具有Holing-Ⅲ反应项的捕食系统的平衡态共存解

具有Holing-Ⅲ反应项的捕食系统的平衡态共存解 Coexistence Solution on the Steady-State of a Predator-Prey System with Holing-Ⅲ Functional Response first-author 冯孝周, 男，博士, 副教授, 研究领域为偏微分方程应用及计算. E-mail:flxzfxz8@163.com 冯孝周, 男，博士, 副教授, 研究领域为偏微分方程应用及计算. E-mail: flxzfxz8@163.com flxzfxz8@163.com 西安工业大学 理学院，陕西 西安 710062 西安工业大学 理学院，陕西 西安 710062 School of Science, Xi'an Technological University, Xi'an 710032, Shaanxi, China School of Science, Xi'an Technological University, Xi'an 710032, Shaanxi, China 研究了一类具有Holling-Ⅲ反应项的捕食模型在Dirichlet边界条件下平衡态解的共存性.利用比较原理、Leray-Schauder度理论及不动点指标可加性，得到了具有Holling-Ⅲ反应项的捕食模型共存解存在的充分必要条件.利用数值模拟的方法研究了一维方程的平衡态解，验证了理论结果的正确性.研究表明该捕食系统中参数满足一定的条件时，系统中的两种生物可以产生共存现象. In this paper, the coexistence on the steady-state of a predator-prey model with Holling-Ⅲ functional response under the Dirichlet boundary condition is studied. Using the comparison principle, Leray-Schauder degree theory and additivity fixed point index, the sufficient and necessary conditions for the coexistence of the equilibrium equations on the predator-prey model with Holling-Ⅲ functional response are established. By using the numerical simulation on one-dimensional equilibrium system, we can verify the correctness of the theoretical results. The results show that the two species in the system can coexist if some parameters of the predator-prey model with Holling-Ⅲ functional response satisfy certain conditions. 捕食模型 共存解 不动点指标 数值模拟 predator-prey model coexistence solution fixed point index numerical simulation 引言 1 生态系统中相互作用物种关于空间和时间的动力学行为是生态学的一个中心问题.具有多物种相互作用的反应扩散模型成为许多学者研究的焦点.特别地，关于种群系统解的共存性、持续性及稳定性已成为该领域研究的热门课题.当物种空间均匀分布时, 一般通过常微分方程系统的正解来刻画.而物种在空间不均匀的情况下, 通常利用非线性反应扩散方程的共存解来刻画物种的演变规律.关于非线性偏微分方程模型的动力学行为的研究在种群生态学中具有重要意义.近几十年来, 学者们广泛地研究了由非线性反应扩散导出的具有空间项的捕食模型, 得到了许多重要研究成果(参见文献[ 1 1 ~ 9 9 1~9 ]). 在文献[ 9 9 ]中，我们研究了具有Holling-Ⅲ反应项的捕食系统在Neumann条件下的正平衡态解的局部分歧与全局分歧解的存在性.而在本文中, 考虑以下具有Holling-Ⅲ功能性反应项捕食系统在齐次Dirichlet边界条件下的平衡态模型： http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257362&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257362&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257362&type=middle 其中 Ω 是 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257533&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257533&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257533&type=middle 中的有界区域( N ≥1为整数)且具有光滑的边界 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257536&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257536&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257536&type=middle ；两个未知函数 u 1 , u 2 分别代表食饵和捕食者在空间的分布密度； l 代表食饵最大饱和承载量；常数 d 1 , d 2 ＞0是相应物种的扩散系数, k 表示食饵的承载能力, 而 δ 是捕食者的死亡率, 并且 m 可视为两个物种之间的相互作用强度的量度.从数学计算的角度考虑，为简化运算, 我们对系统(1)进行无量纲化处理，构造出以下变换： http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257367&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257367&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257367&type=middle 因此, 参数 a , b , c , d , k 均为正常数，故系统(1)可以转化如下形式： http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257372&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257372&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257372&type=middle 本文考虑系统(2)的更一般的情况, 即参数 a , b , d , k 为正常数和 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257539&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257539&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257539&type=middle .系统(2)的一个解( u , v )如果满足： http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257377&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257377&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257377&type=middle 其中 n x 表示单位外法线方向向量，则称( u , v )为系统(2)的共存解. 为了得到主要结果, 我们首先引入一些符号 [ 10 10 ] .对 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257542&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257542&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257542&type=middle , 令 λ 1 ( q )为下列特征值问题的主要特征值 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257382&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257382&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257382&type=middle 并简记 λ 1 (0)为 λ 1 , 易知 λ 1 ( q )是严格增加的，即当 q 1 ( x )≤ q 2 ( x )且 q 1 ( x )≠ q 2 ( x )时， λ 1 ( q 1 )＜ λ 1 ( q 2 )(参见文献[ 5 5 , 7 7 ]). 定义 Θ ( p ( x ))为下面方程的惟一解， http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257387&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257387&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257387&type=middle 且 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257545&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257545&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257545&type=middle .