生态学和传染病学是两个重要的研究领域.生态流行病学是理论生物学的 一个比较新的研究分支,通过处理生态学和传染病学的问题来处理这种情 况. 它可以被看做是生态学中的捕食者-食饵(竞争)模型和传染病学中 的SI,SIS或SIRS模型的结合. 了解捕食者与食饵的动力学关系是种群 动力学的主要目的之一. 传染病在捕食者-食饵系统中也有很重要的 影响[1, 2, 3, 4]. 捕食者-食饵的另一个重要关系是捕食者捕食食饵 的捕食率,也就是捕食者的功能性反应. 文献[5]研究了一个 具有传染率,疾病发作率等等的捕食者功能性反应的寄主-寄生虫-捕食者 模型动力学行为. 根据下面的假设生态流行病模型 {dSdt=rS(1−S+IK)−λIS−h(S)P,[3mm]dIdt=λIS−βIP−μI,[3mm]dPdt=−θβIP−δP+θh(S)P.(1.1) 系统(1.1)在初值条件 S(0)>0; I(0)>0; P(0)>0 情况下进行分析. 系统的平衡点已经由文献[5]给出,也是系统稳定性的条件.
众所周知,扩散在决定捕食者-食饵模型的动力学行为中扮演重 要的角色[6, 7, 8]. 因为种群密度通常不是齐次分布,实际上捕食者和 食饵的自然发展策略是幸存. 扩散可以描述捕食者和食饵种群密度不同时 种群的移动也不相同这一个复杂情况. 文献[]研究了一个具有疾 病和扩散的Holling I捕食者-食饵模型. 文献[11]研究了一个具有疾 病和扩散的比率依赖捕食者-食饵模型.
考虑到在一个固定的有界区域Ω内,对于任意给定的时刻 t种群的分布不均匀,这里Ω⊂RN,(N≥1).而且, 根据下面的假设,h(u1)是系统(1.1)的Holling II功能性反应项. 对系统(1.1)进行无量纲化 u1=SK,u2=IK, u3=PK,τ=λKt,b=rλK, d=βλ,e=μλK, g=δλK,l=Ka, m=αλa. 我们把τ仍记为t,得到 下面反应扩散类型的PDE系统 {u1t−d1△u1=bu1[1−(u1+u2)]−u1u2−mu1u31+lu1,[3mm]u2t−d2△u2=u1u2−du2u3−eu2,x∈Ω,t>0,[3mm]u3t−d3△u3=mu1u31+lu1−du2u3−gu3,[3mm]∂u1∂v=∂u2∂v=∂u3∂v=0,x∈∂Ω,t>0,[2mm]u1(x,0),u2(x,0),u3(x,0)≥0,x∈Ω,(1.2) 这里△是Laplace算子,Ω是RN中边界光滑的区域, ν是Ω的单位外法向量. 正常数d1, d2, d3 是扩散系数,并且有初值ui(x,0) (i=1,2,3)是连续函数. 通 过文献[5]可知系统(1.2)存在五个平衡点,分别为 E0=(0,0,0), E1=(1,0,0), E2=(e,b(1−e)b+1,0), E3=(gm−gl,0,b(m−gl−g)(m−gl)2), E∗=(u∗1,u∗2,u∗3), 其中 u∗2=1d[mu∗11+lu∗1−g], u∗3=u∗1−ed, 其中u∗1满足方程 bdlu∗12−[bd(l−1)+(b+1)gl−m(b+2)]u∗1−[me+(b+1)g+bd]=0.(1.3) 平衡点E0,E1对于所有的参数值总是存在的; E2存在当且仅当e<1; E3存在当且仅当m>g(l+1); 系统的唯一正平衡点E∗存在当且仅当下列条件成立 g(l+1)<m<gl(b+1)+bd(l−1)b+2, u∗1>max
本文的主要目的之一是研究系统(1.2)的正稳态解 的存在性和不存在性,也就是下面椭圆系统的非常值正解的存在性与不存在性 \left\{\begin{array}{ll} -d_{1}\triangle u_{1}=bu_{1}(1-u_{1})-(b+1)u_{1}u_{2}-\frac{mu_{1}u_{3}}{1+lu_1},\\ [3mm] -d_{2}\triangle u_{2}=u_{1}u_{2}-du_{2}u_{3}-eu_{2}, &x\in\Omega,\\ [3mm] -d_{3}\triangle u_{3}=\frac{mu_{1}u_{3}}{1+lu_1}-du_{2}u_{3}-gu_{3},\\ [3mm] \frac{\partial u_{1}}{\partial {\nu}}=\frac{\partial u_{2}}{\partial {\nu}}=\frac{\partial u_{3}}{\partial {\nu}}=0,&x\in\partial\Omega. \end{array} \right.(1.5)
文章剩余部分大体由以下几部分构成. 在第二部分, 通过线性化局部分析证明非负平衡点的局部渐进稳定性. 在第三部分, 对正稳态解的上下界进行先验估计. 在第四部分, 通过构造适当的Lyapunov函数证明了非负平衡点的全局渐进稳定性. 在第五、 六部分,我们研究了非常值正解的存在性和不存在性.
文献[5]中, 作者研究了系统(1.1)中非负平衡点的稳定性.本文研究系统(1.2)非负 平衡点的稳定性. 令0=\mu_{1}<\mu_{2}<\mu_{3}<\cdots是 -\Delta在\Omega中的特征根,并且具有齐次Neumann边界条件,令 X=\bigg\{(u_1,u_2,u_3)\in[C^{1}(\bar\Omega)]^{3}|\frac{\partial u_1}{\partial\nu}=\frac{\partial u_2}{\partial\nu}=\frac{\partial u_3}{\partial\nu}=0,~x\in\partial\Omega\bigg\}, 且X=\bigoplus\limits_{i=1}^{\infty}X_{i},这里 X_i是\mu_{i}对应的特征空间.~
对系统(1.2)的右端求导可得 \begin{eqnarray*} &&G_{u}(\cdot)=: \left( \begin{array}{ccc} a_{11}&~~a_{12}~~&a_{13}\\a_{21}&a_{22}&a_{23}\\a_{31}&a_{32}&a_{33} \end{array} \right) \\ &=&\left( \begin{array}{ccc} b-2bu_{1}-(b+1)u_{2}-\frac{mu_{3}}{1+lu_{1}}+\frac{mu_{1}u_{3}}{(1+lu_{1})^{2}} &-(b+1)u_{1}& -\frac{mu_{1}}{1+lu_{1}}\\ [2mm] u_{2}&~u_{1}-du_{3}-e~&-du_{2}\\[2mm] \frac{mu_{3}}{1+lu_{1}}-\frac{mu_{1}u_{3}}{(1+lu_{1})^{2}}& -du_{3}& \frac{mu_{1}}{1+lu_{1}}-du_{2}-g\end{array} \right). \end{eqnarray*}
定理2.1 系统(1.2)的平衡点具有以下性质
(i)~ E_0是不稳定的;
(ii)~ 当e>1,m<g(1+l)成立时,E_1是渐进稳定的;
(iii)~ 当m<\frac{g(1+le)}{e}成立时,E_2是渐进稳定的;
(iv)~ 当\frac{g(1+el)}{e}<m<\frac{gl(1+l)}{l-1}, \frac{l-1}{2l}<e<1, 或 g(1+l)<m<\frac{gl(1+l)}{l-1}, e>1成立时,E_3是渐进稳定的.
