Ratio-Dependent Predator-Prey Models of Interacting

http : ! www. insect. wg. cndoi : 10?16380 Zj kc ib?2021?02?009：.. 2 月 Febmay2021, 64(2) : 223 -229昆虫学报ACTAENTOMOLOGICASINICA异色瓢虫对与本土c 虫共存的外来扶桑绵粉k 的搜寻和捕食行为刘静雅，李卓苗，李保平，孟玲(南京农业大学植物保护学院，农作物生物灾害综合治理教育部重点实验室,南京210095)摘要：【目的】外来入侵物种作为猎物可能影响本土广谱捕食者的搜寻和捕食行为。

本研究旨在揭示异色瓢e Harmonia axyrPis 对与本土豌豆修尾Megoorajaponica 混合发生的外来入侵扶桑绵粉蜕Phenacoccus solenopsis 的搜寻和捕食行为。

【方法】试验前对异色瓢虫设两个饲喂猎物种类处理：用3代以上(前期饲喂粉斯的瓢e )和用豌豆修尾财连续饲养3代以上(前期e 的瓢虫)。

在盆栽蚕豆Vicia faba 苗上随机选一片叶接扶桑绵粉<3龄 虫和3或4日龄豌豆修尾财若:在总猎物数量20 -30头(具体数量随机确定)范围内随机组配两种猎物数量使粉斯占比从5.6%至83.3%不等,待猎物在叶片上稳定后接入1头饥饿24 h 的异色瓢e 4龄幼虫， 连续观察2 h ,统计异色瓢e 首次捕食是否选择粉斯、捕食的粉斯数量占总捕食猎物数量的比例、搜索用时占总搜捕用时的比例以及处理粉斯用时占总搜捕用时的比例；以扶桑绵粉斯在猎物斑块中的比例和饲喂猎物种类处理为自变量因素，用回归模型拟合其对异色瓢e 搜寻和捕食行为等观测 变量的影响。

【结果】异色瓢e 4龄幼e 首次捕食选择粉斯概率随粉斯比例增大而增大，粉斯比例每增大10% ,首次捕食粉斯概率增大77.2% ,而且此概率不受此前饲喂猎物种类的影响。

捕食的 粉斯数量占比与粉斯比例的关系受饲喂猎物种类的影响：粉斯比例每增加10%，前期饲喂粉斯的瓢e 捕食的粉斯数量占比增大137%,而前期饲喂酚e 的瓢e 捕食的粉斯数量占比增大60%$异 色瓢e 搜索用时比例受粉斯比例的影响，但不受饲喂猎物种类的影响：粉斯比例每增加10%，搜索 用时比例增大13% ；前期饲喂粉斯的瓢e 搜索用时比例略低于前期饲喂斬e 的瓢e (降低7%)$异色瓢e 处理粉斯用时比例受粉斯比例的影响，但不受饲喂猎物种类的影响：粉斯比例每增加 10% ,处理粉斯用时比例增大41%$【结论】本研究结果说明在遭遇外来入侵扶桑绵粉斯与本土酚e 共存的猎物斑块时，2色瓢e 对粉斯的捕食选择依赖于其相对比例，依赖程度因此前取食粉< 而 增强。