其中 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257548&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257548&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257548&type=middle (0＜ α ＜1)是一个正函数, 且仅当 λ 1 (－ p )＜0时解存在 [ 11 11 ~ 13 13 11~13 ] . 为了简化运算, 记 Θ ( a )为 Θ .在本文中, 我们首先建立系统(2)共存态解存在的充分必要条件，主要结果如定理1. 定理1   系统(2)存在一个共存态解的充分必要条件是 a ＞ λ 1 且 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257552&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257552&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257552&type=middle 第1节给出了一些预备知识；第2节利用比较原理，Leray-Schauder度理论及不动点指标可加性，给出了定理1的证明；第3节利用数值模拟的方法研究了系统(2)一维方程的平衡态解，验证了理论结果的正确性. 预备知识 1 在本节中, 我们主要给出一些本文后面证明所需要的基本定理. 定理2 [ 11 11 , 15 15 ]   设 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257555&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257555&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257555&type=middle , 存在一个充分大的 p , 使得 p ＞ q ( x ), x ∈ Ω .定义如下正的紧算子 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257391&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257391&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257391&type=middle r (L)表示算子L的谱半径.则下列结论成立： (ⅰ) λ 1 ( q )＞0⇔ r (L)＜1; (ⅱ) λ 1 ( q )＜0⇔ r (L)＞1; (ⅲ) λ 1 ( q )=0⇔ r (L)=1. 从定理2, 可以推导出以下定理，参见文献[ 3 3 , 11 11 , 12 12 , 14 14 ]. 定理3   设 q ( x )∈L ∞ ( Ω ), φ ≥0, φ ≠0, x ∈ Ω 且 φ =0, http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257558&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257558&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257558&type=middle 则下面结论成立： (ⅰ)若0 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257560&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257560&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257560&type=middle －Δ φ + q ( x ) φ ≤0, 则 λ 1 ( q )＜0； (ⅱ)若0 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257564&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257564&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257564&type=middle －Δ φ + q ( x ) φ ≥0, 则 λ 1 ( q )＞0； (ⅲ)若－Δ φ + q ( x ) φ ≡0, 则 λ 1 ( q )=0. 考虑下面单个方程： http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257394&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257394&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257394&type=middle 其中 Ω 是 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257567&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257567&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257567&type=middle 中具有光滑边界 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257569&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257569&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257569&type=middle 的有界区域.设函数 f ( x , u ): http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257571&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257571&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257571&type=middle 满足以下假设： (H1) f ( x , u )是关于 x 的 C α 函数, 其中0＜ α ＜1； (H2) f ( x , u )为 C ′函数, 且关于 u , f u ( x , u )＜0，( x , u )∈ http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257573&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257573&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257573&type=middle ； (H3) 对于某个正常数 C , 在 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257574&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257574&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257574&type=middle 中， f ( x , u )≤0. 定理4 [ 14 14 , 16 16 ] (ⅰ)系统(5)的非负解 u ( x )满足 u ( x )≤ C , http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257577&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257577&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257577&type=middle (ⅱ)若 λ 1 (－ f ( x , 0))≥0, 则系统(5)没有正解，且平凡解 u ( x )=0是全局渐近稳定的; (ⅲ)若 λ 1 (－ f ( x , 0))＜0, 则系统(5)具有惟一的全局渐近稳定的正解，且 u ( x )=0的平凡解是不稳定的. 为了利用不动点指标理论, 下面给出不动点指标理论基本框架, 这也是后续证明的基本理论. 设 E 为Banach空间, 且为闭凸集.若对所有 γ ≥0, λW ⊂ W , 且 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257580&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257580&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257580&type=middle , 则称 W ∈ E 为总楔形.对 y ∈ W , 定义: http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257397&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257397&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257397&type=middle 且 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257400&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257400&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257400&type=middle 故 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257584&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257584&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257584&type=middle 是包含 W , y , － y 的楔, 而 S y 是包含 y 的 E 的封闭子集.