证 下面仅对(ii)进行证明,其余的证明类似.
(ii)~ 令 D={\rm diag}(d_{1},d_{2},d_{3}),L=D\Delta+G_{u}(E_1). 系统(1.2)在E_1的线性化系统为 u_{t}=Lu. 对于任意的 i\geqslant 1,X_i在算子L下是不变的. 这意味着\lambda是L 在X_i上的特征值当且仅当\lambda是矩阵 -\mu_iD+f_u(u_1,u_2,u_3)的特征值. -\mu_iD+f_u(u_1,u_2,u_3)的特征多项式为 \Psi_i(\lambda)=\lambda^3+B_{1i}\lambda^2+B_{2i}\lambda+B_{3i}, 这里 B_{1i}=\mu_i(d_1+d_2+d_3)+A_1, B_{2i}=\mu_i^2(d_1d_2+d_1d_3+d_1d_3) -\mu_i[d_1(a_{22}+a_{33})+d_3(a_{11}+a_{22})+d_2(a_{11}+a_{33})]+A_2, \begin{eqnarray*} B_{3i}&=&\mu_i^3d_1d_2d_3-\mu_i^2(d_1d_2a_{33}+d_2d_3a_{11} +d_1d_3a_{22})\\ &&+ \mu_i(d_1a_{22}a_{33}+d_{2}a_{11}a_{33}+d_3a_{11}a_{22})+A_{3}, \end{eqnarray*} A_1=-(a_{11}+a_{22}+a_{33}), A_2=a_{11}a_{22}+a_{22}a_{33}+a_{11}a_{33}, A_3=-\det\{G_u(E_1)\}, 其中 \left\{ \begin{array}{l} a_{11}=-b<0,~~a_{12}=-(b+1)<0, ~~a_{13}=-\frac{m}{1+l}<0,\\[2mm] a_{21}=0,~~a_{22}=1-e<0,~~a_{23}=0,\\ [2mm] a_{31}=0,~~a_{32}=0,~~a_{33}=\frac{m}{1+l}-g<0,\end{array} \right. (2.1) 所以有A_{1}>0,A_{2}>0,A_{3}>0,B_{1i}>0,B_{3i}>0, B_{1i}B_{2i}-B_{3i}=M_1\mu_i^3+M_2\mu_i^2+M_3\mu_i+A_1A_2-A_3, 其中 M_1=(d_2+d_3)(d_1d_2+d_1d_3+d_2d_3)+d_1(d_1d_2+d_1d_3)>0, \begin{eqnarray*} M_2&=&-[a_{11}(d_1d_2+d_2d_3)+a_{22}(d_1d_2+d_2d_3)+a_{33}(d_1d_3+d_2d_3)]\\ && -(d_1+d_2+d_3)[d_1(a_{22}+a_{33})+d_2(a_{11}+a_{33})+d_3(a_{11}+a_{22})]>0, \end{eqnarray*} \begin{eqnarray*} M_3&=&2(d_1+d_2+d_3)(a_{11}a_{22}+a_{11}a_{33}+a_{22}a_{33})\\ &&+(d_2+d_3)a_{11}^{2}+(d_1+d_3)a_{22}^{2}+(d_1+d_2)a_{33}^{2}>0, \end{eqnarray*} A_1A_2-A_3>0.
于是,对于任意的i\geq 0,B_{1i}B_{2i}-B_{3i}>0, 则由Routh-Hurwitz判据可知 \Psi_i(\lambda)=0的三个根\lambda_{i,1},\lambda_{i,2}, \lambda_{i,3}都具有负实部.
下面证明存在一个正常数\delta使得 {\rm Re}\{\lambda_{i,1}\},{\rm Re}\{\lambda_{i,2}\},{\rm Re}\{\lambda_{i,3}\}\leqslant-\delta, \forall i\geq 1. (2.2) 因此,令\lambda=\mu_{i}\zeta,则 \Psi_i(\lambda)=\mu_{i}^3\zeta^3+B_{1i}\mu_{i}^2\zeta^2+B_{2i}\mu_{i}\zeta+B_{3i}\triangleq\tilde{\Psi_i}(\zeta). 因为当i\rightarrow\infty时,\mu_{i}\rightarrow\infty,故 \lim_{i\rightarrow+\infty}\{\tilde{\Psi_i}(\zeta)/\mu_{i}^3\} =\zeta^3+(d_1+d_2+d_3)\zeta^2+(d_1d_2+d_2d_3+d_3d_1)\zeta+d_1d_2d_3\triangleq\bar{\Psi}(\zeta). 应用Routh-Hurwitz判据可得\bar{\Psi}(\zeta)=0的三个根都具有负实部. 因此,存在一个正常数\bar{\delta}使得 {\rm Re}\{\zeta_1\}, ~{\rm Re}\{\zeta_2\},~{\rm Re}\{\zeta_3\}\leqslant-\bar{\delta}. 由连续性, 我们有\exists i_{0}, 使得\tilde{\Psi_i}(\zeta)=0的三个根\zeta_{i,1},\zeta_{i,2}, \zeta_{i,3}满足 {\rm Re}\{\zeta_{i,1}\}, {\rm Re}\{\zeta_{i,2}\}, {\rm Re}\{\zeta_{i,3}\}\leqslant-\frac{\bar{\delta}}{2}, \forall i\geqslant i_{0}. 从而有 {\rm Re}\{\lambda_{i,1}\}, ~{\rm Re}\{\lambda_{i,2}\}, ~{\rm Re}\{\lambda_{i,3}\}\leqslant-\frac{\mu_{i}\bar{\delta}}{2}, ~\forall i\geqslant i_{0}. 令-\tilde{\delta}= \max_{1\leqslant i\leqslant i_{0}}\{{\rm Re}\{\lambda_{i,1}\}, ~{\rm Re}\{\lambda_{i,2}\},~{\rm Re}\{\lambda_{i,3}\}\}, 则\tilde{\delta}>0, 当\delta=\min\{\tilde{\delta},~\frac{\bar{\delta}}{2}\} 时, (2.2)式成立.因此, L的包含特征值的谱位于\{{\rm Re}\lambda\leqslant-\delta\}, 则E_1的局部稳定性得证.
