西南大学学报 (自然科学版)  2018, Vol. 40 Issue (11): 35-40.  DOI: 10.13718/j.cnki.xdzk.2018.11.006
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  • 空间异质环境下带交错扩散项的Lotka-Volterra模型分岔解的稳定性    [PDF全文]
    徐茜     
    北京联合大学 基础部, 北京 100101
    摘要:主要研究空间异质环境下带交错扩散项的Lotka-Volterra方程组分岔解的局部渐近稳定性.由分岔方向及细致的谱分析,证明了分岔平衡解是局部渐近稳定的.
    关键词谱分析    稳定性    分岔解    

    本文主要研究下列空间异质环境下Lotka-Volterra交错扩散方程组

    $ \left\{ {\begin{array}{*{20}{c}} \begin{array}{l} {u_t} = \Delta \left[ {\left( {1 + k\rho \left( x \right)v} \right)u} \right] + u\left( {a - u - c\left( x \right)v} \right)\\ {v_t} = \Delta v + v\left( {b + d\left( x \right)u - v} \right)\\ {\partial _\nu }u = {\partial _\nu }v = 0\\ u\left( { \cdot ,0} \right) = {u_0}\left( x \right) \ge 0,v\left( { \cdot ,0} \right) = {v_0}\left( x \right) \ge 0 \end{array}&\begin{array}{l} x \in \mathit{\Omega ,}t > 0\\ x \in \mathit{\Omega ,}t > 0\\ x \in \partial \mathit{\Omega ,}t > 0\\ x \in \mathit{\Omega } \end{array} \end{array}} \right. $ (1)

    其中:Ω$\mathbb{R}$N(N≤3)中的有界区域;c(x)>0和d(x)≥0都是连续函数,ρ(x)>0是光滑函数并且$\partial $νρ=0,x$\partial $Ωuv是被捕食者和捕食者;ak是正常数,b是实数,ab代表被捕食者和捕食者的出生率.交错扩散项Δ[ρ(x)vu]=▽[u▽(ρ(x)v)+ρ(x)vu]描述u扩散到ρ(x)v浓度低的区域的一种趋势.已有一些文章研究了关于空间异质环境对种群浓度的影响.文献[1-3]对于一些扩散的Lotka-volterra方程组研究了种内之间的退化效果;文献[4-7]研究了对于一些带扩散项的竞争方程组,空间异质环境下的出生率问题;文献[8]对于方程组(1)所对应的平衡解的方程组,证明了当ρcd是常数的时候,方程组由分岔参数b分岔出来的正解的集合Γp形成一个有界的曲线,并且当a和|b|小,k很大时,ρ(x),d(x)使得Γp关于b形成一个型的曲线;文献[9]研究了带两个趋化参数的趋化模型非常数平衡解的存在性.本文主要研究文献[8]中得到的分岔平衡解在分岔点(0,b*b*)处的稳定性.

    1 预备知识

    方程组(1)所对应的平衡解问题为:

    $ \left\{ {\begin{array}{*{20}{c}} \begin{array}{l} \Delta \left[ {\left( {1 + k\rho \left( x \right)v} \right)u} \right] + u\left( {a - u - c\left( x \right)v} \right) = 0\\ \Delta v + v\left( {b + d\left( x \right)u - v} \right) = 0\\ {\partial _\nu }u = {\partial _\nu }v = 0 \end{array}&\begin{array}{l} x \in \mathit{\Omega }\\ x \in \mathit{\Omega }\\ x \in \partial \mathit{\Omega } \end{array} \end{array}} \right. $ (2)

    $ U = \left( {1 + k\rho \left( x \right)v} \right)u $

    则方程组(2)变为

    $ \left\{ \begin{array}{l} \Delta U + \frac{U}{{1 + k\rho \left( x \right)v}}\left( {a - \frac{U}{{1 + k\rho \left( x \right)v}} - c\left( x \right)v} \right)\;\;\;\;\;\;\;\;\;\;x \in \mathit{\Omega }\\ \Delta v + v\left( {b + \frac{{d\left( x \right)U}}{{1 + k\rho \left( x \right)v}} - v} \right) = 0\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;x \in \mathit{\Omega }\\ {\partial _\nu }U = {\partial _\nu }v = 0\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;x \in \partial \mathit{\Omega } \end{array} \right. $ (3)

