ V $的无量纲浓度, 通常被认为是非负的. $ d_{1}, d_{2} $分别代表反应物$ U $和催化剂$ V $的扩散系数, 均为正常数. $ \Delta $为拉普拉斯算子, $ a, b, c $均为正常数.

在系统(1.1)不考虑可逆反应的情况下, 文献[3]研究了施加电场对反应动力学的影响, 并利用线性稳定性分析和数值模拟说明了电场对空间结构的影响.文献[4]讨论了唯一正平衡点的稳定性, 并利用Lyapunov-Schmidt技术和奇异理论, 研究了空间非均匀平稳结构, 特别是双特征值分支结构.生化反应过程是一个复杂的过程, 受到许多复杂机制的影响, 常常伴随着多重平衡态、时间震荡行为和稳定性发生改变的分支现象. Hopf分支理论作为研究非线性方程的经典工具, 被广泛应用.文献[5]研究了任意阶自催化模型Hopf分支的存在性和稳定性.文献[6]研究了具有自催化和饱和规律的双分子反应扩散模型, 考虑了由惟一正平衡点产生的Hopf分支和稳态分支.更多关于Hopf分支的研究有兴趣的读者可参见文献[7-10]及其中的参考文献.

本文主要研究系统(1.1)Hopf分支的存在性、稳定性及由扩散引起的Turing不稳定性.

易知, 系统(1.1)所对应的常微分系统

(2.1) $ \begin{equation} \left\{ \begin{array}{ll} u_{t} = a-uv^{2}+cv^{3}, &\quad t>0, \\ v_{t} = uv^{2}-cv^{3}-bv, &\quad t>0 \end{array} \right. \end{equation} $

存在惟一的正平衡点$ (u_{*}, v_{*}) $, 其中$ u_{*} = \frac{b^{3}+a^{2}c}{ab} $, $ v_{*} = \frac{a}{b} $.系统(2.1)在$ (u_{*}, v_{*}) $处的雅可比矩阵为

$ L_{0} = \left( \begin{array}{ccc} { } -\frac{a^{2}}{b^{2}}{\quad} & { } \frac{a^{2}c-2b^{3}}{b^{2}} \\ { } \frac{a^{2}}{b^{2}}{\quad} & { } \frac{b^{3}-a^{2}c}{b^{2}} \end{array} \right) : = \left( \begin{array}{cccc} A & B\\ M & N \end{array} \right ). $

设其特征方程为$ \lambda^{2}-T\lambda+D = 0 $, 其中

$ T = \frac{b^{3}-a^{2}c-a^{2}}{b^{2}}, \quad D = \frac{a^{2}}{b}>0. $

设$ b>a^{\frac{2}{3}} $.令$ c^{H} = \frac{b^{3}-a^{2}}{a^{2}} $, 可得如下结论.

定理2.1  设$ b>a^{\frac{2}{3}} $.

(ⅰ) 若$ 0<c<c^{H} $, 则系统(2.1)的惟一正平衡点$ (u_{*}, v_{*}) $不稳定;

(ⅱ) 若$ c>c^{H} $, 则系统(2.1)的惟一正平衡点$ (u_{*}, v_{*}) $局部渐近稳定;

(ⅲ) 若$ c = c^{H} $, 则系统(2.1)在正平衡点$ (u_{*}, v_{*}) $处产生Hopf分支, 且该Hopf分支为次临界方向, 周期闭轨渐近稳定.

证  (ⅰ) 当$ 0<c<c^{H} $时, $ T>0 $, 又因$ D>0 $, 此时$ L_{0} $存在具有正实部的特征值, 正平衡点$ (u_{*}, v_{*}) $不稳定;

(ⅱ) 当$ c>c^{H} $时, $ T<0 $, 又因$ D>0 $, 此时$ L_{0} $的特征值均具有负实部, 正平衡点$ (u_{*}, v_{*}) $局部渐近稳定;

(ⅲ) 当$ c = c^{H} $时, $ L_{0} $存在一对纯虚特征根.令$ \bar{\lambda}(c) = \lambda_{1}(c)\pm i\lambda_{2}(c) $为$ L_{0} $在$ c = c^{H} $附近的一对共轭复特征根, 其中

$ \lambda_{1}(c) = \frac{b^{3}-a^{2}c-a^{2}}{2b^{2}}, \quad \lambda_{2}(c) = \frac{1}{2}\sqrt{\frac{4a^{2}}{b}-\frac{(b^{3}-a^{2}c-a^{2})^{2}}{b^{4}}}. $

经计算

$ \lambda_{1}(c^{H}) = 0, \quad \lambda_{1}'(c^{H}) = -\frac{a^{2}}{2b^{2}}<0, \quad \lambda_{2}(c^{H}) = \frac{a}{\sqrt{b}}\neq 0. $