注 正平衡点E_*的稳定性需要进一步研究, 这里只给出关于G_u(E_*)的一些结果. 通过直接计算有 a_{11}=\frac{u_1^{*}}{d(1+lu_1^{*})^2}[-bdl^2{u_1^{*}}^2+u_1^{*}(m-2bdl)-(bd+me)]. 而且,a_{11}<0等价于 -bdl^2{u_1^{*}}^2+u_1^{*}(m-2bdl)-(bd+me)<0. (2.3) \left\{ \begin{array}{l} a_{11}<0,~~a_{12}=-(b+1)u_1^{*}<0,~~a_{13}=-\frac{mu_1^{*}}{1+lu_1^{*}}<0,\\ [3mm] a_{21}=u_2^{*}>0,~~a_{22}=u_1^{*}-d\frac{u_1^{*}-e}{d}-e=0,~~a_{23}=-du_2^{*}<0,\\ [3mm] a_{31}=\frac{m(u_1^{*}-e)[1+(l-1)u_1^{*}]}{d(1+lu_1^{*})^2}>0,~~a_{32}=-du_3^{*}<0,\\ [3mm] a_{33}=\frac{mu_1^{*}}{1+lu_1^{*}}-d\frac{1}{d}[\frac{mu_1^{*}}{1+lu_1^{*}}-g]-g=0. \end{array} \right.(2.4) 同时有 \det G_{u}(E_*)=-a_{11}a_{23}a_{32}+a_{12}a_{23}a_{31}+a_{13}a_{21}a_{32}>0.
这部分的主要目的是给出系统(1.5)正解的上下界的先验估计. 为了这个目的,我们首先引用下面已知的一些结果:
引理3.1 [12] 令\omega\in C^{2}(\Omega)\cap C^{1}(\bar{\Omega})是\triangle \omega(x)+c(x)\omega(x)=0的解,其中c\in C(\bar{\Omega}), \omega满足齐次Neumann边界条件,则存在一个正常数C^{\ast}, 其中C^{\ast}只依赖于\alpha,当\|c\|_{\infty}\leq\alpha时, 有\max{\Omega}\bar{\omega}\leq C^{\ast}\min_{\bar{\Omega}}\omega.
引理3.2 [13] 令g(x,\omega)\in C(\Omega\times R^{1}),b_{j}(x)\in C(\bar\Omega),j=1,2,\cdots N,
(I)~ 如果\omega(x)\in C^{2}(\Omega)\cap C^{1}(\bar{\Omega})在 \Omega上满足\triangle \omega(x)+\sum\limits_{j=1}^N b_{j}(x)\omega_{x_{j}}+g(x,\omega(x)) \geq0,在\partial\Omega上满足\partial\omega/\partial\nu\leq0 且\omega(x_{0})=\max_{\bar{\Omega}}\omega,则g(x_{0}, \omega(x_{0}))\geq0.
(II)~ 如果\omega(x)\in C^{2}(\Omega)\cap C^{1}(\bar{\Omega}) 在\Omega上满足\triangle \omega(x)+\sum\limits_{j=1}^N b_{j}(x) \omega_{x_{j}}+g(x,\omega(x))\leq0,在\partial\Omega上满足 \partial\omega/\partial\nu\geq0且\omega(x_{0})= \min_{\bar{\Omega}}\omega,则g(x_{0},\omega(x_{0}))\leq0.
为了简便,定义常数(d_{1},d_{2},d_{3})=d_*,(b,~d_*,~e,g, m_1)=\Lambda.下面给出系统(1.5)正稳态解的有界性.
定理3.1 (上界) 如果(u_1,u_2,u_3) 是系统(1.5)的正解, 则 \max_{\bar{\Omega}}u_{1}(x)\leqslant1, \max_{\bar{\Omega}}u_{2}(x)\leqslant\frac{4ed_{1}+bd_{2}} {4ed_{2}(b+1)}, \max_{\bar{\Omega}}u_{3}(x)\leqslant\frac{4gd_{1}+bd_{3}}{4gd_{3}}. (3.1)
证 首先直接对系统(1.5)用最大值原理有 -d_{1}\triangle u_{1}=bu_{1}(1-u_{1})-(b+1)u_{1}u_{2} -\frac{mu_{1}u_{3}}{1+lu_1}\leqslant{bu_{1}(1-u_{1})} 使得\max_{\bar{\Omega}}u_{1}(x)\leqslant1.
令\omega_{1}=d_{1}u_{1}+(b+1)d_{2}u_{2}, 则 \left\{\begin{array}{ll} -\triangle\omega_{1}=bu_{1}(1-u_{1})- \frac{mu_{1}u_{3}}{1+lu_1}-d(b+1)u_{2}u_{3}-e(b+1)u_{2}, ~~&x\in\Omega,\\[3mm] \frac{\partial \omega_{1}}{\partial{\nu}}=0, &x\in\partial\Omega.\end{array} \right.(3.2) 如果\omega_{1}(x_{0})=\max_{\Omega}\omega_{1}(x),由引理3.2可得 e(b+1)u_{2}(x_{0})\leqslant{bu_{1}(1-u_{1})-\frac{mu_{1}u_{3}}{1+lu_1}-d(b+1)u_{2}u_{3}}\leqslant{\frac{b}{4}}, 进而可得 (b+1)d_{2}\max_{\bar{\Omega}}u_{2}(x)\leqslant{\max_{\Omega}\omega_{1}(x)}= d_{1}u_{1}(x_{0})+(b+1)d_{2}u_{2}(x_{0})\leqslant{d_{1}+{\frac{bd_{2}}{4e}}}=\frac{4ed_{1}+bd_{2}}{4e}, 因此 \max_{\bar{\Omega}}u_{2}(x)\leqslant\frac{4ed_{1}+bd_{2}}{4ed_{2}(b+1)}. 令\omega_{2}=d_{1}u_{1}+d_{3}u_{3}, 我们有 \left\{\begin{array}{ll} -\triangle\omega _{2}=bu_{1}(1-u_{1})-(b+1)u_{1}u_{2}-du_{2}u_{3}-gu_{3}, ~~&x\in\Omega,\\[2mm] \frac{\partial \omega_{2}}{\partial {\nu}}=0, &x\in\partial\Omega. \end{array} \right. (3.3) 如果\omega_{2}(x_{0})=\max_{\Omega}\omega_{2}(x), 由引理3.2可得 gu_{3}(x_{0})\leqslant{bu_{1}(1-u_{1})-(b+1)u_{1}u_{2}-du_{2}u_{3}}\leqslant{\frac{b}{4}} , 进而可得 d_{3}\max_{\bar{\Omega}}u_{3}(x)\leqslant{\max_{\Omega}\omega_{2}(x)}= d_{1}u_{1}(x_{0})+d_{3}u_{3}(x_{0})\leqslant{d_{1}+{\frac{bd_{3}}{4g}}}=\frac{4gd_{1}+bd_{3}}{4g}, 因此 \max_{\bar{\Omega}}u_{3}(x)\leqslant\frac{4gd_{1}+bd_{3}}{4gd_{3}}~. 证毕.
