一类食物链模型正解的存在性和惟一性

一类食物链模型正解的存在性和惟一性 Existence and Uniqueness of Positive Solutions for a Food-Chain Model 1 first-author 李海侠，女，博士，副教授，主要从事偏微分方程及其可视化方面的研究.E-mail:xiami0820@163.com 李海侠，女，博士，副教授，主要从事偏微分方程及其可视化方面的研究.E-mail: xiami0820@163.com xiami0820@163.com 2 宝鸡文理学院 数学与信息科学学院, 陕西 宝鸡 721013 宝鸡文理学院 数学与信息科学学院, 陕西 宝鸡 721013 Institute of Mathematics and Information Sciences, Baoji University of Arts and Sciences, Baoji 721013, Shaanxi, China Institute of Mathematics and Information Sciences, Baoji University of Arts and Sciences, Baoji 721013, Shaanxi, China 陕西科技大学 文理学院，陕西 西安 710021 陕西科技大学 文理学院，陕西 西安 710021 School of Science and Arts, Shaanxi of Science and Technology, Xi'an 710021, Shaanxi, China School of Science and Arts, Shaanxi of Science and Technology, Xi'an 710021, Shaanxi, China 研究了一类带有Crowley-Martin反应函数的食物链模型.运用不动点指数理论给出了正解存在的充分条件, 进而利用扰动理论和分歧理论讨论了正解的惟一性.结果表明参数 c 和 d 对正解的惟一性有很大的影响. A food-chain model with Crowley-Martin functional response is discussed. The sufficient conditions for the existence of positive solutions are given by means of the fixed point index. Furthermore, by using the perturbation techniques and bifurcation theory, the uniqueness of positive solutions is investigated. The results show that parameters c and d have important effect on the uniqueness of positive solutions. 食物链模型 不动点指数 分歧理论 存在性 惟一性 food-chain model the fixed point index bifurcation theory existence uniqueness 引言 1 1989年Crowley和Martin [ 1 1 ] 提出Crowley-Martin (C-M)反应函数 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458824&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458824&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458824&type=middle C-M反应函数是依赖捕食者的经典函数之一，它与B-D反应函数很相似，不同之处在于分母多了体现物种间相互干扰的 cduw .显然，若 c = d =0，则C-M反应函数为Holling-Ⅰ反应函数；若 c ＞0, d =0, 则C-M反应函数为Holling-Ⅱ反应函数.C-M反应函数认为无论某个捕食者目前是否寻找或处理食饵, 都允许存在捕食者间的干扰, 这是比B-D反应函数优越之处, 也非常符合现实中的一些生物现象.因此研究带有C-M反应函数的模型具有很大的生物意义.基于此，本文在齐次Dirichlet边界条件下讨论如下食物链模型： http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458829&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458829&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458829&type=middle 其中 Ω 是 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458966&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458966&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458966&type=middle 中带有光滑边界∂ Ω 的有界区域, u , v 和 w 分别表示食饵、第一、第二个捕食者的浓度, r 1 , r 2 和 r 3 分别是 u , v 和 w 的最大增长率.系统(1)中的参数 c , d , e , a i , r i ( i =1, 2, 3)都是正常数. 目前两物种的捕食-食饵模型得到了广泛的研究 [ 2 2 ~ 14 14 2~14 ] , 其中文献[ 11 11 ~ 14 14 11~14 ]研究了带有改进Leslie-Gower项的捕食-食饵模型, 得到了正解的惟一性、稳定性、多解性、Hopf分歧和非常数正解的存在性.然而三物种捕食-食饵模型的动力学行为非常复杂, 在齐次Dirichlet边界条件下对三物种的捕食-食饵模型的研究很少见, 尤其讨论参数充分大时对系统正解影响的工作更是甚少.因此本文主要研究参数 c 和 d 对系统(1)正解的影响, 从而得到系统(1)正解的惟一性条件. 令 λ 1 ( q )是-Δ+ q ( x )在 Ω 上关于齐次Dirichlet边界条件下的主特征值.简单起见记 λ 1 (0)= λ 1 且对应的特征函数为 φ 1 > 0. 如下问题 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458833&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458833&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458833&type=middle r > λ 1 时, (2)式有惟一正解, 记为 Θ ( r , a ) , 且 Θ ( r , a ) < r / a .特别地, 当 r 1 > λ 1 , r 2 > λ 1 , r 3 > λ 1 时, 为简单起见, 我们记 Θ ( r 1 , 1) 为 Θ r 1 , Θ ( r 2 , 1) 为 Θ r 2 , Θ ( r 3 , a 3 ) 为.显然系统(1)有4类形式的解： 平凡解(0, 0, 0)； 弱半平凡解( u , 0, 0), (0, v , 0)和(0, 0, w )； 强半平凡解( u , v , 0), (0, v , w )和( u , 0, w )； 正解( u , v , w ). 共存解的存在性和不存在性 1 本小节利用度理论给出系统(1)正解存在的条件.设 r 1 > λ 1 , 若 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458970&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458970&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458970&type=middle 记 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458837&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458837&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458837&type=middle 的惟一正解为 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458972&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458972&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458972&type=middle 同理, 若 r 2 > λ 1 , r 3 > λ 1 , 记 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458841&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458841&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458841&type=middle 的惟一正解为 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458974&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458974&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458974&type=middle 由主特征值的比较原理易给出系统(1)正解存在的必要条件. 引理1   若( u , v , w )是系统(1)的正解, 则 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458844&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458844&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458844&type=middle 利用上下解方法给出系统(1)正解的先验估计. 