    易知方程组(3)有半平凡解集为Γv=(0,bb):b>0.定义空间

    $ \begin{array}{*{20}{c}} {X: = W_\nu ^{2,p}\left( \mathit{\Omega } \right) \times W_\nu ^{2,p}\left( \mathit{\Omega } \right)}\\ {Y: = {L^p}\left( \mathit{\Omega } \right) \times {L^p}\left( \mathit{\Omega } \right)\left( {p > N} \right)} \end{array} $

    其中

    $ W_v^{2,p}\left( \mathit{\Omega } \right) = \left\{ {u \in {W^{2,p}}\left( \mathit{\Omega } \right):{\partial _v}u\left| {_{\partial \mathit{\Omega }}} \right. = 0} \right\} $

    对于固定的(kρ(x),c(x),d(x)),引进集合如下:

    $ {S_v}: = \left\{ {\left( {a,b} \right) \in {\mathbb{R}^2}:{\lambda _1}\left( {\frac{{bc\left( x \right) - a}}{{1 + bk\rho \left( x \right)}}} \right) = 0,a \geqslant 0} \right\} $

    为了后面证明的需要,先给出引理1.

    引理1[8]  任意固定(kρ(x),c(x),d(x)),则存在一个单调递增的光滑函数b=b*(a)满足

    $ {b^ * }\left( 0 \right) = 0,\mathop {\lim }\limits_{a \to \infty } {b^ * }\left( a \right) = \infty $

    使得

    $ {S_v} = \left\{ {\left( {a,b} \right) \in {\mathbb{R}^2}:b = {b^ * }\left( a \right),a \geqslant 0} \right\} $

    定义正函数φ*为下列线性椭圆方程解

    $ \begin{array}{*{20}{c}} { - \Delta {\varphi ^ * } + \frac{{{b^ * }c\left( x \right) - a}}{{1 + {b^ * }k\rho \left( x \right)}}{\varphi ^ * } = 0}&{x \in \mathit{\Omega }}&{{\partial _\nu }{\varphi ^ * }\left| {_{\partial \mathit{\Omega }}} \right. = 0}&{{{\left\| {{\varphi ^ * }} \right\|}_{{L^2}\left( \mathit{\Omega } \right)}} = 1} \end{array} $ (4)

    引理2[8] 任意固定(kρ(x),c(x),d(x)),则下列结论成立:

    方程组(3)从Γv处分岔当且仅当b=b*>0,存在一个正数δ*和函数ψ*X使得在(Uvb)=(0,b*b*)∈X×$\mathbb{R}$附近所有正解有如下形式:

    $ \left\{ {\left( {U\left( s \right),\mathit{v}\left( s \right),b\left( s \right)} \right) = \left( {s\left( {{\varphi ^ * } + s\bar U\left( s \right)} \right),{b^ * } + s\left( {{\psi ^ * } + s\bar v\left( s \right)} \right),b\left( s \right) \in X \times \mathbb{R}:0 < s \leqslant {\delta ^ * }} \right)} \right\} $

    其中

    $ {\psi ^ * } = {\left( { - \Delta + {b^ * }} \right)^{ - 1}}\left( {\frac{{{b^ * }d\left( x \right)}}{{1 + {b^ * }k\rho \left( x \right)}}{\varphi ^ * }} \right) > 0,\left( {\bar U\left( s \right),\mathit{\bar v}\left( s \right),b\left( s \right)} \right) $

    是有界函数满足

    $ \begin{array}{*{20}{c}} {b\left( 0 \right) = {b^ * }}&{\int_\mathit{\Omega } {\bar U{\varphi ^ * }{\rm{d}}x} = 0} \end{array} $

    $ \begin{array}{*{20}{c}} {\bar b\left( 0 \right) < 0}&{\bar b\left( 0 \right) = \frac{{\rm{d}}}{{{\rm{d}}s}}b\left( s \right)} \end{array} $
    2 在分岔点(0,b*b*)附近的分岔平衡解的局部渐近稳定性