令T为满足 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257591&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257591&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257591&type=middle 在 E 上的紧线性算子.若存在 t ∈(0, 1)和 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257596&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257596&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257596&type=middle , 使得(I－ t T) ω ∈ S y , 则称T在 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257598&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257598&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257598&type=middle 上具有性质 α .设A: W → W 是具有不动点 y ∈ W 的紧算子, 则A在 y 处Fréchet可微分的.令L=A′( y )是A在 y 处的Fréchet导数.故L将 W y 映射到本身.我们用deg W (I－A, D )表示 D 在 W 上算子I－A的度, 在 y 相对于 W 的A的不动点指标记作index W (A, y ).参考文献[ 17 17 ]的引理4.1，我们给出定理5. 定理5 [ 13 13 , 15 15 ]   设I－L在 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257601&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257601&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257601&type=middle 上是可逆的.则有: (ⅰ)若L在 W y 上具有性质 α , 则index W (A, y )=0； (ⅱ)若L在 W y 上不具有性质 α , 则 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257403&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257403&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257403&type=middle 其中 σ 是L在 W 的所有大于1的特征值的代数重数之和. 注1   由定理4(ⅱ)可知, 当 c ＞- λ 1 时, 系统(2)没有形如(0, v )的半平凡解. 系统(2)共存态解 1 本节主要利用第1节中的理论成果证明定理1.首先给出一个关于系统(2)具有共存解的必要条件. 定理6   若系统(2)具有共存态解, 则 a ＞ λ 1 且 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257604&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257604&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257604&type=middle 证   假设( u , v )是系统(2)的一个共存态解.那么 a ＞ λ 1 和0＜ u ＜ Θ .由于( u , v )满足： http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257406&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257406&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257406&type=middle 所以, http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257408&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257408&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257408&type=middle 故 a ＞ λ 1 且 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257609&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257609&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257609&type=middle .证毕. 为了证明 a ＞ λ 1 和 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257612&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257612&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257612&type=middle 是系统(2)存在共存解的充分条件, 我们利用Leray-Schauder度理论和不动点指标可加性来证明结论.下面，利用极值原理, 先给出系统(2)的共存解一个先验估计. 定理7   假定 c ＞－ λ 1 , 那么系统(2)的任何共存态解( u , v )都有如下先验边界： http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257410&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257410&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257410&type=middle 证   由系统(2)第一个方程可知, u 满足： http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257412&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257412&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257412&type=middle 因此, 根据极大值原理知, u ( x )≤ a .通过直接计算, 由系统(2)的两个方程可知： http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257416&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257416&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257416&type=middle 因此， http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257420&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257420&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257420&type=middle 故 a + c ＞0，否则，若 a + c ≤0，则可导出 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257421&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257421&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257421&type=middle 从而 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257424&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257424&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257424&type=middle 所以矛盾.因此, a + c ＞0是系统(2)存在共存态解的必要条件.在这种情况下 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257427&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257427&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257427&type=middle 定理7得证. 