令$ \tilde{u} = u-u_{*}, \tilde{v} = v-v_{*} $.为方便计算, 仍用$ u, v $表示$ \tilde{u}, \tilde{v} $, 则系统(2.1)变为

$ \left\{\begin{array}{ll} u_{t} = a-(u+u_{*})(v+v_{*})^{2}+c(v+v_{*})^{3}, \\ v_{t} = (u+u_{*})(v+v_{*})^{2}-c(v+v_{*})^{3}-b(v+v_{*}). \end{array}\right. $

将上述系统泰勒展开为

(2.2) $ \begin{equation} \left( \begin{array}{cccc} u_{t}\\ v_{t} \end{array} \right ) = L_{0}\left( \begin{array}{cccc} u\\ v \end{array} \right)+\left( \begin{array}{cccc} f^{1}(u, v, c)\\ f^{2}(u, v, c) \end{array} \right), \end{equation} $

其中

$ f^{1}(u, v, c) = -\frac{2a}{b}uv+\frac{2a^{2}c-b^{3}}{ab}v^{2}-uv^{2}+cv^{3}+O(|u||v|^{3}, |v|^{4}), $

$ f^{2}(u, v, c) = -f^{1}(u, v, c). $

定义

$ S = \left( \begin{array}{cccc} 1 & 0\\ W & Q \end{array} \right), $

其中$ W = \frac{b^{3}-a^{2}c+a^{2}}{2a^{2}c-4b^{3}} $, $ Q = -\frac{\sqrt{4a^{2}b^{3}-(b^{3}-a^{2}c-a^{2})^{2}}}{2a^{2}c-4b^{3}} $.当$ c = c^{H} $时, 有

$ W_{0}: = W|_{c = c^{H}} = -\frac{a^{2}}{b^{3}+a^{2}}, \quad Q_{0}: = Q|_{c = c^{H}} = \frac{ab\sqrt{b}}{b^{3}+a^{2}}. $

作变换

$ \left( \begin{array}{cccc} u\\ v \end{array} \right) = S\left( \begin{array}{cccc} x\\ y \end{array} \right), $

代入系统(2.2)得

$ \left( \begin{array}{cccc} x_{t}\\ y_{t} \end{array} \right ) = \left( \begin{array}{cccc} \lambda_{1}(c) {\quad}& -\lambda_{2}(c)\\ \lambda_{2}(c) {\quad}& \lambda_{1}(c) \end{array} \right)\left( \begin{array}{cccc} x\\ y \end{array} \right) + \left( \begin{array}{cccc} \tilde{f}^{1}(u, v, c)\\ \tilde{f}^{2}(u, v, c) \end{array} \right), $

其中

$ \tilde{f}^{1}(x, y, c) = f^{1}(x, Wx+Qy, c), \quad \tilde{f}^{2}(x, y, c) = -\frac{W+1}{Q}\tilde{f}^{1}(x, y, c). $

由文献[5]知, 分支的方向以及分支周期解的稳定性由$ d(c^{H}) $的符号给出, 其中

$ \begin{eqnarray*} d(c^{H})& = &\frac{1}{16}\left[\tilde{f}^{1}_{xxx}+\tilde{f}^{1}_{xyy}+\tilde{f}^{2}_{xxy}+\tilde{f}^{2}_{yyy}\right]\\ &&+\frac{1}{16\lambda_{2}(c^{H})}\left[\tilde{f}^{1}_{xy}(\tilde{f}^{1}_{xx}+\tilde{f}^{1}_{yy})-\tilde{f}^{2}_{xy}(\tilde{f}^{2}_{xx}+\tilde{f}^{2}_{yy})-\tilde{f}^{1}_{xx}\tilde{f}^{2}_{xx}+\tilde{f}^{1}_{yy}\tilde{f}^{2}_{yy}\right]. \end{eqnarray*} $

由$ \tilde{f}^{1}(x, y, c) $和$ \tilde{f}^{2}(x, y, c) $在$ (0, 0, c^{H}) $处的各阶偏导数及$ W_{0}, Q_{0} $计算得

$ \tilde{f}^{1}_{xx} = \frac{6a^{3}b^{2}}{(b^{3}+a^{2})^{2}}, \quad \tilde{f}^{1}_{xy} = \frac{2a^{2}\sqrt{b}(a^{2}-2b^{3})}{(b^{3}+a^{2})^{2}}, \quad \tilde{f}^{1}_{yy} = -\frac{2ab^{2}(2a^{2}-b^{3})}{(b^{3}+a^{2})^{2}}, $

$ \tilde{f}^{2}_{xx} = -\frac{6a^{2}b^{3}\sqrt{b}}{(b^{3}+a^{2})^{2}}, \quad \tilde{f}^{2}_{xy} = -\frac{2ab^{2}(a^{2}-2b^{3})}{(b^{3}+a^{2})^{2}}, \quad \tilde{f}^{2}_{yy} = \frac{2b^{3}\sqrt{b}(2a^{2}-b^{3})}{(b^{3}+a^{2})^{2}}, $

