一类具饱和项和毒素影响的扩散问题的持久性与稳定性

一类具饱和项和毒素影响的扩散问题的持久性与稳定性 Persistence and Stability of a Diffusion System with Saturation Term and Toxin Term first-author 冯庆红，女，副教授，主要从事生物数学方面的研究.E-mail:fqh3090095@sina.com 冯庆红，女，副教授，主要从事生物数学方面的研究.E-mail: fqh3090095@sina.com fqh3090095@sina.com 常州工学院 数理化工学院，江苏 常州 213032 常州工学院 数理化工学院，江苏 常州 213032 School of Mathematical Sciences and Chemical Engineering, Changzhou Institute of Technology, Changzhou 213032, Jiangsu, China School of Mathematical Sciences and Chemical Engineering, Changzhou Institute of Technology, Changzhou 213032, Jiangsu, China 研究了一类具饱和项和毒素项的互惠模型.由比较原理得到问题是持久的；用线性近似法证明了在一定条件下系统的正平衡点是局部稳定的；利用Lyapunov直接法证明了正平衡点是全局渐近稳定的. A cooperative model with saturation term and toxin term was considered. The persistence of the system was proved by using the comparison principle. Based on the liner approximation method, the positive equilibrium solution is proved to be local stationary under some conditions. The global asymptotic stability of its positive equilibrium is presented by using Lyapunov function. 互惠模型 饱和项 毒素项 持久性 稳定性 cooperative model saturation term toxin term persistence stability 引言 1 本文讨论下面带饱和项和毒素项的扩散问题 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458679&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458679&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458679&type=middle 其中 Ω 是 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458821&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458821&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458821&type=middle 中有界区域, 边界∂ Ω 光滑，Δ为 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458821&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458821&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458821&type=middle 中Laplace算子； a , b , c , d , r i , a i , d i , γ i ( i =1, 2)为正常数；根据生物学概念 u ( t ), v ( t )分别表示 t 时刻 Ω 区域中两个种群的密度， http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458826&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458826&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458826&type=middle 是饱和项，其生物意义参见文献[ 1 1 ]； r i 表示第 i 个种群的内禀增长率， a i 是第 i 个种群的密度制约系数， d i 表示第 i 个种群的扩散速度； γ 1 u 2 v 和 γ 2 uv 2 是毒素污染，来自两种群自身， γ 1 和 γ 2 是污染系数； η 是边界上的单位外法向量；初值 u 0 ( x ), v 0 ( x )非负且是Hölder连续的；齐次Neumann边界条件表示两种群的活动限制在 Ω 内，且在边界上没有种群移动. 问题(1)描述的是两种群的互惠模型，即一种群的出现对另一种群有好的作用.对于生物模型的持久性或稳定性，国内外学者已取得了丰富的研究成果.向红等 [ 2 2 ] 利用比较原理讨论了一类具饱和项和毒素项的互惠模型的持久性和稳定性，但由于模型是常微分系统，不能反应种群的空间扩散.近年来，对带扩散项的生物模型的研究成为热点.张睿等 [ 3 3 ] 证明了具扩散的食饵-捕食者模型在其惟一平衡点处的局部以及全局渐近稳定性.安海龙等 [ 4 4 ] 讨论了一类具饱和项和扩散项的两种群互惠模型，证明了模型整体解的存在性、惟一性和一致有界性；利用Lyapunov函数法，给出了模型正平衡点的全局渐近稳定性的条件.然而文献[ 3 3 ]和[ 4 4 ]都没有考虑模型的持久性问题，而持久性是研究种群动力系统最重要的问题之一.黄运金等 [ 5 5 ] 给出了一类具有无穷时滞竞争模型永久持续生存的充分条件，但并没有对模型的稳定性进行分析.另外，毒素对生物种群的威胁日益严重，因此在分析种群性态时常常需要考虑毒素的影响.朱晶 [ 6 6 ] 在研究一类带功能反应函数的两种群模型的一致持久性时讨论了毒素的作用.以上分析构成了本文思想的出发点. 系统的持久性 1 注意到问题(1)中方程右端函数在 u ≥0, v ≥0时是连续光滑的.故由经典的偏微分方程理论可知, 问题(1)一定存在局部解，且解是非负的.下面证明解有界，故解是整体的. 定理1   问题(1)存在惟一整体古典解，且 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458684&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458684&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458684&type=middle 证   假设解在[0, T 0 ]上存在，其中 T 0 < +∞， u 在 Ω ×[0, T 0 ]上可取到最大值，则只可能存在三种情况： 1) 如果 u 在( x 0 , t 0 )∈ Ω ×(0, T 0 ]上取得最大值，则有 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458691&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458691&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458691&type=middle 即 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458831&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458831&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458831&type=middle 成立，所以 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458694&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458694&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458694&type=middle 因此 u ≤ C 1 ; 2) 如果 u 在 t =0时取得最大值，即 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458698&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458698&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458698&type=middle 则显然 u ≤ C 1 ; 3) 如果 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458835&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458835&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458835&type=middle , 只在( x 0 , t 0 )∈∂ Ω ×(0, T 0 ]取得最大值，则由Hopf边界引理得 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458839&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458839&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458839&type=middle 从而矛盾.故 u ( x , t )≤ C 1 , ( x , t )∈ Ω ×[0, T 0 ].同理可得 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458699&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458699&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458699&type=middle 又因为 C 1 , C 2 与 T 0 无关，因此解有界且是全局的. 定理2   反应扩散问题(1)对非负非平凡初值是持久的，即对 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458843&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458843&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458843&type=middle 存在正常数 m i , M i ( i =1, 2), 使得 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458702&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458702&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458702&type=middle http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458705&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458705&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458705&type=middle 证   首先构造( u ( x , t ), v ( x , t ))的上解 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458846&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458846&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458846&type=middle http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458849&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458849&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458849&type=middle 注意到( u ( x , t ), v ( x , t ))满足不等式 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458708&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458708&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458708&type=middle 