定理3.2 (下界) 设\Lambda,\underline{d}_{1}, \underline{d}_{2},\underline{d}_{3}是正数,如果 \frac{A}{b}<1-e,~(b+1)A^2-4bl[mB-b+A(b+1)]<0,~(3.4) 其中 A=\frac{4ed_{1}+bd_{2}}{4ed_{2}(b+1)} , B=\frac{4gd_{1}+bd_{3}}{4gd_{3}}, ~(d_{1},d_{2},d_{3})\in{{[\underline{d}_{1},{\infty})} \times{[\underline{d}_{2},{\infty})}\times{[\underline{d}_{3}, {\infty})}}. 那么,存在一个正常数\underline{C}=\underline{C}(\Lambda,\underline{d}_{1},\underline{d}_{2},\underline{d}_{3}),使得系统(1.5)的正解u_1,u_2,u_3满足\min_{\bar{\Omega}}u_{i}>\underline{C} ,~i=1,2,3.
证 令 c_{1}(x)=d_{1}^{-1} \bigg[b(1-u_{1})-(b+1)u_{2}-\frac{mu_{3}}{1+lu_1}\bigg], c_{2}(x)=d_{2}^{-1}[u_{1}-du_{3}-e] , c_{3}(x)=d_{3}^{-1}\bigg[\frac{mu_{1}}{1+lu_1}-du_{2}-g\bigg]. 由(3.1)式可知,当d_{1},d_{2},d_{3}>\overline{d}时, 存在一个常数\overline{C}(\overline{d}, \Lambda)使得\|c_{1}\|_{\infty},\|c_{2}\|_{\infty}, \|c_{3}\|_{\infty}\leqslant\overline{C}. 由于u_{1},u_{2},u_{3} 满足\triangle{u_{i}}+c_{i}(x)u_{i}=0,\frac{\partial u_{i}}{\partial {\nu}}=0,(i=1,2,3). 根据引理3.1可知存在一个正常数C_{*}=C_{*}(\Lambda,\overline{d}) 使得\max_{\bar{\Omega}}u_{i}\leqslant{C_{*}\min_{\bar{\Omega}}u_{i}}, ~i=1,2,3.
应用反证法, 假设存在一个序列(d_{1i},d_{2i},d_{3i}), 使得系统(1.5)对应的正解(u_{1i},u_{2i},u_{3i})满足\max_{\bar{\Omega}}u_{1i}\rightarrow0 或者\max_{\bar{\Omega}}u_{2i}\rightarrow0 或者\max_{\bar{\Omega}}u_{3i}\rightarrow0,i\rightarrow{\infty}.\\ 由引理3.2可得u_{1i}\leqslant1. 分部积分可得 \left\{\begin{array}{l} \int_{\Omega}bu_{1i}(1-u_{1i})-(b+1)u_{1i}u_{2i}-\frac{mu_{1i}u_{3i}}{1+lu_{1i}}{\rm d}x=0,\\ [3mm] \int_{\Omega}u_{1i}u_{2i}-du_{2i}u_{3i}-eu_{2i}{\rm d}x=0,\\ [3mm] \int_{\Omega}\frac{mu_{1i}u_{3i}}{1+lu_{1i}}-du_{2i}u_{3i}-gu_{3i}=0, ~~~i=1,2,3.\end{array} \right.(3.5) 由椭圆方程的标准正则定理可知存在{(u_{1i},u_{2i},u_{3i})} 的一个子序列,这里仍记为{(u_{1i},u_{2i},u_{3i})}, 以及三个非负函数u_{1},u_{2},u_{3}\in{C^{2}(\overline{\Omega})}, 使得在[{C^{2}(\overline{\Omega})}]^{3}中, {(u_{1i},u_{2i},u_{3i})\rightarrow{(u_{1},u_{2},u_{3})}}, i\rightarrow{\infty}. 由(3.5)式可得u_{1}\equiv0 或u_{2}\equiv0 或u_{3}\equiv0.
进一步,令(d_{1i},d_{2i},d_{3i})\rightarrow ({\overline{d}_{1}, \overline{d}_{2},\overline{d}_{3}}) \in{{[\underline{d}_{1},{\infty})}\times{[\underline{d}_{2}, {\infty})}\times{[\underline{d}_{3},{\infty})}}, 其中\overline{d}_{1},\overline{d}_{2},\overline{d}_{3}满足(3.4)式. 由(3.5)式可得,当 i\rightarrow{\infty}时, \left\{\begin{array}{l} \int_{\Omega}bu_{1}(1-u_{1})-(b+1)u_{1}u_{2}-\frac{mu_{1}u_{3}}{1+lu_1}{\rm d}x=0,\\ [3mm] \int_{\Omega}u_{1}u_{2}-du_{2}u_{3}-eu_{2}{\rm d}x=0,\\ [3mm] \int_{\Omega}\frac{mu_{1}u_{3}}{1+lu_1}-du_{2}u_{3}-gu_{3}=0.\end{array} \right.(3.6)
我们考虑以下三种情况.
第一种情况: u_{1}\equiv0,~u_2>0,u_3>0.
由于当i\rightarrow{\infty}时,u_{1i}\rightarrow{u_{1}}, 故u_{1i}-du_{3i}-e<0, 0=d_{2i}\int_{\partial{\Omega}}\partial{\nu}u_{2i}{\rm d}S =-d_{2i}\int_{\Omega}\Delta{u_{2i}}{\rm d}x=\int_{\Omega}u_{2i}(u_{1i}-du_{3i}-e){\rm d}x<0{ , }i\gg1, 产生矛盾.
第二种情况: u_{2}\equiv0,u_{1}>0,u_3>0.
u_{1},u_{2},u_{3}满足 -d_{1}\triangle u_{1}=u_{1}\bigg[b(1-u_{1})-(b+1)u_{2} -\frac{mu_{3}}{1+lu_1}\bigg],~\frac{\partial u_1}{\partial\nu}=0. 令u_{1}(x_{0})=\min_{\bar{\Omega}}u_{1}(x), 由引理3.2可得 b(1-u_{1}(x_{0}))-(b+1)u_{2}(x_{0}) -\frac{mu_{3}(x_{0})}{1+lu_1(x_0)}\leqslant0, 于是bu_{1}(x_{0})\geqslant{b-(b+1)u_{2}(x_{0}) -\frac{mu_{3}(x_{0})}{1+lu_1(x_0)}}\geqslant b-(b+1)A -\frac{mB}{1+lu_1(x_0)}, 则\frac{blu_1(x_0)^2+(b+1)Au_1(x_0)+mB-[b-(b+1)A]} {1+lu_1(x_0)}\geqslant0. 对u_{2i}的微分方程在\Omega上分部积分有 0=d_{2i}\int_{\partial{\Omega}}\partial{\nu}u_{2i}{\rm d}S =-d_{2i}\int_{\Omega}\Delta{u_{2i}}{\rm d}x=\int_{\Omega}u_{2i}(u_{1i} -du_{3i}-e){\rm d}x>0 ,~~i\gg1, 产生矛盾.