    在方程组(1)中,令

    $ U = \left( {1 + k\rho \left( x \right)v} \right)u $

    则方程组(1)变为

    $ \left\{ \begin{array}{l} {\left( {\frac{U}{{1 + k\rho \left( x \right)v}}} \right)_t} = \Delta U + \frac{U}{{1 + k\rho \left( x \right)v}}\left( {a - \frac{U}{{1 + k\rho \left( x \right)v}} - c\left( x \right)v} \right)\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;x \in \mathit{\Omega },t > 0\\ {v_t} = \Delta v + v\left( {b + \frac{{d\left( x \right)U}}{{1 + k\rho \left( x \right)v}} - v} \right)\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;x \in \mathit{\Omega },t > 0\\ {\partial _\nu }U = {\partial _\nu }v = 0\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;x \in \partial \mathit{\Omega },t > 0\\ U\left( { \cdot ,0} \right) = \left( {1 + k\rho \left( x \right){v_0}} \right){u_0}\left( x \right) \ge 0,v\left( { \cdot ,0} \right) = {v_0}\left( x \right) \ge 0\;\;\;\;\;\;\;\;\;\;\;x \in \mathit{\Omega } \end{array} \right. $ (5)

    在(U(s),v(s))处线性化方程组(5),相应的特征值问题为

    $ \left\{ \begin{array}{l} \frac{1}{{1 + k\rho \left( x \right)v\left( s \right)}}\sigma U - \frac{{k\rho \left( x \right)U\left( s \right)}}{{{{\left( {1 + k\rho \left( x \right)v\left( s \right)} \right)}^2}}}\sigma v = \Delta U + \\ \frac{a}{{1 + k\rho \left( x \right)v\left( s \right)}}U - \frac{{2U\left( s \right)}}{{{{\left( {1 + k\rho \left( x \right)v\left( s \right)} \right)}^2}}}U - \frac{{c\left( x \right)v\left( s \right)}}{{1 + k\rho \left( x \right)v\left( s \right)}}U - \\ \frac{{ak\rho \left( x \right)U\left( s \right)}}{{{{\left( {1 + k\rho \left( x \right)v\left( s \right)} \right)}^2}}} + \frac{{2k\rho \left( x \right){U^2}\left( s \right)}}{{{{\left( {1 + k\rho \left( x \right)v\left( s \right)} \right)}^2}}}v - \frac{{c\left( x \right)U\left( s \right)}}{{{{\left( {1 + k\rho \left( x \right)v\left( s \right)} \right)}^2}}}v,x \in \mathit{\Omega }\\ \sigma v = \Delta v + bv - 2v\left( s \right)v + \frac{{d\left( x \right)U\left( s \right)}}{{{{\left( {1 + k\rho \left( x \right)v\left( s \right)} \right)}^2}}}v + \frac{{d\left( x \right)v\left( s \right)}}{{1 + k\rho \left( x \right)v\left( s \right)}}U,x \in \mathit{\Omega }\\ {\partial _\nu }U = {\partial _\nu }v = 0,x \in \partial \mathit{\Omega } \end{array} \right. $ (6)

    定义算子HX×$\mathbb{R}$Y

    $ \begin{array}{l} H\left( {U,v,b} \right) = \left( {\begin{array}{*{20}{c}} {{H_1}\left( {U,v,b} \right)}\\ {{H_2}\left( {U,v,b} \right)} \end{array}} \right) = \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\left( \begin{array}{l} \Delta U + \frac{U}{{1 + k\rho \left( x \right)v}}\left( {a - \frac{U}{{1 + k\rho \left( x \right)v}} - c\left( x \right)v} \right)\\ \Delta v + v\left( {b + \frac{{d\left( x \right)U}}{{1 + k\rho \left( x \right)v}} - v} \right) \end{array} \right) \end{array} $

    经过简单计算可得:

    $ {H_{\left( {U,v} \right)}}\left( {0,{b^ * },{b^ * }} \right)\left( {\begin{array}{*{20}{c}} U\\ v \end{array}} \right) = \left( \begin{array}{l} \Delta U + \frac{{a - {b^ * }c\left( x \right)}}{{1 + {b^ * }k\rho \left( x \right)}}U\\ \Delta v + \frac{{{b^ * }d\left( x \right)}}{{1 + {b^ * }k\rho \left( x \right)}}U - bv \end{array} \right) $