引理2   系统(1)的任意正解( u , v , w )满足 u ≤ http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458976&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458976&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458976&type=middle 这里 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458979&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458979&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458979&type=middle http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458981&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458981&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458981&type=middle 进而, 若 r 2 > λ 1 , 则 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458983&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458983&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458983&type=middle 为了利用度理论我们引入如下空间: http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458847&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458847&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458847&type=middle 定义算子 Α τ : X → X 为 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458850&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458850&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458850&type=middle 其中 τ ∈[0, 1], q 是充分大的正常数.显然 Α τ 是紧算子.记 Α = Α 1 , 则易知 Α : D → W 是连续可微算子.显然, 系统(1)有正解当且仅当 Α 在 D 中有正不动点. 利用Dancer指数定理 [ 15 15 ] 可得如下引理, 在此省略其证明. 引理3 (ⅰ)若 r 1 > λ 1 , 或 r 2 > λ 1 , 或 r 3 > λ 1 , 则 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458853&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458853&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458853&type=middle 若 r 1 < λ 1 , r 2 < λ 1 且 r 3 < λ 1 , 则 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458856&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458856&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458856&type=middle (ⅱ) index W ( Α , D )=1. (ⅲ)设 r 1 > λ 1 .若 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458985&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458985&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458985&type=middle 或 r 3 > λ 1 , 则index W ( Α , ( Θ r 1 , 0, 0))=0; 若 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458987&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458987&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458987&type=middle 则 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458861&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458861&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458861&type=middle (ⅳ)设 r 2 > λ 1 .若 r 1 > λ 1 ( a 1 Θ r 2 )或 r 3 > λ 1 , 则 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458865&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458865&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458865&type=middle 若 r 1 < λ 1 ( a 1 Θ r 2 ), r 3 < λ 1 , 则 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458869&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458869&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458869&type=middle (ⅴ)设 r 3 > λ 1 .若 r 1 > λ 1 或 r 2 > λ 1 , 则 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458872&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458872&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458872&type=middle 若 r 1 < λ 1 , r 2 < λ 1 , 则 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458989&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458989&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458989&type=middle (ⅵ)设 r 1 > λ 1 , r 3 > λ 1 .若 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458876&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458876&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458876&type=middle 则 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458878&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458878&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458878&type=middle 若 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458881&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458881&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458881&type=middle (ⅶ)设 r 2 > λ 1 , r 3 > λ 1 .若 r 1 > λ 1 ( a 1 Θ r 2 ), 则 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458882&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458882&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458882&type=middle 若 r 1 < λ 1 ( a 1 Θ r 2 ), 则 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458992&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458992&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458992&type=middle http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458886&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458886&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458886&type=middle 这里( u , v )是如下问题 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458891&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458891&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458891&type=middle 的正解.