一类带负交叉扩散项二维系统的空间Turing斑图张道祥;赵李鲜;孙光讯;周文;于艳【摘要】考虑一类带负交叉扩散项二维系统的Turing斑图生成及其选择问题.先利用稳定性理论和Hopf分支理论得到Turing斑图的存在区域,再利用多重尺度分析法推导系统的振幅方程,并给出Turing斑图的选择结果.最后考虑一个具有比率依赖的Holling-Tanner捕食模型生态系统,利用MATLAB软件对该模型的斑图生成及选择结果进行数值模拟,得到了包括点状、条状以及二者共存等不同类型的Turing斑图.%We considered the generation and selection of Turing pattern of a class of two dimensional system with negative cross-diffusion.Firstly,the existence region of Turing pattern was obtained by using stability theory and Hopf bifurcation theory.Secondly,the amplitude equations of the system were derived by using multi-scales analysis method,and the selection result of Turing pattern was given.Finally,we considered a specific ecosystem with a ratio dependent Holling-Tanner predator-prey model.MATLAB software was used to simulate the pattern generation and selection results of the model,and the different types of Turing patterns,such as dot,strip and the coexistence of the two types were obtained.【期刊名称】《吉林大学学报（理学版）》【年(卷),期】2017(055)003【总页数】10页(P537-546)【关键词】二维系统;负交叉扩散系数;振幅方程;Turing斑图【作者】张道祥;赵李鲜;孙光讯;周文;于艳【作者单位】安徽师范大学数学计算机科学学院,安徽芜湖 241003;赫尔辛基大学数学与统计学院,芬兰赫尔辛基 00014;安徽师范大学数学计算机科学学院,安徽芜湖 241003;安徽师范大学数学计算机科学学院,安徽芜湖 241003;安徽师范大学数学计算机科学学院,安徽芜湖 241003;安徽师范大学数学计算机科学学院,安徽芜湖 241003【正文语种】中文【中图分类】O175.21考虑如下二维系统:其中参数d11,d12,d21,d22均为正常数, d11,d22称为自扩散系数, d12,-d21称为交叉扩散系数. 目前, 关于带自扩散与交叉扩散系统的研究已取得很多成果[1-16]. 文献[1]研究了一类带群体行为的捕食-食饵模型中交叉扩散引起的Turing不稳定性, 并得到了一系列不同类型的Turing斑图；文献[2]研究了一类具有比率依赖捕食-食饵模型空间斑图的存在与不存在性; 文献[3]研究了FitzHugh-Nagumo模型的模式生成问题, 通过推导系统的振幅方程做出斑图选择, 得到了不同类型的Turing斑图, 包括点状斑图、点条混合斑图、复杂斑图和Z字斑图等; 文献[4]研究了具有自扩散的捕食-食饵模型中, 捕食者同类残杀效应对Turing斑图生成问题的影响; 文献[5]研究了具有自扩散的传染病模型的动力学形态. 上述研究所带的交叉扩散系数均为正数, 其生物学意义为一个种群总是从另一个种群的高密度区域向低密度区域移动. 以二维捕食-食饵模型为例, 捕食者为躲避成群的食饵伤害, 将向低密度食饵方向移动, 如狮子与野水牛组成的捕食-食饵系统. 但在自然界中, 一些猛兽群体总会积极追捕特定的食饵, 如老虎追捕鹿群, 狼追捕羊群, 即捕食者向食饵的高密度区域移动, 这种现象在模型中表现为交叉扩散系数为负数. 目前, 关于带负交叉扩散系数的系统Turing斑图行为在生态模型中的研究尚未见文献报道. 本文考虑一类带负交叉扩散系数的一般二维模型的Turing空间斑图生成与选择问题. 假设系统(1)有唯一的平衡点(u*,v*), 即f(u*,v*)=g(u*,v*)=0. 为了研究Turing斑图的生成问题, 首先, 考虑系统(1)对应的常微分方程:系统(2)在平衡点(u*,v*)处的Jacobi矩阵为通过稳定性分析知, 当平衡点(u*,v*)满足时, 系统(2)在平衡点(u*,v*)局部稳定, 其中tr J和det J分别表示J的迹和行列式. 其次, 考虑系统(1)平衡点(u*,v*)的Turing不稳定性. 