$ \tilde{f}^{1}_{xxx} = -\frac{12a^{4}b^{3}}{(b^{3}+a^{2})^{3}}, \quad \tilde{f}^{1}_{xyy} = \frac{4a^{2}b^{3}(a^{2}-2b^{3})}{(b^{3}+a^{2})^{3}}, $

$ \tilde{f}^{2}_{xxy} = \frac{2a^{2}b^{3}(a^{2}-5b^{3})}{(b^{3}+a^{2})^{3}}, \quad \tilde{f}^{2}_{yyy} = \frac{6b^{6}(a^{2}-b^{3})}{(b^{3}+a^{2})^{3}}. $

进一步计算得

$ \tilde{f}^{1}_{xxx}+\tilde{f}^{1}_{xyy}+\tilde{f}^{2}_{xxy}+\tilde{f}^{2}_{yyy} = -\frac{6b^{3}}{b^{3}+a^{2}}, $

$ \tilde{f}^{1}_{xy}(\tilde{f}^{1}_{xx}+\tilde{f}^{1}_{yy}) = \frac{4a^{3}b^{2}\sqrt{b}(a^{2}-2b^{3})}{(b^{3}+a^{2})^{3}}, \quad \tilde{f}^{2}_{xy}(\tilde{f}^{2}_{xx}+\tilde{f}^{2}_{yy}) = \frac{4ab^{5}\sqrt{b}(a^{2}-2b^{3})}{(b^{3}+a^{2})^{3}}, $

$ \tilde{f}^{1}_{xx}\tilde{f}^{2}_{xx} = -\frac{36a^{5}b^{5}\sqrt{b}}{(b^{3}+a^{2})^{4}}, \quad \tilde{f}^{1}_{yy}\tilde{f}^{2}_{yy} = -\frac{4ab^{5}\sqrt{b}(2a^{2}-b^{3})^{2}}{(b^{3}+a^{2})^{4}}. $

将上述等式代入$ d(c^{H}) $的表达式得

$ \begin{eqnarray*} d(c^{H})& = &\frac{1}{16}\bigg[-\frac{6b^{3}}{b^{3}+a^{2}}+\frac{4a^{2}b^{3}(a^{2}-2b^{3})}{(b^{3}+a^{2})^{3}}-\frac{4b^{6}(a^{2}-2b^{3})}{(b^{3}+a^{2})^{3}}+\frac{36a^{4}b^{6}}{(b^{3}+a^{2})^{4}}-\frac{4b^{6}(2a^{2}-b^{3})^{2}}{(b^{3}+a^{2})^{4}}\bigg]\\ & = &\frac{1}{16}\bigg[-\frac{6b^{3}}{b^{3}+a^{2}}+\frac{4b^{3}}{b^{3}+a^{2}}\bigg]\\ & = &-\frac{b^{3}}{8(b^{3}+a^{2})}<0. \end{eqnarray*} $

由Poincare-Andronov-Hopf分支定理^[11]可知, 系统(2.1)在正平衡点$ (u_{*}, v_{*}) $处产生Hopf分支.注意到$ \lambda_{1}'(c^{H})<0 $, 因此系统(2.1)在$ (u_{*}, v_{*}) $处产生的Hopf分支为次临界方向且周期闭轨渐近稳定.证毕.

本节在一维空间$ \Omega = (0, \pi) $上考虑可逆生化反应扩散系统

(3.1) $ \begin{equation} \left\{ \begin{array}{ll} u_{t}-d_{1}\Delta u = a-uv^{2}+cv^{3}, &\quad x\in (0, \pi), t>0, \\ v_{t}-d_{2}\Delta v = uv^{2}-cv^{3}-bv, &\quad x\in (0, \pi), t>0, \\ u_{x}(x, t) = v_{x}(x, t) = 0, &\quad x = 0, \pi, t>0, \\ u(x, 0) = u_{0}(x)\geq 0, v(x, 0) = v_{0}(x)\geq 0, &\quad x\in (0, \pi) \end{array} \right. \end{equation} $

的Hopf分支.

定义实Sobolev空间

$ X = \left\{(u, v)\in H^{2}(0, \pi)\times H^{2}(0, \pi):(u_{x}, v_{x})|_{x = 0, \pi} = 0\right\}, $

并且定义$ X $的复延拓空间$ X_{C} = X\oplus {\rm i}X = \{x_{1}+{\rm i}x_{2}|x_{1}, x_{2}\in X\}. $

系统(3.1)在$ (u_{*}, v_{*}) $处的线性化算子为

$ L = \left( \begin{array}{cccc} A+d_{1}\Delta{\quad} & B\\ M{\quad} & N+d_{2}\Delta \end{array} \right). $

在齐次Neumann边界条件下, 算子$ -\Delta $的特征值为$ \mu_{k} = k^{2}\ (k = 0, 1, 2, \cdots ) $, 满足

$ 0 = \mu_{0}<\mu_{1}\leq \mu_{2}\leq \mu_{3}\leq\cdot\cdot\cdot, $

且$ \cos(kx)(k\in N) $为对应$ \mu_{k} $的特征函数, 取函数序列$ \{\cos(kx)\}_{k = 0}^{\infty} $使其成为空间$ L^{2}(0, \pi) $的标准正交基.