    由式(4)可得

    $ {\rm{Ker}}\left\{ {{H_{\left( {U,v} \right)}}\left( {0,{b^ * },{b^ * }} \right)} \right\} = {\rm{span}}\left\{ {\left( {{\varphi ^ * },{\psi ^ * }} \right)} \right\} $

    其中

    $ {\psi ^ * } = {\left( { - \Delta + {b^ * }} \right)^{ - 1}}\left( {\frac{{{b^ * }d\left( x \right)}}{{1 + {b^ * }k\rho \left( x \right)}}{\varphi ^ * }} \right) > 0 $

    方程组(6)可被改写为

    $ {H_{\left( {U,v} \right)}}\left( {U\left( s \right),v\left( s \right),b} \right)\left( {\begin{array}{*{20}{c}} U\\ v \end{array}} \right) = \left( {\begin{array}{*{20}{c}} {\frac{1}{{1 + k\rho \left( x \right)v\left( s \right)}}}&{ - \frac{{k\rho \left( x \right)U\left( s \right)}}{{{{\left( {1 + k\rho \left( x \right)v\left( s \right)} \right)}^2}}}}\\ 0&1 \end{array}} \right)\left( {\begin{array}{*{20}{c}} {\sigma U}\\ {\sigma v} \end{array}} \right) $

    引进一算子LX×$\mathbb{R}$Y

    $ L\left( {U\left( s \right),v\left( s \right),b} \right) = {\left( {\begin{array}{*{20}{c}} {\frac{1}{{1 + k\rho \left( x \right)v\left( s \right)}}}&{ - \frac{{k\rho \left( x \right)U\left( s \right)}}{{{{\left( {1 + k\rho \left( x \right)v\left( s \right)} \right)}^2}}}}\\ 0&1 \end{array}} \right)^{ - 1}}{H_{\left( {U,v} \right)}}\left( {U\left( s \right),v\left( s \right),b} \right) $

    则方程组(6)可记为

    $ L\left( {U\left( s \right),v\left( s \right),b} \right)\left( {\begin{array}{*{20}{c}} U\\ v \end{array}} \right) = \left( {\begin{array}{*{20}{c}} {\sigma U}\\ {\sigma v} \end{array}} \right) $

    根据文献[10]中定理2.1和式(4.5)定义函数如下:

    $ {l_1}:X \to \mathbb{R},\left\langle {\left[ {f,g} \right],{l_1}} \right\rangle = \int_\mathit{\Omega } {f{\varphi ^ * }{\text{d}}x} $ (7)

    定理1  对任意固定的(akρ(x),c(x),d(x)),由引理2定义的方程组(5)的分岔解(U(s),v(s))是局部渐近稳定的.

      首先证明0是L(0,b*b*)的第一特征值.对任意固定的(Uv)∈X满足

    $ \begin{array}{l} L\left( {0,{b^ * },{b^ * }} \right)\left( {\begin{array}{*{20}{c}} U\\ v \end{array}} \right) = {\left( {\begin{array}{*{20}{c}} {\frac{1}{{1 + k\rho \left( x \right){b^ * }}}}&0\\ 0&1 \end{array}} \right)^{ - 1}}{H_{\left( {U,v} \right)}}\left( {0,{b^ * },{b^ * }} \right)\left( {\begin{array}{*{20}{c}} U\\ v \end{array}} \right) = \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\left( {\begin{array}{*{20}{c}} {1 + k\rho \left( x \right){b^ * }}&0\\ 0&1 \end{array}} \right)\left( \begin{array}{l} \Delta U + \frac{{a - {b^ * }c\left( x \right)}}{{1 + {b^ * }k\rho \left( x \right)}}U\\ \Delta v + \frac{{{b^ * }d\left( x \right)}}{{1 + {b^ * }k\rho \left( x \right)}}U - bv \end{array} \right) = \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\left( \begin{array}{l} \left( {1 + k\rho \left( x \right){b^ * }} \right)\left( {\Delta U + \frac{{a - {b^ * }c\left( x \right)}}{{1 + {b^ * }k\rho \left( x \right)}}U} \right)\\ \Delta v + \frac{{{b^ * }d\left( x \right)}}{{1 + {b^ * }k\rho \left( x \right)}}U - bv \end{array} \right) = 0 \end{array} $ (8)