对于问题(3)有 引理4   若 r 1 > λ 1 ( a 1 Θ r 2 )且 r 2 > λ 1 , 则系统(3)至少存在一个正解. 利用文献[ 16 16 ]可得 引理5   设 r 1 > λ 1 ( a 1 Θ r 2 ), r 2 > λ 1 .若 r 3 > λ 1 , 则index W ( Α , S )=0. 由度的可加性得系统(1)正解存在的充分条件. 定理1   若 r 1 > λ 1 ( a 1 Θ r 2 ), r 2 > λ 1 , r 3 > λ 1 , 则系统(1)至少存在一个正解. 证   由引理3和引理5有 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458894&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458894&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458894&type=middle 根据度的可加性知, 系统(1)至少存在一个正解. 最后给出系统(1)正解的不存在性条件. 定理2   若下列条件之一成立, 则系统(1)没有共存解： http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458897&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458897&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458897&type=middle 共存解的稳定性和惟一性 1 c 充分大且 r 1 , r 2 远离 λ 1 2 首先利用上下解方法也可以得到系统(1)当参数 c 充分大时正解存在的条件. 引理6   设 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458995&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458995&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458995&type=middle 对于充分小的 ε 存在充分大的 C = C ( ε )使得若 c ≥ C 且 r 2 > λ 1 , 则系统(1)有正解 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458996&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458996&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458996&type=middle 且满足 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458997&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458997&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458997&type=middle http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458998&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458998&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458998&type=middle 这里 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458999&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458999&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458999&type=middle 分别是如下问题的惟一正解 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458899&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458899&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458899&type=middle 证   根据上下解方法, 只需要证明( u , v , w )= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459000&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459000&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459000&type=middle 是系统(1)的一对有序上下解即可, 即证明如下不等式成立 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458901&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458901&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458901&type=middle http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458904&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458904&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458904&type=middle http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458906&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458906&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458906&type=middle (4) 和(5)式显然成立.至于(6)式直接验证只要 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458909&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458909&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458909&type=middle 成立即可.取 C ( ε )=2 a 2 / ε , 则当 c ≥ C ( ε )时, (6)式成立.故对于充分大的 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459001&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459001&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459001&type=middle 和 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459002&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459002&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459002&type=middle 是系统(1)的一对有序上下解.由椭圆系统的存在比较原理知系统(1)存在正解. 引理7   设 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459003&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459003&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459003&type=middle 则对于充分小的 δ 和 ε 存在充分大的 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459004&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459004&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459004&type=middle 使得当 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459005&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459005&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459005&type=middle 且 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459007&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459007&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459007&type=middle http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459009&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459009&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459009&type=middle 这里( u , v , w )是系统(1)的任意正解. 