令

$ \left( \begin{array}{cccc} \phi\\ \varphi \end{array} \right) = \sum\limits_{k = 0}^{\infty}\left( \begin{array}{cccc} a_{k}\\ b_{k} \end{array} \right)\cos(kx) $

为$ L $对应于特征值$ \beta $的特征函数, 即$ L(\phi, \varphi)^{T} = \beta(\phi, \varphi)^{T} $.通过直接计算得

$ L_{k}\left( \begin{array}{cccc} a_{k}\\ b_{k} \end{array} \right) = \beta\left( \begin{array}{cccc} a_{k}\\ b_{k} \end{array} \right), k = 0, 1, 2, \cdots , $

其中

$ L_{k} = \left( \begin{array}{cccc} { } -\frac{a^{2}}{b^{2}}-d_{1}\mu_{k} {\quad}&{ } \frac{a^{2}c-2b^{3}}{b^{2}} \\ { } \frac{a^{2}}{b^{2}} {\quad}&{ } \frac{b^{3}-a^{2}c}{b^{2}}-d_{2}\mu_{k} \end{array} \right). $

$ L $的所有特征值可由$ L_{k} $的特征值给出.设$ L_{k} $的特征方程为

$ \beta^{2}-T_{k}\beta+D_{k} = 0, $

其中

$ T_{k} = T-(d_{1}+d_{2})\mu_{k}, $

$ D_{k} = d_{1}d_{2}\mu_{k}^{2}+\frac{a^{2}}{b^{2}}\left[d_{2}+(c-c^{H}-1)d_{1}\right]\mu_{k}+\frac{a^{2}}{b}. $

显然, 当$ 0<c<c^{H} $时, 系统(3.1)的正平衡点$ (u_{*}, v_{*}) $不稳定.当$ c\geq c^{H}+1 $时, 对任意的$ k\geq0 $, 都有$ D_{k}\geq0 $且$ T_{k}<0 $, 此时系统(3.1)的正平衡点$ (u_{*}, v_{*}) $局部渐近稳定.

下面考虑系统(3.1)当$ c^{H}<c<c^{H}+1 $时正平衡点$ (u_{*}, v_{*}) $的稳定性.当$ \frac{d_{2}}{d_{1}}\geq c^{H}+1-c $时, 对任意的$ k\geq0 $, 都有$ D_{k}\geq0 $且$ T_{k}<0 $, 则正平衡点$ (u_{*}, v_{*}) $局部渐近稳定.

当$ 0<\frac{d_{2}}{d_{1}}<c^{H}+1-c $时, 令

$ \begin{eqnarray*} \Delta& = &\left[\frac{a^{2}}{b^{2}}\left(d_{2}+(c-c^{H}-1)d_{1}\right)\right]^{2}-\frac{4a^{2}d_{1}d_{2}}{b} \\ & = &\frac{a^{4}}{b^{4}}d_{2}^{2}+\left[\frac{2a^{4}(c-c^{H}-1)}{b^{4}}-\frac{4a^{2}}{b}\right]d_{1}d_{2}+\frac{a^{4}(c-c^{H}-1)^{2}}{b^{4}}d_{1}^{2}. \end{eqnarray*} $

注意到二次函数

$ f(z) = \frac{a^{4}}{b^{4}}z^{2}+\left[\frac{2a^{4}(c-c^{H}-1)}{b^{4}}-\frac{4a^{2}}{b}\right]z+\frac{a^{4}(c-c^{H}-1)^{2}}{b^{4}} $

的判别式为

$ \begin{eqnarray*} \tilde{\Delta}& = &\left[\frac{2a^{4}(c-c^{H}-1)}{b^{4}}-\frac{4a^{2}}{b}\right]^{2}-\frac{4a^{8}(c-c^{H}-1)^{2}}{b^{8}} \\ & = &\frac{16a^{4}[b^{3}-a^{2}(c-c^{H}-1)]}{b^{5}}>0, \end{eqnarray*} $

因此$ f(z) = 0 $存在两个实根

$ z_{1} = c^{H}+1-c+\frac{2b^{2}}{a^{2}}\bigg[b-\sqrt{\frac{b^{3}-a^{2}(c-c^{H}-1)}{b}}\bigg], $

$ z_{2} = c^{H}+1-c+\frac{2b^{2}}{a^{2}}\bigg[b+\sqrt{\frac{b^{3}-a^{2}(c-c^{H}-1)}{b}}\bigg]. $

如果$ z_{1}<z<z_{2} $, 那么$ f(z)<0 $, 则当$ z_{1}<\frac{d_{2}}{d_{1}}<z_{2} $时, $ \Delta<0 $, 从而有$ D_{k}>0, k\geq0 $.由于$ z_{1}<c^{H}+1-c<z_{2} $并且当$ \frac{d_{2}}{d_{1}}\geq c^{H}+1-c $时, 正平衡点$ (u_{*}, v_{*}) $局部渐近稳定, 所以当$ \frac{d_{2}}{d_{1}}>z_{1} $时, $ (u_{*}, v_{*}) $局部渐近稳定.