三种情况: u_{3}\equiv0,u_{1}>0,u_{2}>0.
u_{1},u_{2}满足-d_{1}\triangle u_{1}=u_{1}[b(1-u_{1})-(b+1)u_{2}],{~~}\frac{\partial {u_{1}}}{\partial{\nu}}=0. 令u_{1}(x_{0})=\min_{\bar{\Omega}}u_{1}(x),通过引理3.2有 b(1-u_{1}(x_{0}))-(b+1)u_{2}(x_{0})\geqslant{b-\frac{4ed_{1}+bd_{2}}{4ed_{2}}}, 则有\min_{\bar{\Omega}}u_{1}(x)\geqslant{1-\frac{4ed_{1}+bd_{2}}{4bed_{2}}}. 由于\overline{d}_{1},\overline{d}_{2}满足(3.4)式, 易得u_{1i}-du_{3i}-e>0,i\gg1. 对u_{2i}的微分方程在\Omega上分部积分有 0=d_{2i}\int_{\partial{\Omega}}\partial{\nu}u_{2i}{\rm d}S =-d_{2i}\int_{\Omega}\Delta{u_{2i}}{\rm d}x=\int_{\Omega}u_{2i}(u_{1i} -du_{3i}-e){\rm d}x>0 ,~i\gg1, 矛盾产生,证毕.
定理4.1 当e>1,~m<g(1+l)成立时, 系统(1.2)的常值解(1,0,0)是全局渐近稳定的.
证 为了研究系统(1.2)的稳定性,考虑下面辅助系统 \left\{ \begin{array}{l} \omega_{1t}-d_1\Delta\omega_1=b\omega_1(1-\omega_1)-(b+1)\omega_1\omega_2,\\ \omega_{2t}-d_2\Delta \omega_2=\omega_1\omega_2-e\omega_2,\\ [2mm] \frac{\partial \omega_1}{\partial\nu}=\frac{\partial\omega_2}{\partial\nu}=0,~~~~x\in\partial\Omega,\\ [2mm] \omega_1(x,0)\geqslant0,\omega_2(x,0)\geqslant0. \end{array} \right.(4.1) 定义下面Lyapunov函数 V(t)=\int_{\Omega}(\omega_1-1-\ln \omega_1){\rm d}x+(b+1)\int_{\Omega}\omega_2{\rm d}x, 则我们有 \begin{eqnarray*} V'(t)&=&\int_{\Omega}\frac{\omega_{1}-1}{\omega_{1}}\omega_{1t}{\rm d}x+(b+1)\int_{\Omega}\omega_{2t}{\rm d}x\\ &=&\int_{\Omega} \bigg\{\frac{d_{1}(\omega_{1}-1)\triangle\omega_{1}}{\omega_{1}}+(\omega_{1}-1)[b(1-\omega_1)-(b+1)\omega_2] \bigg\}{\rm d}x\\ &&+(b+1)\int_{\Omega}(d_{2}\triangle \omega_{2}+\omega_1\omega_2-e\omega_2){\rm d}x\\ &\leq&-b\int_{\Omega}(\omega_1-1)^2{\rm d}x-(e-1)(b+1)\int_{\Omega}\omega_2{\rm d}x\\ &\leqslant &0. \end{eqnarray*} 由定理3.1有 \lim_{t\rightarrow+\infty}\|\omega_{1}-1\|^{2}=0, \quad \lim_{t\rightarrow+\infty}\|\omega_{2}-0\|^{2}=0. 因此,取\varepsilon>0充分小,\exists T, 使得当t>T时有\omega_{1}(x,t)\leq1+\varepsilon, ~\omega_{2}(x,t)\leq0+\varepsilon=\varepsilon. 通过比较原理, (1.2)式的任意正解满足u_{1}(x,t)\leq \omega_{1}(x,t)\leq1+\varepsilon, ~ u_{2}(x,t)\leq \omega_{2}(x,t)\leq0+\varepsilon. 因此 \left\{ \begin{array}{ll} u_{3t}-d_{3}\triangle u_{3}= u_{3}\bigg(\frac{mu_1}{1+lu_1}-du_2-g\bigg)\leqslant u_3\bigg[\frac{m(1+\varepsilon)}{1+l(1+\varepsilon)}-g\bigg], ~~ &x\in\Omega,\\ [3mm] \frac{\partial u_{3}}{\partial\nu}=0 ,&x\in\partial\Omega,~t>T,\\ [3mm] u_{3}(x,T)>0 ,&x\in\Omega. \end{array} \right. 令\omega_3'(t)=\omega_3[\frac{m(1+\varepsilon)}{1+l(1+\varepsilon)}-g], 可得\omega_3(t)=e^{c[\frac{m(1+\varepsilon)}{1+l(1+\varepsilon)}-g]t}, 其中 \frac{m(1+\varepsilon)}{1+l(1+\varepsilon)}-g<0如果m<g(1+l)并且\varepsilon充分小. 故 \lim_{t\rightarrow+\infty} \omega_3(t)=0. 因为u_3(x,t)\leqslant \omega_3(t), 通过比较原理可得\lim_{t\rightarrow+\infty} u_3(x,\cdot)=0. 因此 \lim_{t\rightarrow+\infty} \|(u_1(x,t),u_2(x,t),u_3(x,t))-(1,0,0)\|=0. 证毕.
下面定理的证明与定理4.1类似.
定理4.2 当e>1,m<\frac{g(1+le)}{e}时,(1.2)式的 平衡点(e,\frac{b(1-e)}{b+1},0)是全局渐近稳定的.
这部分,研究系统(1.5)非常值正解的不存在性. 定理5.1 令d_{2}^{\ast},d_{3}^{\ast}是 固定的常数, 并且满足d_{2}^{\ast}\mu_1>1-e,~d_{3}^{\ast}\mu_1>\frac{m}{1+l}-g, 则存在一个正常数d_{1}=d_{1}(\Lambda,d_{2}^{*},d_{3}^{*}), 使得当d_{i}\geq d_{i}^{\ast}(i=1,2,3)时, 系统(1.5)没有非常值正解.