证   采用反证法. 则存在 c i →∞, r 2, i → r 2 ∈[ λ 1 + ε , +∞)和系统(1)的正解( u i , v i , w i )使得( u i , v i , w i )远离 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459011&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459011&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459011&type=middle http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459012&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459012&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459012&type=middle 根据引理2、 L p 估计和Sobolev嵌入定理可设 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459013&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459013&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459013&type=middle 是如下问题的惟一正解 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458914&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458914&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458914&type=middle 因此, 在 C 1 中1/(1+ c i u i )(1+ dw i )→0.故( u , v , w )满足方程组 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458917&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458917&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458917&type=middle 因 r 2 ≥ λ 1 + ε , 故 v ≡ Θ r 2 .由此得 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459014&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459014&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459014&type=middle 矛盾.于是结论成立. 接下来给出 c 充分大且 r 2 远离 λ 1 时系统(1)正解的惟一性和稳定性. 定理3   设 r 1 > λ 1 ( a 1 ), r 3 > λ 1 , 则对充分小的 ε 存在充分大的 C = C ( ε )使得若 c ≥ C 且 r 2 ≥ λ 1 + ε , 则系统(1)存在惟一的非退化且线性稳定的正解. 证   由定理1只证明惟一性即可. 先证明系统(1)的任意正解非退化和线性稳定, 需证明系统(1)在正解处的线性化特征值问题没有 Re ( μ )≤0的特征值 μ 即可.相反, 假设对于 c i →∞, r 2. i → r 2 ≥ λ 1 + ε 的序列 c i 和 r 2. i , 系统(1)存在退化或线性不稳定的正解( u i , v i , w i ), 则存在Re( μ i )≤0的 μ i 以及满足‖ ξ i ‖ L 2 +‖ η i ‖ L 2 +‖ χ i ‖ L 2 =1的( ξ i , η i , χ i )≠0使得对于系统(1)的任意正解( u i , v i , w i )有 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458921&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458921&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458921&type=middle 由引理7知, 当 c i →∞时, http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458924&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458924&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458924&type=middle 记 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459015&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459015&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459015&type=middle 分别是 ξ i , η i 和 χ i 的共轭算子, 则由(7)式得 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458926&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458926&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458926&type=middle http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458929&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458929&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458929&type=middle 因 u i , v i 和 w i 有界, 故Re( μ i )和Im( μ i )有界, 即 μ i 有界, 设 μ i → μ , 则Re( μ )≤0.又由 L p 估计可知 ξ i 、 η i 和 χ i 有界, 设 ξ i → ξ , η i → η , χ i → χ .对(7)式取极限得 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458934&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458934&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458934&type=middle 若 η ≡0, 则 μ ≥ λ 1 (- r 2 +2 Θ r 2 ) > 0, 矛盾.因此, η ≡0.同理可得 ξ ≡0, χ ≡0, 产生矛盾.因此系统(1)的任意正解非退化且线性稳定. 下面证明惟一性. 首先系统(1)的平凡解和半平凡解都远离正解, 所以根据线性算子的紧性理论可知系统(1)最多有有限个正解, 记为{( u i , v i , w i ): i =1, 2, …, n }.又由前面的分析知index W ( Α , ( u i , v i , w i ))=1.所以, 根据度的可加性、引理3和引理5得 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458938&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458938&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458938&type=middle 此即证明了惟一性.这说明系统(1)存在惟一的非退化和线性稳定的正解. 同理可得当参数 d 充分大时系统(1)正解的稳定性和惟一性. 定理4   设 r 1 > λ 1 ( a 1 Θ r 2 ), r 2 > λ 1 , r 3 > λ 1 , 则对于充分大的 d , 系统(1)存在惟一的非退化且线性稳定的正解. c 充分大且 r 1 远离 λ 1 , r 2 接近于 λ 1 2 本小节利用分歧理论给出 c 充分大且 r 2 接近于 λ 1 时系统(1)正解的惟一性和稳定性. 定理5   设 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459016&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459016&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459016&type=middle 对于充分小的 ε 存在 C 充分大使得若 c ≥ C 且 r 2 > r 2 , 则系统(1)存在惟一非退化且线性稳定的正解, 这里 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458941&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458941&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458941&type=middle 证   首先证明当 r 2 ∈ r ( r 2 , r 2 + ε 0 ]且 c 充分大时系统(1)存在惟一正解, 这里 ε 0 在后面给出.