定理3.1  设$ b>a^{\frac{2}{3}} $.

(ⅰ) 当$ 0<c<c^{H} $时, 系统(3.1)的正平衡点$ (u_{*}, v_{*}) $不稳定; 当$ c\geq c^{H}+1 $时, 系统(3.1)的正平衡点$ (u_{*}, v_{*}) $局部渐近稳定;

(ⅱ) 当$ c^{H}<c<c^{H}+1 $时, 若$ \frac{d_{2}}{d_{1}}>z_{1} $, 则系统(3.1)的正平衡点$ (u_{*}, v_{*}) $局部渐近稳定.

下面讨论$ 0<\frac{d_{2}}{d_{1}}<z_{1} $的情形, 此时$ \Delta>0 $, 从而方程

$ d_{1}d_{2}\mu^{2}+\frac{a^{2}}{b^{2}}\left[d_{2}+(c-c^{H}-1)d_{1}\right]\mu+\frac{a^{2}}{b} = 0 $

存在两个正实根

$ \mu_{-}(d_{1}, d_{2}) = \frac{R-\sqrt{R^{2}-4d_{1}d_{2}D}}{2d_{1}d_{2}}, \quad \mu_{+}(d_{1}, d_{2}) = \frac{R+\sqrt{R^{2}-4d_{1}d_{2}D}}{2d_{1}d_{2}}, $

其中

$ R = -\frac{a^{2}}{b^{2}}d_{2}-\frac{a^{2}c-b^{3}}{b^{2}}d_{1} = d_{2}A+d_{1}N, \quad D = \frac{a^{2}}{b}>0. $

当$ c^{H}<c<c^{H}+1 $时, $ N>0 $.注意到

$ \mu_{+}(d_{1}, d_{2}) = \frac{1}{2d_{2}}\left[\frac{d_{2}}{d_{1}}A+N+\sqrt{\left(\frac{d_{2}}{d_{1}}A+N\right)^{2}-\frac{4d_{2}D}{d_{1}}}\right]. $

令

$ H(d_{1}) = \frac{d_{2}}{d_{1}}A+N+\sqrt{\left(\frac{d_{2}}{d_{1}}A+N\right)^{2}-\frac{4d_{2}D}{d_{1}}}, $

则

$ H'(d_{1}) = -\left[\frac{d_{2}}{d_{1}^{2}}A+\frac{\left(\frac{d_{2}}{d_{1}}A+N\right)\frac{d_{2}}{d_{1}^{2}}A-\frac{2d_{2}D}{d_{1}^{2}}}{\sqrt{\left(\frac{d_{2}}{d_{1}}A+N\right)^{2}-\frac{4d_{2}D}{d_{1}}}}\right]. $

由于$ A<0, N>0 $并且当$ 0<\frac{d_{2}}{d_{1}}<z_{1} $时, $ \frac{d_{2}}{d_{1}}A+N>0 $.因此对所有的$ d_{1}>0 $, 有$ H'(d_{1})>0 $, 从而$ H(d_{1}) $关于$ d_{1} $是单调递增的, 即$ \mu_{+}(d_{1}, d_{2}) $关于$ d_{1} $单调递增.

定义

$ \Phi_{1} = \{\mu|\mu\geq 0, \ \mu_{-}(d_{1}, d_{2})<\mu<\mu_{+}(d_{1}, d_{2})\}, \; \; \; \; \Phi_{2} = \{\mu_{0}, \mu_{1}, \mu_{2}, \mu_{3}, \cdot\cdot\cdot\}. $

要使不等式$ 0<\frac{d_{2}}{d_{1}}<z_{1} $成立, 可以固定$ d_{2} $而取$ d_{1} $足够大, 或固定$ d_{1} $而取$ d_{2} $足够小.