取 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458852&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458852&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458852&type=middle 是满足问题 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458711&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458711&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458711&type=middle 的解，则由比较原理知 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458714&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458714&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458714&type=middle 对上述常微分方程组求解，得 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458716&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458716&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458716&type=middle 显然 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458718&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458718&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458718&type=middle 接着，构造( u ( x , t ), v ( x , t ))的下解 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458855&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458855&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458855&type=middle 由上面结论可知，∀ ε > 0，∃ T ε ，当 t ≥ T ε 时， http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458720&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458720&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458720&type=middle 显然 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458723&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458723&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458723&type=middle 取 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458858&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458858&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458858&type=middle 为满足问题 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458727&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458727&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458727&type=middle 的解，则同样由比较原理知 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458859&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458859&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458859&type=middle http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458863&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458863&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458863&type=middle 求解上述方程，得解 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458730&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458730&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458730&type=middle 又显然 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458732&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458732&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458732&type=middle 故 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458733&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458733&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458733&type=middle 再令 ε →0，则有 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458736&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458736&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458736&type=middle 因为存在不依赖于问题(1)任何正解的正常数 m i , M i ( i =1, 2), 使得(2)与(3)成立，所以对非负非平凡初值，问题(1)是持久的；同时也说明毒素对问题(1)的持久性没有直接的影响，只影响了种群数量. 平衡点的局部稳定性 1 显然，问题(1)有平凡平衡点 S 0 (0, 0), 边界平衡点 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458868&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458868&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458868&type=middle , 关于正平衡点有以下定理. 定理3   反应扩散问题(1)至少存在一个正平衡点. 证   设问题(1)存在正平衡点( u * , v * ), 满足 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458741&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458741&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458741&type=middle 消去 v * ，得 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458745&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458745&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458745&type=middle 设 f ( x ):= Ax 5 + Bx 4 + Cx 3 + Dx 2 + Ex + F 其中 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458748&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458748&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458748&type=middle 显然 A > 0, F < 0. 由 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458873&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458873&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458873&type=middle 知，∃ u * > 0使 f ( u * )=0, 从而问题(1)至少存在一个正平衡点. 下面我们采用文献[ 7 7 ]中的线性近似法，讨论问题(1)在各个平衡点处的稳定性. 问题(1)对应的常微分系统的Jacobi矩阵如下： http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458750&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458750&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458750&type=middle 在平凡平衡点 S 0 (0, 0)处，特征方程为 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458756&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458756&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458756&type=middle 它有两个正根 λ = r 1 , λ = r 2 .故在 S 0 (0, 0)点处问题(1)是不稳定的. 对于偏微分问题(1)，我们需要考虑空间分解. 设 μ 1 , μ 2 , μ 3 , …(0= μ 1 < μ 2 < μ 3 < …)是问题(1)满足齐次Neumann边界条件的区域 Ω 上的-Δ算子的特征值， E ( μ i )是 C 1 ( Ω )中对应 μ i 的特征子空间. 令 X ={ p =( u , v )∈[ C 1 ( Ω )] 2 |∂ η p =0, x ∈∂ Ω }且{ φ ij , j =1, 2, …, dim E ( μ i )}是 E ( μ i )的正交基. 又令 D =diag( d 1 , d 2 ), L = D Δ+ F p ( p * )= D Δ+( a ij ) 2×2 ，其中 p * =( u * , v * ), 则反应扩散方程(1)中的Jacobi矩阵为 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458761&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458761&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458761&type=middle 在 S 0 (0, 0)处的特征方程为 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458764&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458764&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458764&type=middle 当 i =0时， μ i =0.该特征方程有两个正根 λ i 1 = r 1 和 λ i 2 = r 2 ，故问题(1)在 S 0 (0, 0)点是不稳定的. 