对于固定的 c > 0, 以 r 2 为分歧参数得 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459017&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459017&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459017&type=middle 是(1)的简单分歧点, 于是，根据Crandall- Rabinowitz分歧定理 [ 17 17 ] 知存在 δ > 0和 C 1 曲线 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458944&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458944&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458944&type=middle 使得系统(1)在 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458946&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458946&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458946&type=middle 附近的所有正解都在这条曲线 Τ c 上.因此我们只需要证明对于适当选择的 ε 0 , Τ c 一致覆盖 r 2 区间( r 2 , r 2 + ε 0 ]且对于固定的 c 仅覆盖一次.另一方面, 容易计算 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458949&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458949&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458949&type=middle 此处 ψ 1 是 r 2 对应的主特征函数. 于是存在充分大的 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459018&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459018&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459018&type=middle 和充分小的0 < δ 0 < δ , 使当0 < s ≤ δ 0 且 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459019&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459019&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459019&type=middle 因此, r 2 ( c , δ 0 )- r 2 = r 2 ( c , δ 0 )- r 2 ( c , 0) > 0. 令 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459020&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459020&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459020&type=middle 于是若 r 2 ≤ r 2 + ε 0 , 则对于 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459021&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459021&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459021&type=middle 这说明对于 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459022&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459022&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459022&type=middle 覆盖 r 2 区间( r 2 , r 2 + ε 0 ].又由于当0 < s ≤ δ 0 且 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459023&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459023&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459023&type=middle 时 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459024&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459024&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459024&type=middle 故 Τ c 仅覆盖 r 2 区间( r 2 , r 2 + ε 0 ]一次. 最后证明正解的稳定性. 设 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459025&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459025&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459025&type=middle 和 L ( r 2 ( c , s ), u ( c , s ), v ( c , s ), w ( c , s ))分别是系统(1)在 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459026&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459026&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459026&type=middle 和( r 2 ( c , s ), u ( c , s ), v ( c , s ), w ( c , s ))处的线性化算子. http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458952&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458952&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458952&type=middle 此处0 < α < 1. i : E 0 → E 1 是包含映射.则由 i -简单特征值的定义可知0是 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459027&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459027&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459027&type=middle 的 i -简单特征值且 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459028&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459028&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459028&type=middle 的其余特征值都在右半复平面.于是根据文献[ 18 18 ]的引理1.3, 存在 κ > 0和 C 1 函数 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458955&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458955&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458955&type=middle 使得对于 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459029&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459029&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459029&type=middle 有 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458959&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458959&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458959&type=middle 这里 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458962&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458962&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458962&type=middle 而且, 若 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459030&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459030&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459030&type=middle 则 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458963&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458963&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4458963&type=middle 另一方面, 易得 γ ( r 2 , c )= r 2 - r 2 .