    φ*ψ*的定义易知

    $ L\left( {0,{b^ * },{b^ * }} \right)\left( \begin{array}{l} {\varphi ^ * }\\ {\psi ^ * } \end{array} \right) = \left( \begin{array}{l} 0\\ 0 \end{array} \right) $

    因此0是L(0,b*b*)的一个特征值,下面证明0是L(0,b*b*)的第一特征值.反证法,假设0不是L(0,b*b*)的第一特征值,则存在L(0,b*b*)的一个正的特征值λ1及相应的特征函数U1v1X使得

    $ L\left( {0,{b^ * },{b^ * }} \right)\left( {\begin{array}{*{20}{c}} {{U_1}}\\ {{v_1}} \end{array}} \right) = \left( {\begin{array}{*{20}{c}} {{\lambda _1}{U_1}}\\ {{\lambda _1}{v_1}} \end{array}} \right) $

    $ \left\{ \begin{array}{l} \left( {1 + k\rho \left( x \right){b^ * }} \right)\left( {\Delta {U_1} + \frac{{a - {b^ * }c\left( x \right)}}{{1 + {b^ * }k\rho \left( x \right)}}{U_1}} \right) = {\lambda _1}{U_1}\;\;\;\;\;\;\;\;\;\;\;\;\;x \in \mathit{\Omega }\\ \Delta {v_1} + \frac{{{b^ * }d\left( x \right)}}{{1 + {b^ * }k\rho \left( x \right)}}{U_1} - {b^ * }{v_1} = {\lambda _1}{v_1}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;x \in \mathit{\Omega }\\ {\partial _\nu }{U_1} = {\partial _\nu }{v_1} = 0\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;x \in \partial \mathit{\Omega } \end{array} \right. $ (9)

    如果U1=0且v1≠0,则方程组(9)的第二个方程推出

    $ \Delta {v_1} - {b^ * }{v_1} = {\lambda _1}{v_1}\left( {{\lambda _1} > 0} \right) $

    进而

    $ {v_1} = {\left( { - \Delta + {b^ * }} \right)^{ - 1}}\left( { - {\lambda _1}{v_1}} \right) $

    这是矛盾的,因为当-λ1v1为负时(-Δ+b*)-1(-λ1v1)是负的,因此U1≠0,则方程组(9)的第一个方程为

    $ \Delta {U_1} + \frac{{a - {b^ * }c\left( x \right)}}{{1 + {b^ * }k\rho \left( x \right)}}{U_1} = \frac{{{\lambda _1}}}{{1 + {b^ * }k\rho \left( x \right)}}{U_1} $

    由式(4)及单个椭圆方程定理,0是式(4)的第一特征值与λ1>0矛盾,因此0是L(0,b*b*)的第一特征值且其它特征值都是负的.应用文献[11]中的命题I.7.2,对0<sδ,存在扰动特征值σ(s)及连续可微函数φ1(s),φ2(s)∈X∩Range(L(Uv)(0,b*b*))满足

    $ L\left( {U\left( s \right),v\left( s \right),b\left( s \right)} \right)\left( {\begin{array}{*{20}{c}} {{\varphi ^ * } + {\varphi _1}\left( s \right)}\\ {{\psi ^ * } + {\varphi _2}\left( s \right)} \end{array}} \right) = \sigma \left( s \right)\left( {\begin{array}{*{20}{c}} {{\varphi ^ * } + {\varphi _1}\left( s \right)}\\ {{\psi ^ * } + {\varphi _2}\left( s \right)} \end{array}} \right) $

    $ \sigma \left( 0 \right) = {\varphi _1}\left( 0 \right) = {\varphi _2}\left( 0 \right) = 0 $