固定$ d_{2} $且令$ d_{1}\rightarrow\infty $, 则

$ \lim\limits_{d_{1}\rightarrow\infty} \mu_{-}(d_{1}, d_{2}) = 0, \quad \lim\limits_{d_{1}\rightarrow\infty} \mu_{+}(d_{1}, d_{2}) = \frac{b^{3}-a^{2}c}{d_{2}b^{2}}. $

从而对所有的$ d_{1}>0 $, 有

$ 0<\mu_{+}(d_{1}, d_{2})<\frac{b^{3}-a^{2}c}{d_{2}b^{2}}. $

如果$ \mu_{1}>\frac{b^{3}-a^{2}c}{d_{2}b^{2}} $, 那么$ \Phi_{1}\cap\Phi_{2} = \varnothing $, 即对所有的$ k\in N, D_{k}>0 $且$ T_{k}<0 $, 故系统(3.1)的正平衡点$ (u_{*}, v_{*}) $局部渐近稳定.由上述分析可得如下结论.

定理3.2  设$ c^{H}<c<c^{H}+1 $.若

(3.2) $ \begin{equation} \mu_{1}>\frac{b^{3}-a^{2}c}{d_{2}b^{2}}, \end{equation} $

则对于固定的$ d_{2}>0 $和所有的$ d_{1}>0 $, 系统(3.1)的正平衡点$ (u_{*}, v_{*}) $局部渐近稳定.

固定$ d_{1} $且令$ d_{2}\rightarrow 0 $, 则

$ \lim\limits_{d_{2}\rightarrow 0} \mu_{-}(d_{1}, d_{2}) = \frac{a^{2}b}{d_{1}(b^{3}-a^{2}c)}, \quad \lim\limits_{d_{2}\rightarrow 0}\mu_{+}(d_{1}, d_{2}) = \infty. $

因此, 存在$ \tilde{d}>0 $, 使得当$ 0<d_{2}<\tilde{d} $时, $ \Phi_{1}\cap\Phi_{2}\neq\varnothing $, 从而系统(3.1)正平衡点$ (u_{*}, v_{*}) $不稳定.

定理3.3  设$ c^{H}<c<c^{H}+1 $.对固定的$ d_{1}>0 $, 存在$ \tilde{d}>0 $使得当$ 0<d_{2}<\tilde{d} $时, 系统(3.1)的正平衡点$ (u_{*}, v_{*}) $是Turing不稳定的.

定理3.4  令

(3.3) $ \begin{equation} \bar{z} = 1+\frac{2b^{2}}{a^{2}}\left[b-\sqrt{b^{2}+\frac{a^{2}}{b}}\right], \end{equation} $

如果$ \frac{d_{2}}{d_{1}}>\bar{z} $, 那么系统(3.1)在$ (u_{*}, v_{*}, c^{H}) $处产生Hopf分支, 且该Hopf分支为次临界方向, 周期闭轨渐近稳定.

证  如果$ c = c^{H} $, 那么$ T_{0} = 0 $且$ D_{0}>0 $.由于$ \mu_{k}>0(k\geq1) $且$ d_{1}, d_{2}>0 $, 于是对所有的$ k\geq1 $, 有$ T_{k}(c^{H})<0 $.因为

$ D_{k}(c^{H}) = d_{1}d_{2}\mu_{k}^{2}+\frac{a^{2}}{b^{2}}(d_{2}-d_{1})\mu_{k}+\frac{a^{2}}{b}, $

由前面的计算可知, 当$ \frac{d_{2}}{d_{1}}>\bar{z} $时, 对任意的$ k\geq 1 $, 都有$ D_{k}(c^{H})>0 $.因此, 当$ c = c^{H} $时, 算子$ L $除一对纯虚特征根外, 其他特征值均具有负实部.

由文献[12]知, 分支的方向和分支周期解的稳定性由$ \rho''(0) $的符号给出, 其中

$ \rho''(0) = -\frac{1}{\lambda_{1}'(c^{H})}{\rm Re}(d_{1}(c^{H})), $

$ d_{1}(c^{H}) = \frac{\rm i}{2\lambda_{2}(c^{H})}\langle q^{*}, Q_{qq}\rangle\cdot \langle q^{*}, Q_{q\bar{q}}\rangle+\langle q^{*}, Q_{\tilde{q}q}\rangle+ \frac{1}{2}\langle q^{*}, Q_{\hat{q}\bar{q}}\rangle+\frac{1}{2} \langle q^{*}, C_{qq\bar{q}}\rangle. $

由文献[12]的定理2.1可知, 如果

$ \frac{1}{\lambda_{1}'(c^{H})}{\rm Re}(d_{1}(c^{H}))<0(>0), $

那么分支是超临界的(或次临界的).另外, 若$ L $的所有其它特征值都有负实部且$ {\rm Re}(d_{1}(c^{H}))<0 $($ >0 $), 则分支周期解是稳定的(不稳定的).

