Variable-territory捕食模型正解的惟一性和稳定性

Variable-territory捕食模型正解的惟一性和稳定性 The Uniqueness and Stability of Positive Solution for Variable-Territory Predator-Prey Model first-author 姜洪领, 男, 副教授, 现从事偏微分方程应用方面的研究. E-mail:jhonglings@163.com 姜洪领, 男, 副教授, 现从事偏微分方程应用方面的研究. E-mail: jhonglings@163.com jhonglings@163.com 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 Science, Baoji University of Arts and Sciences, Baoji, Shaanxi 721013, China Institute of Mathematics and Information Science, Baoji University of Arts and Sciences, Baoji, Shaanxi 721013, China 研究了具有扩散的variable-territory捕食模型的平衡态正解.利用椭圆方程正则化理论和扰动理论, 给出当参数 b 充分小时, 正解惟一且线性稳定的充分条件.最后利用数值模拟验证了结论. The steady state positive solutions for a Variable-territory predator-prey model with diffusion are studied. By the elliptic equation regularity theory and perturbation theory, a sufficient condition of the uniqueness and linear stability of positive solution is obtained as the parameter b is small enough. At the end of this paper, some numerical simulations are given to check the theoretical results. 捕食食饵模型 平衡态正解 惟一性 稳定性 predator-prey model steady state positive solution uniqueness stability 引言 1 Variable-territory模型描述的是一类具有自控能力的捕食食饵模型, 它是文献[ 1 1 ]在研究耗费者与资源系统(consumer-resource systems)时, 通过改进Bazykin捕食模型建立的.一般的具有扩散的variable-territory模型可表示为 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459772&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459772&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459772&type=middle 模型中 u , v 分别表示食饵和捕食者的密度; 参数 a ＞0表示食饵的生长率; http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459833&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459833&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459833&type=middle , d ≥0表示捕食者增长率, d ＜0表示其死亡率; b 表示捕食常数； m = bT h , 这里 T h 表示捕食者处理一个单位食饵所用的时间; v 2 / u 项表示捕食者的捕食能力与食饵密度成反比; f ( s )= s /(1+ ms )为HollingⅡ型功能函数.详细模型背景见文献[ 1 1 ].文献[ 2 2 ~ 5 5 2~5 ]在齐次Neumann边界条件下, 研究了平衡态系统正常数解的稳定性以及正解的渐近行为.文献[ 6 6 , 7 7 ]在齐次Dirichlet边界条件下, 分别在 Ω 为一维区域和多维空间下, 讨论了平衡态正解的存在性和稳定性.特别的, 文献[ 6 6 ]利用扰动理论研究了 m 充分大时正解的惟一性.本文在文献[ 6 6 ]基础上继续研究平衡态系统 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459774&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459774&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459774&type=middle 这里 l 为给定正数.讨论参数 b 充分小时正解的惟一性和稳定性.并利用数值模拟来验证理论结果. 预备工作 1 首先介绍两个记号.记 λ 1 ( q )为 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459779&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459779&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459779&type=middle 的主特征值, 这里 q ∈ C ( Ω ), 对应特征函数在 Ω 上具有正性.特别当 q ≡0时, 记 λ 1 = λ 1 (0), 对应的特征函数记为 φ 1 ( x )＞0, x ∈(0, l ). 记 θ a 为 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459782&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459782&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459782&type=middle 的正解.由文献[ 8 8 ~ 10 10 8~10 ]可知, 当且仅当 a ＞ λ 1 时正解 θ a 存在且惟一, 同时 θ a ＜ a 且关于 a 是( λ 1 , +∞)到 C 2 ( Ω )∩ C 0 ( Ω )上的严格递增, 连续可微映射. 下面是关于系统(1)正解的结论, 详细证明参见文献[ 6 6 ]. 定理1   若系统(1)有正解( u , v ), 则 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459785&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459785&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459785&type=middle 且 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459787&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459787&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459787&type=middle 进而当 a ＞ λ 1 时, u ＜ θ a . 定理2   当且仅当 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459789&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459789&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459789&type=middle 系统(1)有正解( u , v ). 下面介绍 v / u 在[0, l ]上的一些性质, 详情见文献[ 6 6 ]引理2.2. 定理3   定义函数类 Λ ={ h ( x )∈ C 1 [0, l]: h ＞0, x ∈(0, l). h (0)= h ( l )=0, h ′(0)＞0, h ′( l )＜0} 若 u ( x ), v ( x ), V ( x )∈ Λ , 则 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459791&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459791&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459791&type=middle 是 C 1 (0, l )∩ C [0, l ]上的正函数，且 V ( x ) w ∈ C 1 [0, l ]. 注1   由定理3可知, 若 u , v 是(1)的正解, 则 v / u 在 x =0, l 处的值分别是 v ′(0)/ u ′(0)和 v ′( l )/ u ′( l ), 并且 v 2 ( x )/ u ( x )是 C 1 [0, l ]函数. b 充分小时系统(1)正解的惟一性和稳定性 1 首先考虑系统(1)对应的一种极限系统 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459793&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459793&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459793&type=middle 由系统(2)的结论可知 a ＞ λ 1 时, 系统(3)中 u 的方程有惟一正解 θ a , 因此系统(3)的正解只依赖于下列边值问题 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459797&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459797&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459797&type=middle 定理4   对任意(4)式的正解 v (若存在), 必有 v ＜( d + ac ) θ a , x ∈(0, l ). 