证 \forall u_i\in{{L}^{1}(\Omega)}, 令 u_i=\frac{1}{|\Omega|}\int_{\Omega}{u_i}{\rm d}x. 首先, 在(1.5)式两端同时乘以{u}_{i}-\bar{u}_{i}, 然后积分有 \sum_{i=1}^{3}\int_{\Omega}d_{i}|\nabla u_{i}|^{2}{\rm d}x = \sum_{i=1}^{3}\int_{\Omega}G_{i}(u_{i})(u_{i}-\bar{u}_{i}){\rm d}x = \sum_{i=1}^{3}\int_{\Omega} (G_{i}(u_{i})-G_{i}(\bar{u}_{i}))(u_{i}-\bar{u}_{i}){\rm d}x. (5.1) 利用Young's不等式 |ab|\leqslant\frac{1}{p}|a|^{p}+\frac{1}{q}|b|^{q},~~ \frac{1}{p}+\frac{1}{q}=1, 有 \begin{eqnarray*} && \sum_{i=1}^{3}\int_{\Omega} (G_{i}(u_{i})-G_{i}(\bar{u}_{i}))(u_{i}-\bar{u}_{i}){\rm d}x\\ &=&\int_{\Omega}\bigg[b-bu_{1}-b\bar{u}_{1}-(b+1)u_{2} -\frac{mu_{3}}{(1+lu_1)(1+l\bar{u}_1)}\bigg](u_{1}-\bar{u}_{1})^{2}{\rm d}x \\ && +\int_{\Omega}(u_{1}-du_{3}-e)(u_{2}-\bar{u}_{2})^{2}{\rm d}x +\int_{\Omega} \bigg(\frac{mu_{1}}{1+lu_1}-du_{2}-g\bigg)(u_{3}-\bar{u}_{3})^{2}{\rm d}x \\ &&+\int_{\Omega}[-(b+1)\bar{u}_{1}+\bar{u}_{2}](u_{1}-\bar{u}_{1})(u_{2}-\bar{u}_{2}){\rm d}x\\ &&+\int_{\Omega} \bigg(-\frac{m\bar{u}_{1}}{1+l\bar{u}_1}+\frac{m\bar{u}_{3}} {(1+lu_1)(1+l\bar{u}_1)}\bigg)(u_{1}-\bar{u}_{1}) (u_{3}-\bar{u}_{3}){\rm d}x \\ && -\int_{\Omega}(d\bar{u}_{2}+d\bar{u}_{3})(u_{2}-\bar{u}_{2}) (u_{3}-\bar{u}_{3}){\rm d}x\\ &\leqslant & \int_{\Omega}b(u_{1}-\bar{u}_{1})^{2}{\rm d}x +\int_{\Omega}(u_{1}-e)(u_{2}-\bar{u}_{2})^{2}{\rm d}x+\int_{\Omega} \bigg(\frac{mu_{1}}{1+lu_1}-g\bigg)(u_{3}-\bar{u}_{3})^{2}{\rm d}x\\ &&+\int_{\Omega}[-(b+1)\bar{u}_{1}+\bar{u}_{2}] (u_{1}-\bar{u}_{1})(u_{2}-\bar{u}_{2}){\rm d}x \\ && +\int_{\Omega} \bigg(-\frac{m\bar{u}_{1}}{1+l\bar{u}_1} +\frac{m\bar{u}_{3}}{(1+lu_1)(1+l\bar{u}_1)}\bigg) (u_{1}-\bar{u}_{1})(u_{3}-\bar{u}_{3}){\rm d}x\\ &&-\int_{\Omega}(d\bar{u}_{2}+d\bar{u}_{3}) (u_{2}-\bar{u}_{2})(u_{3}-\bar{u}_{3}){\rm d}x\\ &\leqslant & \int_{\Omega}\bigg[(b+C_{1}+C_{2})(u_{2}-\bar{u}_{2})^{2}+(1-e+\varepsilon_{1})(u_{2}-\bar{u}_{2})^{2} +\bigg(\frac{m}{1+l}-g+\varepsilon_{2}\bigg) (u_{3}-\bar{u}_{3})^{2}\bigg]{\rm d}x, \end{eqnarray*} 其中\varepsilon_{1},\varepsilon_{2}是由Young's不等式引 起的任意小的正常数, C_{1}=C_{1}(\Lambda,d_{2}^{*},d_{3}^{*},\varepsilon_{1}), C_{2}=C_{2}(\Lambda, d_{2}^{*},d_{3}^{*},\varepsilon_{2}).
由Poincare不等式\mu_{1}\int_{\Omega}(f-\bar{f})^{2}{\rm d}x\leq \int_{\Omega}|\nabla f|^{2}{\rm d}x,我们有 \begin{eqnarray*} \mu_{1} \sum_{i=1}^{3}\int_{\Omega}d_{i}|\nabla u_{i}|^{2}{\rm d}x & \leqslant& \int_{\Omega}\bigg[(b+C_{1}+C_{2})(u_{2}-\bar{u}_{2})^{2}+(1-e+\varepsilon_{1})(u_{2}-\bar{u}_{2})^{2}\\ &&+\bigg(\frac{m}{1+l}-g+\varepsilon_{2}\bigg)(u_{3}-\bar{u}_{3})^{2}\bigg]{\rm d}x. \end{eqnarray*} 选取足够小的 \varepsilon_{2},\varepsilon_{2}>0 使得 \mu_{1}d_{2}^{*}\geqslant1-e+\varepsilon_{1}, \mu_{1}d_{3}^{*}\geqslant \frac{m}{1+l}-g+\varepsilon_{2}, 积分上面不等式可得u_{2}\equiv\bar{u}_{2}\equiv 常量, u_{3}\equiv\bar{u}_{3}\equiv 常量, 如果d_{1}>d_{1}^*\triangleq\mu_{1}^{-1}(b+C_{1}+C_{2}), 则u_{1}\equiv\bar{u}_{1}\equiv 常量. 证毕.
d 本部分我们研究系统(1.5)在E_*=u^*=(u_1^*,u_2^*,u_3^*~)的线性化. 令X如第二部分定义,且定义 \begin{array}{l} {\bf X}^{+}=\{{\bf u}\in{\bf X}|u_{i}>0~~\mbox{on}~~{\bar\Omega},i=1,2,3\},\\ {B}(C)=\{{\bf u}\in{\bf X}|C^{-1}<u_{i}<C~~\mbox{on}~~{\bar\Omega},i=1,2,3\}. \end{array} %(1.2) 则系统(1.5)可以被记为 \left\{\begin{array}{ll} -{D}\Delta{\bf u}={\bf G}({\bf u}),~~&x\in\Omega,\\ \partial_{\nu}{\bf u}=0,~~~~&x\in\partial\Omega, \end{array} \right.(6.1) \mathbf u是系统(1.5)的一个正解当且仅当 在{\bf X}^{+}中 ~{\bf F}({\bf u})\triangleq {\bf u}-({\bf I}-\Delta)^{-1}\{{D}^{-1}{\bf G}({\bf u})+{\bf u}\}=0 , 其中({\bf I}-\Delta)^{-1}是({\bf I}-\Delta)的逆. 由于{\bf F}(\cdot)是恒同算子的紧摄动,如果在\partial {B}上{\bf F}(u)\neq 0,对任意的{B}={B}(c) Leray-Schauder度deg({\bf F}(\cdot),0,{B})有定义. 进一步,我们有 {\bf D}_{u}{\bf F}({\bf u}^{*}) ={\bf I}-({\bf I}-\Delta)^{-1}\{{D}^{-1}{\bf G}_{u}({\bf u}^{*})+{\bf I}\} , 如果{\bf D}_{u}{\bf F}({\bf u}^{*})可逆,{\bf F}在{\bf u}^{*}的指数定义为 \mbox{index}({\bf F}(\cdot),{\bf u}^{*})=(-1)^{\gamma},其中\gamma是{\bf D}_{u}{\bf F}({\bf u}^{*})的负特征根的个数.