设$ L^{*} $是线性化算子$ L $的伴随算子, 定义

$ L^{*}\left( \begin{array}{cccc} u_{t}\\ v_{t} \end{array} \right) : = D\left( \begin{array}{cccc} u_{xx}\\ v_{xx} \end{array} \right) + L_{0}^{*}\left( \begin{array}{cccc} u\\ v \end{array} \right), $

其中$ L_{0}^{*} = L_{0}^{T} $, $ L_{0}^{*} = \left( \begin{array}{cccc} -\frac{a^{2}}{b^{2}} & \frac{a^{2}}{b^{2}} \\ \frac{a^{2}c-2b^{3}}{b^{2}} & -\frac{a^{2}c-b^{3}}{b^{2}} \end{array} \right). $

令

$ q: = \left( \begin{array}{cccc} s_{0}\\ t_{0} \end{array} \right) = \left( \begin{array}{cccc} { } -1+\frac{b\sqrt{b}}{a}{\rm i}\\ 1 \end{array} \right) , \quad q^{*}: = \left( \begin{array}{cccc} s^{*}_{0}\\ t^{*}_{0} \end{array} \right) = \frac{1}{2\pi}\left( \begin{array}{cccc} { } \frac{a}{b\sqrt{b}}{\rm i}\\ { } 1+\frac{a}{b\sqrt{b}}{\rm i} \end{array} \right), $

满足$ \langle q^{*}, q\rangle = 1, $$ \langle q^{*}, \bar{q}\rangle = 0, $$ L(c^{H})q = {\rm i}\lambda_{2}(c^{H})q, $$ L^{*}(c^{H})q^{*} = -{\rm i}\lambda_{2}(c^{H})q^{*} $, 其中$ \langle p, q\rangle = \int^{\pi}_{0}\bar{p}^{T}q{\rm d}x $.令

$ f(u, v, c) = a-uv^{2}+cv^{3}, \quad g(u, v, c) = uv^{2}-cv^{3}-bv. $

定义

$ Q_{qq} = \left( \begin{array}{cccc} e_{k}\\ f_{k} \end{array} \right)\cos^{2}(kx), \quad Q_{q\bar{q}} = \left( \begin{array}{cccc} g_{k}\\ h_{k} \end{array} \right)\cos^{2}(kx), \quad C_{qq\bar{q}} = \left( \begin{array}{cccc} m_{k}\\ n_{k} \end{array} \right)\cos^{2}(kx), $

其中

$ \begin{eqnarray*} &&e_{k} = f_{uu}s^{2}_{k}+2f_{uv}s_{k}t_{k}+f_{vv}t^{2}_{k}, \\ &&f_{k} = g_{uu}s^{2}_{k}+2g_{uv}s_{k}t_{k}+g_{vv}t^{2}_{k}, \\ &&g_{k} = f_{uu}|s_{k}|^{2}+f_{uv}(s_{k}\overline{t_{k}}+\overline{s_{k}}t_{k})+f_{vv}|t_{k}|^{2}, \\ &&h_{k} = g_{uu}|s_{k}|^{2}+g_{uv}(s_{k}\overline{t_{k}}+\overline{s_{k}}t_{k})+g_{vv}|t_{k}|^{2}, \\ &&m_{k} = f_{uuu}|s_{k}|^{2}+f_{uuv}(2|s_{k}|^{2}t_{k}+s^{2}_{k}\overline{t_{k}})+f_{uvv}(2|t_{k}|^{2}s_{k}+t^{2}_{k}\overline{s_{k}})+f_{vvv}|t_{k}|^{2}t_{k}, \\ &&n_{k} = g_{uuu}|s_{k}|^{2}+g_{uuv}(2|s_{k}|^{2}t_{k}+s^{2}_{k}\overline{t_{k}})+g_{uvv}(2|t_{k}|^{2}s_{k}+t^{2}_{k}\overline{s_{k}})+g_{vvv}|t_{k}|^{2}t_{k}. \end{eqnarray*} $

当$ k = 0 $时, 计算得

$ Q_{qq} = \left( \begin{array}{cccc} e_{0}\\ -e_{0} \end{array} \right), \quad Q_{q\bar{q}} = \left( \begin{array}{cccc} g_{0}\\ -g_{0} \end{array} \right), \quad C_{qq\bar{q}} = \left( \begin{array}{cccc} m_{0}\\ -m_{0} \end{array} \right), $

其中

$ e_{0} = \frac{2b^{2}}{a}-4\sqrt{b}i = -f_{0}, \quad g_{0} = \frac{2b^{2}}{a} = -h_{0}, \quad m_{0} = \frac{6b^{3}}{a^{2}}-\frac{2b\sqrt{b}}{a}i = -n_{0}. $

按照内积定义计算可得

$ \langle q^{*}, Q_{qq}\rangle = -\frac{e_{0}}{2} = -\frac{b^{2}}{a}+2\sqrt{b}i = \langle\bar{q}^{*}, Q_{qq}\rangle, $

$ \langle q^{*}, Q_{q\bar{q}}\rangle = -\frac{g_{0}}{2} = -\frac{b^{2}}{a} = \langle \bar{q}^{*}, Q_{q\bar{q}}\rangle, $