证   设 z =( d + ac ) θ a － v , 代入系统(3)有 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459799&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459799&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459799&type=middle 注意到－ θ a ″= θ a ( a － θ a ), 代入上式整理可得 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459802&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459802&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459802&type=middle 其中 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459835&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459835&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459835&type=middle 在[0, l ]上非负且有界, 因此由极值原理可知 z ( x )＞0, x ∈(0, l ).即 v ＜( d + ac ) θ a . 定理5   若(4)式有正解, 则 d ＞ d * . 证   设 ψ 1 ＞0是 d * 对应的特征函数, 即 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459804&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459804&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459804&type=middle 对(4)式两端同乘以 ψ 1 并在(0, l )上积分有 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459805&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459805&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459805&type=middle 注意到 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459808&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459808&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459808&type=middle 代入上式整理可得 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459809&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459809&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459809&type=middle 因此有 d ＞ d * . 定理6   若 a ＞ λ 1 , d ＞ d * , 则(4)式有惟一正解, 记为 v * . 证   对给定的 ε ＞0充分小, 令 v =( d + ac ) θ a , v = εψ 1 , 则有 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459811&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459811&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459811&type=middle 所以 v 是(4)式的上解.由定理3或文献[ 11 11 ]第三章习题3.3可知 ψ 1 / θ a 有界, 再结合定理4有 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459814&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459814&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459814&type=middle 所以 v 是(4)式的下解.显然 v ≤ v , 从而, v 是(4)式一对有序上下解.利用上下解方法即可得到正解存在性. 现证惟一性.设 v 1 , v 2 是(4)式的任意两个正解, 记 w = v 1 － v 2 , 则有 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459816&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459816&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459816&type=middle 两边同乘 w 并在(0, l )积分可得 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459818&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459818&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459818&type=middle 又由 v 1 是(4)式的正解, 从而知 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459837&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459837&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459837&type=middle , 由主特征值定义, 对任意的 φ ∈ W 0 1, p , http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459838&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459838&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459838&type=middle 有 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459819&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459819&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459819&type=middle 利用 φ 的任意性取 φ = w , 则有 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459839&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459839&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459839&type=middle , 从而 w ≡0.所以 v 1 = v 2 , 惟一性得证. 利用定理5, 6即可得到 定理7   当且仅当 a ＞ λ 1 , d ＞ d * , 系统(3)有惟一正解, 记为( θ a , v * ). 定理8   如果 a ＞ λ 1 , d ＞ d * , 则当 b →0 + 时, 系统(1)的任意正解( u , v )→( θ a , v * ). 证   设当 i →∞时, b i →0, 记 b = b i 时系统(1)的正解为( u i , v i ), 那么( u i , v i )满足 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459821&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459821&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459821&type=middle 由定理3可知 v i 2 / u i 是 C 1 [0, l ], 因此 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459840&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459840&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459840&type=middle 是 C 1 [0, l ]的, 从而由标准的椭圆型方程正则化理论可知|( u i , v i )|关于 i 一致有界, 进而存在{( u i , v i )}的子列(仍记为自身)以及 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459841&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459841&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459841&type=middle ∈( C 2+ α [0, l ]) 2 , http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459842&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459842&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459842&type=middle , 使得在 C 2+ α [0, l ]中( u i , v i )→ http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459843&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459843&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459843&type=middle .因此在(5)式中当 i →∞可得 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459822&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459822&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459822&type=middle 显然(6)式没有(0, 0)解, 即 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459844&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459844&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459844&type=middle .