首先,对任意的整数i>0和整数1\leq j\leq\dim E(\mu_{i}),{\bf X}_{ij}在{\bf D}_{u}{\bf F}({\bf u}^{*})下是不变的. 并且\lambda是{\bf D}_{u}{\bf F}({\bf u}^{*})在{\bf X}_{ij}上的特征值,当且仅当它是下面矩阵的特征值. {\bf I}-\frac{1}{1+\mu_{i}}[{D}^{-1}{\bf G}_{u}({\bf u}^{*}) +{\bf I}]=\frac{1}{1+\mu_{i}}[\mu_{i}{\bf I}-{D}^{-1}{\bf G}_{u}({\bf u}^{*})]. 因此,{\bf D}_{u}{\bf F}({\bf u}^{*})是可逆的,当且仅当对所有的i\geq0, 矩阵{\bf I}-\frac{1}{1+\mu_{i}}[{D}^{-1}{\bf G}_{u}({\bf u}^{*})+{\bf I}]是非奇异的. 定义 H(\mu)=H({\bf u}^{*};\mu)\triangleq \det\{\mu{\bf I}-{D}^{-1}{\bf G}_{u}({\bf u}^{*})\} =\frac{1}{d_{1}d_{2}d_{3}}\det\{\mu {D}-{\bf G}_{u}({\bf u}^{*})\}. (6.2) 如果H(\mu_{i})\neq 0,则对任意的1\leq j\leq \dim E(\mu_{i}),{\bf D}_{u}{\bf F}({\bf u}^{*})在{\bf X}_{ij}上的负特征根的个数为奇数等价于H(\mu_{i})<0. 因此我们可以得到下面的结果.
引理6.1 假设对\forall i\geq0,矩阵\mu_{i}{\bf I}-{D}^{-1}{\bf G}_{u}({\bf u}^{*})是非奇异的,则 \mbox{index}({\bf F}(\cdot),{\bf u}^{*})=(-1)^\sigma, 其中 \sigma=\Sigma_{i\geq0,H(\mu_{i})<0}\dim E(\mu_{i}).
为了计算({\bf F}(\cdot){\bf u}^{*})的指数, 我们需要考虑H(\mu_i)的符号 \det\{{\mu {D}-{\bf G}_{u}({\bf u}^{*})}\} =A_{3}(d_{2})\mu^{3}+A_{2}(d_{2})\mu^{2}+A_{1}(d_{2})\mu-\det{{{\bf G}_{u}({\bf u}^{*})}}\triangleq {\cal A}(d_{2};\mu), (6.3) 其中 A_{3}(d_{2})=d_{1}d_{2}d_{3}, A_{2}(d_{2})=-a_{11}d_{2}d_{3}, A_{1}(d_{2})=-a_{12}a_{21}d_{3}-a_{13}a_{31}d_{2}-a_{23}a_{32}d_{1}, \det\{G_{u}(u^{*})\}=-a_{11}a_{23}a_{32}+a_{12}a_{23}a_{31}+a_{13}a_{21}a_{32}, 这里a_{ij} 在第二部分中已给出.
首先我们考虑\mathcal {A}与d_2的关系. 令\tilde{\mu}_{1}(d_{2}),\tilde{\mu}_{2}(d_{2}), \tilde{\mu}_{3}(d_{2}) 是{\cal A} (d_2;\mu)=0的三个根, 且Re{\tilde{\mu}_{1}(k)}\leq Re{\tilde{\mu}_{2}(k)}\leq Re{\tilde{\mu}_{3}(k)}. 则 \tilde{\mu}_{1}(d_{2})\tilde{\mu}_{2}(d_{2})\tilde{\mu}_{3}(d_{2})=\det\{{\bf G}_{u}({\bf u}^{*})\}. 由于\det\{{\bf G}_{u}({\bf u}^{*})\}<0,~~A_{3}(d_{2})>0. 因此\tilde{\mu}_{1}(d_{2}),\tilde{\mu}_{2}(d_{2}),\tilde{\mu}_{3}(d_{2})之一是正实数,并且另外两个的乘积是负数. 考虑下面的极限 \lim_{d_2\rightarrow+\infty} \frac{A_{3}(d_{2})}{d_{2}}=d_{1}d_{3} ,~~ \lim_{d_2\rightarrow+\infty}\frac{A_{2}(d_{2})}{d_{2}}=-d_{3}a_{11} ,~~ \lim_{d_2\rightarrow+\infty}\frac{A_{1}(d_{2})}{d_{2}}=-a_{13}a_{31} , \lim_{d_2\rightarrow+\infty}\frac{{\cal A}(\mbox{d}_{2};\mu)}{d_{2}}=d_{1}d_{3}\mu^{3}-d_{3}a_{11}\mu^{2}-a_{13}a_{31}\mu =\mu(d_{1}d_{3}\mu^{2}-d_{3}a_{11}\mu-a_{13}a_{31}) ,
如果参数\Lambda,d_{1},d_{3}满足 {d_{3}}^{2}{a_{11}}^{2}+4d_{1}d_{3}a_{13}a_{31}>0, 我们可以得到下面的引理.
引理6.2nbsp; 如果(1.4),(2.4)式成立,且C^{-1}<\frac{ru_{1}^{*}(\alpha-1)}{bR},则存在一个正数D_{2}, 使得当d_{2}\geq D_{2}时, {\cal A}(\mbox{d}_{2};\mu)=\mbox{0}的三个根\tilde{\mu}_{1}(d_{3}),\tilde{\mu}_{2}(d_{3}),\tilde{\mu}_{3}(d_{3})都是实的且满足 \begin{array}{l} \lim_{d_2\rightarrow+\infty}\tilde{\mu}_1(d_{2})=\frac{1}{2d_{1}d_{2}}[d_{3}a_{11}-\sqrt{{d_{3}}^{2}{a_{11}}^{2}+4d_{1}d_{3}a_{13}a_{31}}]<0,\\ [3mm] \lim_{d_2\rightarrow+\infty}\tilde{\mu}_2(d_{2})=\frac{1}{2d_{1}d_{2}}[d_{3}a_{11}+\sqrt{{d_{3}}^{2}{a_{11}}^{2}+4d_{1}d_{3}a_{13}a_{31}}]<0,\\ [3mm] \lim_{d_2\rightarrow+\infty}\tilde{\mu}_3(d_{2})=0=\tilde{\mu}. \end{array}(6.4) 进一步,如果-\infty<\tilde{\mu}_1(d_{2})<\tilde{\mu}_2(d_{2})<0<\tilde{\mu}_3(d_{2}), 则 \begin{array}{l} \mu\in(-\infty,\tilde{\mu}_{\mbox{1}}(\mbox{d}_{2})) \cup(\tilde {\mu}_{2}(\mbox{d}_{2}),\tilde {\mu}_{3}(\mbox{d}_{2})), ~~{\cal A}(\mbox{d}_{2};\mu)<\mbox{0},\\ \mu\in(\tilde{\mu}_{\mbox{1}}(\mbox{d}_{2}), \tilde{\mu}_{2}(\mbox{d}_{2}))\cup(\tilde{\mu}_{3}(\mbox{d}_{2}), +\infty), ~~{\cal A}(\mbox{d}_{2};\mu)>\mbox{0}. \end{array}(6.5)
当d_{2}足够大,则存在\Lambda,d_{i},i=1,3使得系统(1.5)至 少有一个非常值正解.