由(8)式以及 τ (·, c )的连续性可知对于所有的0 < s ≤ δ 0 和 c ≥ http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459031&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459031&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459031&type=middle ，有 τ ( s , c ) > 0.这便证明了正解是非退化和线性稳定性的. 取 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459032&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459032&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459032&type=middle 则 λ 1 + ε < r 2 + ε 0 < λ 1 + ε 0 .令 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459033&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459033&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459033&type=middle 这里 C ( ε )在定理2中给出, 根据定理2得若 c ≥ C 且 r 2 > r 2 , 则系统(1)存在惟一的正解. 参考文献 CROWLEY P H, MARTIN E K. Functional responses and interference within and between year classes of a dragonfly population [J]. Journal of the North American Benthological Society , 1989, 8: 211-221. WU J H. Stability and persistence for prey-predator model with saturation [J]. Chinese Science Bulletin , 1998, 43 (24): 2102-2103. PENG R, WANG M X. On multiplicity and stability of positive solutions of a diffusive prey-predator model [J]. Journal of Mathematical Analysis and Applications , 2006, 316 (1):256-268. DOI: 10.1016/j.jmaa.2005.04.033. WANG Z G, WU J H. Qualitative analysis for a ratio-dependent predator-prey model with stage structure and diffusion [J]. Nonlinear Analysis: Real World Applications , 2008, 9 (5):2270-2287. DOI:10.1016/j.nonrwa.2007.08.004. JIA J F, WU J H, NIE H. The coexistence states of a predator-prey model with nonmonotonic functional response and diffusion [J]. Acta Application Mathe-matice , 2009, 108 (2):413-428. DOI: 10.1007/s10440-008-9319-y. PENG R, SHI J P. Non-existence of non-constant positive steady-state of two Holling type Ⅱ predator-prey systems: Strong interaction case [J]. Journal of Differential Equations , 2009, 247 (3):866-886. DOI: 10.1016/j.jde.2009.03.008. NIE H, WU J H. Multiplicity and stability of a predator-prey model with non-monotonic conversion rate [J]. Nonlinear Analysis: Real World Applications , 2009, 10:154-171. DOI: 10.1016/j.nonrwa.2007.08.020. GUO G H, WU J H. Multiplicity and uniqueness of positive solutions for a predator-prey model with B-D functional response [J]. Nonlinear Analysis: Theory, Methods & Applications , 2010, 72 (3): 1632-1646. DOI: 10.1016/j.na.2009.09.003. GUO G H, WU J H. The effect of mutual interference between predators on a predator-prey model with diffusion [J]. Journal of Mathematical Analysis Applications , 2012, 389 (1):179-194. DOI: 10.1016/j.jmaa.2011.11.044. WEI M H, WU J H, GUO G H. The effect of predator competition on positive solutions for a predator-prey model with diffusion [J]. Nonlinear Analysis , 2012, 75 (13):5053-5068. DOI:10.1016/j.na.2012.04.021. ZHOU J, SHI J P. The existence, bifurcation and stability of positive stationary solutions of a diffusive Leslie-Gower predator-prey model with Holling-type Ⅱ functional responses [J]. Journal of Mathematical Analysis and Applications , 2013, 405 (2):618-630. DOI: 10.1016/j.jmaa.2013.03.064. ZHOU J. Positive solutions of a diffusive Leslie-Gower predator-prey model with Bazykin functional response [J]. Zeitschrift für angewandte Mathematik und Physik , 2014, 65 (1):1-18. DOI:10.1007/s00033-013-0315-3. LI H X. Asymptotic behavior and multiplicity for a diffusive Leslie-Gower predator-prey system with Crowley-Martin functional response [J]. Computers and Mathematics with Applications , 2014, 68 (7):693-705. DOI: 10.1016/j.camwa.2014.07.018. LI S B, WU J H, NIE H. Steady-state bifurcation and Hopf bifurcation for a diffusive Leslie-Gower predator-prey model [J]. Computers and Mathematics with Applications , 2015, 70:3043-3056. DOI: 10.1016/j.camwa.2015.10.017. DANCER E N. On the indices of fixed points of mapping in cones and applications [J]. Journal of Mathematical Analysis and Applications , 1983, 91 (1):131-151. DANCER E N, DU Y H. Positive solutions for a three-species competition system with diffusion I. General existence results [J]. Nonlinear Analysis: Theory, Methods and Applications , 1995, 24 (3):337-357. CRANDALL M G, RABINOWITZ P H. Bifurcation from simple eigenvalue [J]. Journal of Functional Analysis , 1971, 8 (2): 321-340. CRANDALL M G, RABINOWITZ P H. Bifurcation, perturbation of simple eigenvalues and linearize stability [J]. Arch Ration Mech Anal , 1973, 52 (2):161-180. 10.14188/j.1671-8836.2018.05.011 O175.26 whdxxblxb-64-5-453 国家自然科学基金(11501496);陕西省自然科学基础研究计划(2015JM1008, 2014JQ2-1003);陕西省教育厅专项科研计划项目(16JK1046);宝鸡文理学院博士科研启动项目(ZK2018069) 2016-07-15 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