再根据 a ＞ λ 1 , d ＞ d * , 可知(6)式只可能有两个非负解( θ a , 0)和( θ a , v * ). 若 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459845&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459845&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459845&type=middle , 则‖ v i ‖→0.记 V i = v i /‖ v i ‖, 那么 V i 满足 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459823&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459823&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459823&type=middle 注意到上式中 v i 2 / u i ∈ C 1 [0, l ], 利用标准的正则化理论可得| V i | 2+ α 关于 i 有界.从而存在子列(仍记为其自身)以及非负函数 V ∈ C 2+ α [0, l ], 使得在 C 2+ α [0, l ]中 V i → V .故 V 满足 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459824&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459824&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459824&type=middle 由 d ＞ d * 可知 V ≡0, 这与‖ V ‖=1矛盾.这说明 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459846&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459846&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459846&type=middle , 再由极值原理可得 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459847&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459847&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459847&type=middle , 所以我们有( u , v )→( θ a , v * ). 定理9   如果 a ＞ λ 1 , d ＞ d * , 则存在 ε ＞0, 使得当0＜ b ＜ ε 时, 系统(1)有惟一的正解且是线性稳定的. 证   记 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459825&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459825&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459825&type=middle 那么求系统(1)的正解等价于求 T ( b , u , v )=0的正解.由 a ＞ λ 1 , d ＞ d * 可知( b , u , v )=(0, θ a , v * )是系统(1)的解.设 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459826&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459826&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459826&type=middle 那么由 L ( φ , ψ )=0可得 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459827&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459827&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459827&type=middle 根据 λ 1 (－ a +2 θ a )＞ λ 1 (－ a + θ a )=0可得到 φ ≡0.又因为 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459828&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459828&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459828&type=middle 所以 ψ ≡0.故0不是 L 的特征值.显然 L 的特征值由它的主对角线上算子的特征值决定, 从上面的讨论中可知 L 的所有特征值都大于0.因此根据标准的稳定性理论可知( θ a , v * )是非退化线性稳定的.利用隐函数定理可知存在 ε ＞0, 当 b ≤ ε 时, 系统(1)存在惟一的正解( u ( b ), v ( b )), 且满足 u (0)= θ a , v (0)= v * .再由定理8和线性算子扰动理论可知正解( u ( a ), v ( a ))是线性稳定的. 数值模拟 1 本节利用数值模拟验证定理8和定理9，结果见 图 1 图 1 和 图 2 图 2 . 曲线从下到上 b 分别取值为10, 1, 0.1, 0.01, 0.001以及 b =0 (a) shows the positive solutions u ( x ); (b) shows the positive solution v ( x ), where b =10, 1, 0.1, 0.01, 0.001 and 0 from bottom to top, respectively (a) The maximum of positive solutions u ( t , x ) varies with time; (b) The maximum of positive solutions v ( t , x ) varies with time 由文献[ 6 6 ]中(6.3)式可知 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459831&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459831&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459831&type=middle 因此可通过上式判断 d 与 d * 的大小.取 l =2π, 直接计算可知 λ 1 =0.25.大量的数值模拟表明, 在定理8的条件下, 当 b →0 + 时, 系统(1)存在正解且趋于系统(3)的正解.比如 a =2, d =2, c =7, m =0.5时, d =-5.875, 所以 d ＞ d * .见 图 1 图 1 . 关于定理9给出的稳定性结论, 我们考虑在上述参数下, b =0.01时, 系统(1)对应的抛物型方程组进行初值扰动, 观察平衡态正解有无变化.大量的数值模拟表明, 当 b 充分小时, 初值没有影响到平衡态正解.这与定理9吻合.为了直观性, 我们选取时间为10个单位, 沿时间轴观测最大值, 显然, 当正解达到平衡态, 其最大值是不随时间变化的，见 图 2 图 2 . 本文在一维空间下利用文献[ 6 6 ]提供连续延拓方法，讨论了Variable-territory捕食模型正解的性质，但是我们尚不能证明这种方法适用于高维区域.因此，有关高维区域下模型的动力学行为还需进一步研究. 参考文献 TURCHIN P, BATZLI G O. 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Applied Mathematics and Computation, 2014, 236(4):647-662. DOI: 10.1016/j.amc.2014.03.080. 姜洪领, 王利娟, 郭改慧.一类Variable-Territory捕食模型正解的分歧及稳定性[J].武汉大学学报(理学版), 2014, 60(4): 363-368. JIANG H L, WANG L J, GUO G H. Bifurcation and stability of positive solutions for a class of variable-territory predator-prey model[J]. Journal of Wuhan University (Nat Sci Ed), 2014, 60(4): 363-368(Ch). DU Y H, LOU Y. Some uniqueness and exact multiplicity results for a predator-prey model[J]. Transactions of the American Mathematical Society, 1997, 349(6):2443-2475. BLAT J, BROWN K J. Global bifurcation of positive solutions in some systems of elliptic equations[J]. SIAM Journal on Mathematical Analysis, 1986, 17: 1339-1353. WANG L J, JIANG H L, LI Y. Positive steady state solutions of a plant-pollinator model with diffusion [J]. Discrete and Continuous Dynamical Systems-Series B, 2015, 20:1805-1819. DOI: 10.3934/dcdsb.2015.20.1805. 叶其孝, 李正元, 王明新, 等.反应扩散方程引论[M].第二版.北京:科学出版社, 2011. YE Q X, LI Z Y, WANG M X, et al. Introduction to Reaction-Diffusion Equations [M]. 2nd Ed. Beijing: Science Press, 2011(Ch). 10.14188/j.1671-8836.2017.06.013 O175.26 whdxxblxb-63-6-543 宝鸡市科技局软科学研究项目(16RKX1-22);宝鸡文理学院重点科研资助项目(ZK15038, ZK12043) 2015-11-26 2017-12-24 2020-12-09 版权所有© 《武汉大学学报(理学版)》编辑部2017 Copyright © 2017 Journal of Wuhan University(Natural Science Edition). All rights reserved. 2017 武汉大学学报(理学版) 6 63