定理6.1 假设参数\Lambda,d_{1},d_{3}固定, \tilde{\mu}\in(\mu_{0},\mu_{1})且\sigma_{0}= \dim_{i\geqslant0,H(\mu_{i})<0}E(\mu_{i})是奇数, 则存在一个正常数D_{2},使得当d_{2}\geq D_{2}时, 系统(1.5)至少有一个非常值正解.
证 通过引理6.2可知,存在一个正常数D_{2}使 得当d_{2}\geq D_{2}时,(6.5)式成立并且 ~~~~0=\mu_{0}<\tilde{\mu}_{3}(d_{2})<\mu_{1}.(6.6) 要证明\forall d_{2}\geq D_{2},系统(1.5)至少有一个非常值正解.
我们将基于拓扑度的同伦不变性,利用反证法来证明. 假设存在d_{2}=\tilde{d_{2}}\geq D_{2}使得结论不成立. 固定d_{2}=\tilde{d_{2}},~{d_{2}}^{*}=\frac{2-e}{\mu_{1}},
由定理5.1, 存在一个正常数D_{1}=D_{1}(\Lambda,d_{2}^{*},d_{3}^{*}), \hat{d_{2}}\geqslant{d_{2}}^{*}, ~\hat{d_{3}}\geqslant\max\{{d_{3}}^{*},d_{3}\}, ~\hat{d_{1}}\geqslant\max\{D_{1},~d_{1}\}. 对于$t\in[0, 1], 定义{D}(t)={\rm diag}(d_{1}(t),~d_{2}(t),~d_{3}(t)) 其中d_{i}(t)=td_{i}+(1-t)\hat{d_{i}}, i=1,2,3. 考虑问题 \left\{\begin{array}{ll} -{D}(t)\Delta{\bf u}={\bf G}({\bf u}), ~~&x\in\Omega,\\ \partial_{\nu}{\bf u}=0, &x\in\partial\Omega. \end{array}\right.(6.7) 则{\bf u}是系统(1.5)的一个非常值解当且仅当{\bf u}是方程(6.7)当t=1时的一个解. 显然对任意的0\leq t\leq 1,{\bf u}^{*}$是方程(6.7)的唯一正常值解.
对任意的0\leq t\leq 1,{\bf u}是方程(6.7) 的正解当且仅当 在{\bf X}^{+}上 {\bf F}(t;{\bf u})\triangleq{\bf u}-({\bf I}-\Delta)^{-1}\{{D}^{-1}(t){\bf G}({\bf u})+{\bf u}\}=0 . 显然{\bf F}(1;{\bf u})={\bf F}({\bf u}), 由定理5.1可知{\bf F}(0;{\bf u})={\bf F}({\bf u})=0在{\bf X}^{+}中只有正解\mathbf u^{*},直接计算可得 {\bf D}_{u}{\bf F}(t;{\bf u}^{*})= {\bf I}-({\bf I}-\Delta)^{-1}\{{D}^{-1}(t){\bf G}_{u}({\bf u}^{*})+{\bf I}\}. 特别地, {\bf D}_{u}{\bf F}(0;{\bf u}^{*})= {\bf I}-({\bf I}-\Delta)^{-1}\{\tilde{{D}}^{-1}{\bf G}_{u}({\bf u}^{*})+{\bf I}\}, {\bf D}_{u}{\bf F}(1;{\bf u}^{*})= {\bf I}-({\bf I}-\Delta)^{-1}\{{D}^{-1}{\bf G}_{u}({\bf u}^{*})+{\bf I}\}={\bf D}_{u}F({\bf u}^{*}). 其中\hat{D}={\rm diag}(\hat{d_{1}},\hat{d_{2}},\hat{d_{3}}). 由(6.2), (6.3)式可知 H(\mu)=\frac{1}{d_{1}d_{2}d_{3}} {\cal A}(\mbox{d}_{2};\mu),(6.8) 由(6.5), (6.6)式通过直接计算有 \left\{\begin{array}{l} H(\mu_{0})=H(0)<0,\\ H(\mu_{i})>0,~~i\geq 1. \end{array}\right.(6.9) 因此, 对任意的i\geq 0,0不是矩阵 \mu_{i}{\bf I}-{D}^{-1}{\bf G}_{u}({\bf u}^{*})的特征值, 且有 \Sigma_{i\geq0,H(\mu_{i})<0}\dim E(\mu_{i})=\sum_{i=1}^n\dim E(\mu_{i})=\sigma_{n}~~~~\mbox{是奇数.} 由引理6.1有 \mbox{index}({\bf F}(1;\cdot),{\bf u}^{*})=(-1)^{\gamma}=(-1)^{\sigma_{n}}=-1. (6.10)
类似地我们可以证明 \mbox{index}({\bf F}(0;\cdot),{\bf u}^{*})=(-1)^{0}=1. (6.11) 通过定理3.1, 存在一个正常数C使得对所有的0\leq t\leq 1, (6.7)的正解满足C^{-1}<u_{1},u_{2},u_{3}<C. 因此,在\partial{\cal B}\mbox{(c)}上,对所有的C^{-1}<u_{1},u_{2},u_{3}<C,{\bf F}(t;{\bf u})\neq 0. 由拓扑度的同伦不变性可知 ~~~~\mbox{deg}({\bf F}(1;\cdot),0,{\cal B}\mbox{(c)})=\mbox{deg}({\bf F}(0;\cdot),0,{\cal B}\mbox{(c)}). (6.12) 另一方面,通过假设,方程 {\bf F}(1;{\bf u})=0 和 {\bf F}(0;{\bf u})=0在 {\cal B}\mbox{(c)}都只有一个正解,因此, 通过(6.10)和(6.11) 式 \mbox{deg}({\bf F}(0;\cdot),0,{\cal B}\mbox{(c)})=\mbox{index}({\bf F}(0;\cdot),{\bf u}^{*})=(-1)^{0}=1, \mbox{deg}({\bf F}(1;\cdot),0,{\cal B}\mbox{(c)})=\mbox{index}({\bf F}(1;\cdot),{\bf u}^{*})=(-1)^{0}=-1. 这与(6.12)式矛盾, 证毕.