下面引入以下符号 E = C 0 1 ( Ω )× C 0 1 ( Ω ) W = K × K , K ={ φ ∈ C 0 1 ( Ω ): φ ≥0, x ∈ Ω } D ={( u , v )∈ W : u ≤ a +1, v ≤ Q +1} 其中 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257616&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257616&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257616&type=middle . 由定理7可知, 系统(2)的非负解必在 D 中.取充分大正常数 p ， p ＞max{ a + bQ / k , c }, 使得 u ( a http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257619&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257619&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257619&type=middle 和 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257623&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257623&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257623&type=middle ，且( u , v )∈[0, a ]×[0, Q ], 分别关于 u 和 v 单调加. 定义一个正紧算子A: E → E ，具体如下： http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257432&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257432&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257432&type=middle 注2   由椭圆方程的正则性 [ 12 12 ] 知, 系统(2)等价于( u , v )=A( u , v ).故, 证明A在 D 中具有正不动点等价于证明系统(2)具有正解.若A在 D 中有一个正不动点( u , v ), 则( u , v )是系统(2)的一个共存态解. 引理1   假设 a ＞ λ 1 和 c ＞－ λ 1 , 那么 (ⅰ) deg W (I－A, D )=1; (ⅱ) index W (A 0 , (0, 0))=0. 证   (ⅰ)显然, A在 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257625&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257625&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257625&type=middle 上没有不动点, 因此deg W (I－A, D )有定义.对 θ ∈[0, 1], 定义一个正紧算子 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257629&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257629&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257629&type=middle , 具体如下： http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257435&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257435&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257435&type=middle 则A 1 =A.对于每个 θ , A θ 的一个不动点是下面问题的解： http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257438&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257438&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257438&type=middle 类似于定理7的证明可知, 对每个 θ ∈[0, 1], A θ 的不动点( u , v )满足 u ≤ a 和 v ≤ Q .因此, A θ 在 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257631&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257631&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257631&type=middle D 上没有不动点, 所以deg W (I－A, D )有定义, 且deg W (I－A, D )与 θ 无关.故, deg W (I－A, D )=deg W (I－A 1 , D )=deg W (I－A 0 , D ).注意到, 当 θ =0时，(8)式仅存在平凡解(0, 0).令 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257440&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257440&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257440&type=middle 假设L( ξ 1 , ξ 2 )=( ξ 1 , ξ 2 ), 对某些( ξ 1 , ξ 2 )∈ W (0, 0) = K × K ，显然( ξ 1 , ξ 2 )=(0, 0).因此I-L在 W (0, 0) 上是可逆的.由于 λ 1 ＞0, 根据定理2知， r (L)＜1.故L没有性质 α .从而由定理5可得： http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257445&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257445&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257445&type=middle (ⅱ)注意到A(0, 0)=(0, 0).令 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257451&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257451&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257451&type=middle 则 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257456&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257456&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257456&type=middle 假设对某个( ξ 1 , ξ 2 )∈ W (0, 0) , L ( ξ 1 , ξ 2 )=( ξ 1 , ξ 2 ), 则 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257459&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257459&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257459&type=middle http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257462&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257462&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257462&type=middle 若 ξ 1 ≠0, 则 a = λ 1 ，这与 λ 1 ＜ a 矛盾.同样, 由于 c ＞ λ 1 , 易知 ξ 2 =0.故( ξ 1 , ξ 2 )=(0, 0).因此, I－L在 W (0, 0) 中是可逆的.由于 a ＞ λ 1 , 根据定理2可知, r a [(-Δ+ p ) －1 ( a + p )]＞1是算子(-Δ+ p ) －1 ( a + p )的主要特征值，且对应的特征函数 φ a ＞0.由于 S (0, 0) ={(0, 0)}, 故( φ a , 0)∈ W (0, 0) S (0, 0) .令 t a = r a －1 , 则(I－ t a L)( φ a , 0)=(0, 0)∈ S (0, 0) .故L具有性质 α .因此, 由定理5知, index W (A 0 , (0, 0))=0.定理得证. 接下来, 我们计算系统(2)的半平凡解的指标数. 引理2   假设 a ＞ λ 1 和 c ＞－ λ 1 , 那么下面结论成立： (ⅰ)若index W (A 0 , (0, 0))=0, 则index W (A 0 , ( Θ , 0))=0； (ⅱ)若 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257634&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257634&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257634&type=middle , 则index W (A 0 , ( Θ , 0))=1. 