对于边界平衡点 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458875&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458875&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458875&type=middle ，其特征方程为 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458767&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458767&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458767&type=middle 特征根为 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458770&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458770&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458770&type=middle 当 i =0时， μ i =0.故特征方程有一个负根 λ i 1 =- r 1 和一个正根 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458879&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458879&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458879&type=middle 所以问题(1)在边界平衡点 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458883&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458883&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458883&type=middle 不稳定. 由于对称性，可得问题(1)在其另一边界平衡点 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458888&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458888&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458888&type=middle 也不稳定. 对于正平衡点( u * , v * ), 由(4)和(5)式得其特征方程为 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458773&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458773&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458773&type=middle 即 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458777&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458777&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458777&type=middle 其中 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458781&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458781&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458781&type=middle http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458784&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458784&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458784&type=middle 即，若 a 1 a 2 > ac ，则 B > 0, 于是问题(1)在正平衡点局部稳定. 由此可得如下定理： 定理4   反应扩散方程(1)满足条件 a 1 a 2 > ac 时，正平衡点是局部稳定的. 正平衡点的全局稳定性 1 下面通过引入Lyapunov泛函，采用文献[ 7 7 ]中的Lyapunov直接法，讨论问题(1)在正平衡点( u * , v * )的全局渐近稳定性. 构造如下Lyapunov泛函 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458788&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458788&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458788&type=middle 沿着解的曲面关于 t 求导，得 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458792&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458792&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458792&type=middle 由于( u * , v * )是正平衡点，因此 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458893&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458893&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458893&type=middle - γ 1 u * v * =0.其中 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458797&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458797&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458797&type=middle 同理可得 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458801&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458801&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458801&type=middle 故 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458805&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458805&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458805&type=middle 其中 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458809&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458809&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458809&type=middle 即当 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458896&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458896&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458896&type=middle 时， G ′( t ) < 0成立. 利用Poincaré不等式且按照文献[ 7 7 ]第二章的方法可以证明正平衡点是全局渐近稳定的，因此正平衡点是惟一的.由此得到如下定理. 定理5   如果反应扩散问题(1)满足条件 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458813&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458813&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458813&type=middle 则它只有惟一的正平衡点，且正平衡点是全局渐近稳定的. 参考文献 WU J H, WEI G S. Coexistence state for cooperative model with diffusion [J]. Computers and Mathematics with Applications , 2002, 43 (10): 1279-1290. DOI: 10.1016/s0898-1221102700100-1. 向红, 苏克所, 姜惠敏, 等.具有饱和项和毒素影响的互惠模型的定性分析[J].兰州交通大学学报, 2010, 29 (3): 142-145. XIANG H, SU K S, JIANG H M, et al . Qualitative analysis of cooperative model with saturation term and toxin term [J]. Journal of Lanzhou Jiaotong University , 2010, 29 (3): 142-145 (Ch). 张睿, 苗亮英.一类具有非线性密度制约的食饵-捕食者扩散模型的稳定性[J].生物数学学报, 2014, 29 (3): 566-570. ZHANG R, MIAO L Y. Stability of a prey-predator model with non-liner density dependent and diffusion[J]. Journal of Biomath , 2014, 29 (3): 566-570 (Ch). 安海龙, 杨芳, 李艳玲.带饱和项的互惠交错扩散模型整体解的存在性和稳定性[J].生物数学学报, 2009, 24(4): 649-656. AN H L, YANG F, LI Y L. Existence and stability of global solutions for a cooperative cross-diffusion model with saturation [J]. Journal of Biomath , 2009, 24(4): 649-656 (Ch). 黄运金, 张朝阳.无穷时滞Lotka-Volterra竞争系统的持久性[J].福建农林大学学报(自然科学版), 2010, 39 (2): 222-224. HUANG Y J, ZHANG Z Y. Permanence of the Lotka-Volterra competitive system with infinite delays [J]. Journal Fujian Agriculture and Forestry University (Natural Science Edition ), 2010, 39 (2): 222-224 (Ch). 朱晶.有毒物影响和Beddington-DeAngelis型功能性反应的捕食系统的全局吸引性[J].生物数学学报, 2013, 28 (4): 716-724. ZHU J. Global attractivity of a predator-prey system with toxicants effect and Beddington-DeAngelis functional response [J]. Journal of Biomath , 2013, 28 (4): 716-724 (Ch). 林支桂.数学生态学导引[M].北京:科学出版社, 2013. LIN Z G. Guide of Mathematical Ecology [M]. Beijing: Science Press, 2013 (Ch). 10.14188/j.1671-8836.2018.05.010 O175.26 whdxxblxb-64-5-448 国家自然科学基金资助项目(11371311) 2017-07-01 2018-10-24 2020-12-09 版权所有© 2018 《武汉大学学报(理学版)》编辑部 Copyright © 2018 Journal of Wuhan University(Natural Science Edition). All rights reserved. 2018 武汉大学学报(理学版) 5 64