$ \langle q^{*}, C_{qq\bar{q}}\rangle = -\frac{m_{0}}{2} = -\frac{3b^{3}}{a^{2}}+\frac{b\sqrt{b}}{a}i. $

令

$ \hat{q}: = [2{\rm i}\lambda_{2}(c^{H})I-L(c^{H})]^{-1}\omega_{20}, {\quad} \tilde{q}: = -[L(c^{H})]^{-1}\omega_{11}, $

其中

$ \omega_{20} = Q_{qq}-\langle q^{*}, Q_{qq}\rangle q-\langle \bar{q}^{*}, Q_{qq}\rangle\bar{q}, $

$ \omega_{11} = Q_{q\bar{q}}-\langle q^{*}, Q_{q\bar{q}}\rangle q-\langle \bar{q}^{*}, Q_{q\bar{q}}\rangle\bar{q}. $

计算得$ \omega_{20} = \omega_{11} = 0 $.从而有

$ \langle q^{*}, Q_{\tilde{q}q}\rangle = \langle q^{*}, Q_{\hat{q}\bar{q}}\rangle = 0. $

进一步计算得

$ \begin{eqnarray*} {\rm Re}(d_{1}(c^{H}))& = &\frac{-{\rm Im}[\langle q^{*}, Q_{qq}\rangle\cdot\langle q^{*}, Q_{q\bar{q}}\rangle]} {2\lambda_{2}(c^{H})}+\frac{{\rm Re}\langle q^{*}, C_{qq\bar{q}}\rangle}{2}\\ & = &-\frac{g_{0}{\rm Im}(e_{0})}{8\lambda_{2}(c^{H})}-\frac{{\rm Re}(m_{0})}{4}\\ & = &-\frac{b^{3}}{2a^{2}}<0. \end{eqnarray*} $

由文献[13]中的Hopf分支定理知, 系统(3.1)在$ (u_{*}, v_{*}, c^{H}) $处产生Hopf分支.注意到$ \lambda_{1}'(c^{H})<0 $, 因此该Hopf分支为次临界方向且周期闭轨渐近稳定.证毕.

本节利用Matlab软件给出具体的数值实例, 以验证补充前面给出的结果.

对于常微分系统(2.1), 取$ a = 0.5, b = 1 $, 则$ c^{H} = 3 $.若取$ c = 3.4>c^{H} $, 则正平衡点$ (u_{*}, v_{*}) $渐近稳定, 见图 1.若取$ c = 2.9<c^{H} $, 由定理2.1可知, 当$ c $经过$ c^{H} $的左侧会失去它的稳定性, 并在$ c^{H} $附近产生Hopf分支, Hopf分支的方向是次临界的且分支周期解渐近稳定, 见图 2.初始值均取$ (u_{0}, v_{0}) = (3.5, 0.5) $.

扩散系统(3.1)包含五个参数: $ a, b, c, d_{1}, d_{2} $.取$ a = 0.5, b = 1, c = 3.5 $, 则$ z_{1} = 0.0147 $.如果$ d_{1} = 1, d_{2} = 0.015 $, 那么$ d_{1}, d_{2} $满足$ d_{2}/d_{1}>z_{1} $.由定理3.1(ⅱ)知系统(3.1)的正平衡点$ (u_{*}, v_{*}) $局部渐近稳定, 见图 3.如果$ d_{1} = 1, d_{2} = 0.01 $, 那么$ d_{1}, d_{2} $满足$ 0<d_{2}/d_{1}<z_{1} $, 但(3.2)式成立, 由定理3.2知系统(3.1)的正平衡点$ (u_{*}, v_{*}) $局部渐近稳定, 见图 4.如果$ d_{1} = 1, d_{2} = 0.005 $, 那么$ d_{1}, d_{2} $满足$ 0<d_{2}/d_{1}<z_{1} $且(3.2)式不成立.此时$ d_{2} $很小, 由定理3.3可知正平衡点可能是Turing不稳定的, 系统可能产生非常数稳态分支, 见图 5.如果$ d_{1} = 1, d_{2} = 0.08 $, 那么$ d_{1}, d_{2} $满足不等式(3.3), 由定理3.4知系统(3.1)产生稳定的Hopf分支周期解, 见图 6.初始值均取$ (u_{0}, v_{0}) = (3.5+0.1254\cos5x, 0.5+0.1254\cos5x) $.

本文在Neumann边界条件下研究一类自催化可逆三分子生化反应模型, 以可逆反应率为参数, 分别给出常微分系统和扩散系统的Hopf分支及其稳定性.特别对扩散系统, 给出扩散系数对系统稳定性的影响.结果表明, 当可逆反应率较小时, 正平衡点不稳定; 当可逆反应率较大时, 正平衡点稳定; 当可逆反应率介于某一范围内时, 扩散系数会对系统的稳定性产生较大影响.此时, 当催化剂的扩散系数较小时, 会产生Turing不稳定性.