证   由定理6和引理1直接得到(ⅱ)的结果.事实上, 由定理6知, 如果 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257638&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257638&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257638&type=middle , 那么系统(2)的非负解是(0, 0)和( Θ , 0).因此, 通过不动点指标可加性 [ 13 13 , 16 16 ] 知, http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257466&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257466&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257466&type=middle 由引理1知， http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257470&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257470&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257470&type=middle 且 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257473&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257473&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257473&type=middle 则由(11)知, http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257477&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257477&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257477&type=middle 接下来, 给出(ⅰ)的证明.观察A( Θ , 0)=( Θ , 0).令L=A′( Θ , 0), 则 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257481&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257481&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257481&type=middle 假设对某个( ξ 1 , ξ 2 )∈ W ( Θ , 0) = C 0 1 × K 有L( ξ 1 , ξ 2 )=( ξ 1 , ξ 2 ), 则 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257485&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257485&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257485&type=middle 考虑到 ξ 2 ∈ K , 由定理3知, 当 ξ 2 ≠0时, 由(12)式的第二个方程得出 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257641&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257641&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257641&type=middle . 这与 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257643&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257643&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257643&type=middle 矛盾, 故 ξ 2 =0.因此，我们由(12)式的第一个方程得到 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257489&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257489&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257489&type=middle 由定理3知，若 ξ 1 ≠0, 则 λ 1 (2 Θ － a )=0.而另一方面, λ 1 (2 Θ － a )＞ λ 1 ( Θ － a )=0, 故两者矛盾.因此( ξ 1 , ξ 2 )=(0, 0), I－L在 W ( Θ , 0) 上是可逆的. 下证L在 W ( Θ , 0) 具有性质 α .事实上, 令 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257493&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257493&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257493&type=middle 由于 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257645&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257645&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257645&type=middle , 由定理2知, r c ( F )＞1是具有对应特征函数 F 的特征值 φ c ＞0.由 S ( Θ , 0) = C 0 1 ( Ω )× K 可知, (0, φ c )∈ W ( Θ , 0) S ( Θ , 0) .设 t c = r c －1 ∈(0, 1), 因此 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257497&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257497&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257497&type=middle 于是证明了L具有性质 α . 故index W (A 0 , ( Θ , 0))=0.引理2得证. 下面证明 a ＞ λ 1 和 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257646&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257646&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257646&type=middle 是系统(2)具有共存态解的充分条件. 定理8   若 a ＞ λ 1 且 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257649&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257649&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257649&type=middle , 则系统(2)具有共存态解. 证   如果 a ＞ λ 1 和 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257653&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257653&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257653&type=middle , 由引理1、2与不动点指标可加性知 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257501&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257501&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257501&type=middle 因此由Kronecker存在定理 [ 14 14 ] ，系统(2)至少有一个共存态解.定理8的证明完成. 注3   由定理6和定理8显然可证明，系统(2)具有共存态解充分必要条件是 a ＞ λ 1 和－ λ 1 ＜ c ＜ http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257655&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257655&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257655&type=middle 数值模拟 1 本节运用MATLAB软件对系统(2)进行数值模拟.