    类似地,存在扰动特征值σ(b)及连续可微函数φ1(b),φ2(b)∈X∩Range(L(Uv)(0,b*b*))满足

    $ L\left( {0,b,b} \right)\left( {\begin{array}{*{20}{c}} {{\varphi ^ * } + {\varphi _1}\left( b \right)}\\ {{\psi ^ * } + {\varphi _2}\left( b \right)} \end{array}} \right) = \sigma \left( b \right)\left( {\begin{array}{*{20}{c}} {{\varphi ^ * } + {\varphi _1}\left( b \right)}\\ {{\psi ^ * } + {\varphi _2}\left( b \right)} \end{array}} \right) $ (10)

    $ \sigma \left( {{b^ * }} \right) = {\varphi _1}\left( {{b^ * }} \right) = {\varphi _2}\left( {{b^ * }} \right) = 0 $

    在式(10)中对b进行求导及利用

    $ \sigma \left( {{b^ * }} \right) = {\varphi _1}\left( {{b^ * }} \right) = {\varphi _2}\left( {{b^ * }} \right) = 0 $

    可推出

    $ \frac{{\rm{d}}}{{{\rm{d}}b}}L\left( {0,{b^ * },{b^ * }} \right)\left( \begin{array}{l} {\varphi ^ * }\\ {\psi ^ * } \end{array} \right) + L\left( {0,{b^ * },{b^ * }} \right)\left( \begin{array}{l} {{\varphi '}_1}\left( {{b^ * }} \right)\\ {{\varphi '}_2}\left( {{b^ * }} \right) \end{array} \right) = \sigma '\left( {{b^ * }} \right)\left( \begin{array}{l} {\varphi ^ * }\\ {\psi ^ * } \end{array} \right) $

    其中

    $ \sigma '\left( b \right) = \frac{{\rm{d}}}{{{\rm{d}}b}}\sigma \left( b \right) $

    由式(7)可推出

    $ \left\langle {\frac{{\rm{d}}}{{{\rm{d}}b}}L\left( {0,{b^ * },{b^ * }} \right)\left( \begin{array}{l} {\varphi ^ * }\\ {\psi ^ * } \end{array} \right),{l_1}} \right\rangle = \sigma '\left( {{b^ * }} \right) $ (11)

    由式(8)可计算如下方程

    $ \frac{{\rm{d}}}{{{\rm{d}}b}}L\left( {0,{b^ * },{b^ * }} \right)\left( \begin{array}{l} {\varphi ^ * }\\ {\psi ^ * } \end{array} \right) = \left( \begin{array}{l} \frac{{ - ak\rho \left( x \right) - c\left( x \right)}}{{1 + k{b^ * }\rho \left( x \right)}}{\varphi ^ * }\\ \frac{{d\left( x \right)}}{{{{\left( {1 + k{b^ * }\rho \left( x \right)} \right)}^2}}}{\varphi ^ * } - {\psi ^ * } \end{array} \right) $ (12)

    由式(11)及(12)可得到

    $ \sigma '\left( {{b^ * }} \right) = \int_\mathit{\Omega } {\frac{{ - ak\rho \left( x \right) - c\left( x \right)}}{{1 + k{b^ * }\rho \left( x \right)}}{{\left( {{\varphi ^ * }} \right)}^2}{\rm{d}}x} < 0 $ (13)

    由文献[11]中的公式得到

    $ - \bar \sigma \left( 0 \right) = \bar b\left( 0 \right)\sigma '\left( {{b^ * }} \right) $

    其中

    $ \bar \sigma \left( 0 \right) = \frac{{\rm{d}}}{{{\rm{d}}s}}\sigma \left( s \right) $

    由引理2及式(13)可知σ(0)<0.由此可推出当s>0且很小时σ(s)<0,因此引理2得到的分岔解(U(s),v(s))是局部渐近稳定的.

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    The Stability of Bifurcating Solution for a Spatially Heterogeneous Lotka-Volterra Model with Cross Diffusion
    XU Qian     
    Department of Basic Courses, Beijing Union University, Beijing 100101, China
    Abstract: In this paper, we concern with the local asymptotical stability of the bifurcating solution for the Lotka-Volterra system with cross diffusion in a spatially heterogeneous environment. By applying a detailed spectral analysis based on the bifurcating direction we prove that the bifurcating steady state solution is locally asymptotically stable.
    Key words: spectral analysis    stability    bifurcating solution    
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