为了研究方便，不失一般性，我们给出平衡态方程的正解在一维情况下的数值解图像( 图 1 图 1 ~ 图 6 图 6 ).通过数值计算得到的图像可直观验证前面定理的正确性.系统(2)对应的一维系统形式如下： http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257505&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257505&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257505&type=middle 系统(15)平衡态方程中的参数满足充分必要条件: a ＞ λ 1 和 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257658&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257658&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257658&type=middle .即当食饵的最大增长率 a ＞ λ 1 ，捕食者的最大增长率满足－ λ 1 ＜ c ＜ http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257659&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257659&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257659&type=middle 时，两物种 u 和 v 可以共存，解共存解及长时行为范数变化趋势见 图 1 图 1 ~ 图 6 图 6 .而且数值模拟表明，当模型中的参数 b =15, k =6, c =－15, d =20时，由 图 1 图 1 ~ 图 3 图 3 可以发现，随着参数 a 的增加，捕食者 v 的浓度逐渐增大，而食饵 u 的浓度逐渐减小.从生物意义上讲，食饵的最大增长率 a 的增加会使系统在单位时间内提供更多的食饵，从而增强捕食-食饵系统的物种数量. 图 4 图 4 ~ 图 6 图 6 是当 a =20, b =15, k =6, d =20, c =－15, －10, －3时， u , v 的共存解及长时变化趋势.数值模验证的结果表明， 图 4 图 4 ~ 图 6 图 6 与定理1的结果相一致，也与该模型的生物意义完全吻合. 结论 1 利用比较原理、Leray Schauder度理论及不动点指标可加性，研究了一类具有Holling-Ⅲ反应项的捕食模型中参数不同取值对系统在Dirichlet边界条件下平衡态解的共存解存在性的影响，得到了具有Holling-Ⅲ反应项的捕食模型共存解存在的充分必要条件，如定理1，定理6与定理7.利用数值模拟的方法研究了一维方程的平衡态解的变化规律，验证了理论结果的正确性，也实现了该模型解的可视化.研究表明该捕食系统中参数满足一定的条件时，两种生物种群可以共同生存的生态现象. 参考文献 CHEN J P. The qualitative analysis of two species predator-prey model with Holling type Ⅲ functional response[J].Applied Mathematics & Mechanics, 1986, 7 (1): 77-86. HOLLING C S.The functional response of predators to prey density and its role in mimicry and populations regulation[J].Memoirs of the Entomological Society of Canada, 1965, 97 (45): 53-60. CANO-CASANOVA S, LÓPEZ-GÓMEZ J. Properties of the principle eigenvalues of a general class if non-classical mixed boundary value problems[J]. Journal Differential Equations, 2002, 178 (1):123-211. CASAL A, EELBECK J C, LÓPEZ-GÓMEZ J. Existence and uniqueness of coexistence states for a predator-prey model with diffusion[J].Differential Integral Equations, 1994, 7 (2): 411-439. DU Y H, LOU Y. Some uniqueness and exact multiplicity results for a predator-prey model[J]. Transactions of the American Mathematical Societ, 1997, 349 (6): 2443-2475. DU Y H, LOU Y. S-shaped global bifurcation curve and Hopf bifurcation of positive solution to a predator-prey model[J]. J Differential Equations, 1998, 144 (2): 390-440. 冯孝周, 李艳玲, 聂华.具有功能反应项的捕食模型解的定性分析[J].西北师范大学学报(自然科学版), 2011, 47 (5): 9-16. FENG X Z, LI Y L, NIE H. Qualitative analysis of solutions of predator-prey model with functional responses[J]. Journal of Northwest Normal University (Natural Science), 2011, 47 (5): 9-16(Ch). 冯孝周, 吴建华.具有饱和与竞争项的捕食系统的全局分歧及稳定性[J].系统科学与数学, 2010, 30 (7):979-989. FENG X Z, WU J H. Global bifurcation and stability for prey-predator model with predator saturation and competition[J]. Journal of Systems Science and Mathematical Sciences, 2010, 30 (7): 979-989(Ch). 冯孝周, 李畅通.具有Holling Type Ⅲ反应项的捕食系统正平衡态解的存在性[J].安徽大学学报(自然科学版), 2013, 37 (3):16-21. FENG X Z, LI C T. Existence of positive steady-state for prey-predator system with Holling Type Ⅲ response[J].Journal of Anhui University (Natural Science Edition), 2013, 37 (3):16-21(Ch). CRANDALLM G, RABINOWITZ P H. Bifurcation from simple eigenvalues[J].Journal of Functional Analysis, 1971, 8: 321-340. DANCER E N. On positive solutions of some pairs of differential equations[J].Transactions of the American Mathematical Society, 1984, 284 (2):729-743. DANCER E N. On positive solutions of some pairs of differential equations, Ⅱ[J]. Journal of Differential Equations, 1985, 60 (2): 236-258. DANCER E N.On the indices of fixed points of mapping in cones and applications[J].Journal of Mathematical Analysis and Applications, 1983, 91: 131-151. PAO C V. Nonlinear Parabolic and Elliptic Equations[M]. New York: Plenum Press, 1992. LI L. Coexistence theorems of steady states for predator-prey interacting systems[J].Transactions of the American Mathematical Society, 1988, 305 (1):143-166. RUAN W H, FENG W. On the fixed point index and multiple steady states of reaction-diffusion system[J].Differential Integral Equations, 1995, 8 (2): 371-391. SMOLLER J. Shock Waves and Reaction-Diffusion Equations[M]. New York: Spring-Verlag, 1994. 10.14188/j.1671-8836.2018.06.007 O175.26 whdxxblxb-64-6-525 国家自然科学基金(61102144)资助项目；陕西省教育厅专项科研计划资助项目(18JK0393);西安工业大学研究生教改项目(2017033);西安工业大学校长基金资助项目(XAGDXJJ17028) 2017-11-20 2018-12-24 2020-11-30 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) 6 64