在(u*,v*)处引入如下小扰动: 其中U(x,y,t)=U0eλtei(kxx+kyy), V(x,y,t)=V0eλtei(kxx+kyy), λ表示时间t处的扰动增长率, kx,ky为相应的振幅, k=表示波长, U0,V0为两个实数. 将式(4)代入系统(1), 可得如下特征方程:其中:通过计算知, 特征值λk有如下形式:其中trk和δk分别表示J-k2D-λI的迹和行列式:在条件(3)成立的情况下, 显然有trk=tr J-k2(d11+d22)<0, 因此只有存在某些k 使得δk<0, 系统(1)的平衡点(u*,v*)才会失稳. 求δk关于k2的极小值, 得到最危险模数kT满足将其代入δk可得Turing不稳定的条件为由条件(3),(9), 可得Turing斑图的存在区域.注1 利用稳定性理论知识, 只能得到Turing斑图的存在空间, 而Turing斑图的类型仍无法确定.本文利用多重尺度分析法推导系统(1)的振幅方程, 并得到Turing斑图的类型. 文献[17]已给出了振幅方程的相关理论, 本文仅给出主要结论.令为方便, 本文仍然使用原变量. 系统(1)在平衡点(u*,v*)处的Taylor展开式为系统(10)可表示为其中:这里c取为控制参数, cT为对应的临界值且p=, q=满足方程:其中为LT的伴随算子.在计算中, 本文只分析控制参数在临界值附近的行为, 因此将控制参数c展成如下形式:其中ε为一个小参数. 同理, 将U和N也按ε展开:其中: c.c.表示复数共轭项;u0=, v0=,u1=,v1=,u*=,v*=,h1=-fuu-fvv-pfuv, h2=-guu-gvv-pguv.利用中心流形理论推导可得如下振幅方程[17]:其中: μ表示到临界值的归一化距离; τ0表示松弛时间. 计算可得系统(13)中系数τ0,μ,h,g1,g2的表达式分别为g1=, g2=,其中:考虑一类具比率依赖的Holling-Tanner捕食-食饵模型. 在系统(1)中取f(u,v)=u(1-u)-buv/(u+v), g(u,v)=cv(1-v/u), 其中u和v分别表示食饵和捕食者的密度. 易知系统(1)有一个常稳态解((2-b)/2,(2-b)/2), b<2. 简单计算得det J=(2-b)c/2>0成立, tr J=3b/4-1-c. 则由条件(3)知, 当b<min{2,4(c+1)/3}时, 系统(2)在((2-b)/2,(2-b)/2)处稳定. 在系统(1)中,δk=(d11d22+d12d21)k4+[(1-3b/4)d22+cd11+bd21/4+cd12]k2+(2-b)c/2. 由于当k=0, Im(λk)≠0, Re(λk)=0时, 系统(2)出现Hopf分支, 此时Hopf分支曲线为当k=kT, Im(λk)=0, Re(λk)=0时, 系统(2)出现Turing分支, 此时因此Turing 分支参数c的临界值cT满足以下Turing分支曲线方程:根据Hopf分支曲线和Turing分支曲线, 可得Hopf分支区域和Turing存在区域, 如图1所示, 其中: b=1.7; d11=0.02; d22=6; d12=0.1. 区域D11位于Turing分支曲线下方及Hopf分支曲线右方, 当参数位于该区域时会出现Turing失稳, 从而出现Turing斑图, 称D11为Turing空间. 当参数位于D12区域时, Hopf不稳定, 但Turing稳定, 称为Hopf空间. 当参数位于D13区域时, Hopf和Turing均不稳定, 称为Hopf-Turing空间.方程组(13)的每个振幅均可分解为模及一个相应的相角ψj的乘积, 将Aj=ρjejφj代入方程组(13)并分离实部和虚部, 可得如下方程:其中φ=φ1+φ2+φ3. 系统(16)有下列4种类型的解[17]:1) 定态解(0): ρ1=ρ2=ρ3=0, 当μ<μ2=0时, 定态解稳定, 当μ>μ2时, 定态解不稳定;2) 条状斑图(S): ρ1=, ρ2=ρ3=0, 当μ>μ3=h2g1/(g2-g1)2时, 该解稳定, 否则不稳定;3) 六边形斑图(H0,Hπ): ρ1=ρ2=ρ3=. 当μ>μ1=时, 该解存在, 当μ<μ4=h2时, 解ρ+=稳定, 而解ρ-=一直不稳定;4) 混合斑图: ρ1=, ρ2=ρ3=, 当时该解存在.负交叉扩散项系数-d21对系统Turing不稳定性的影响如图2所示, 其中b=1.7, c=2, d11=0.02, d22=6, d12=0.1. 由于不影响定性结果, 为方便, 下面将直接讨论正数d21. 固定参数b=1.7, d11=0.02, d22=6, d12=0.1, 当d21=2.2时, 无论k取何值, Re(λ)均为负值, 而当d21=1.885 8, k=0.986 0时, Re(λ)=0. 根据Turing 分支理论可知, Turing不稳定出现的一个必要条件是存在某个波数k, 使得Re(λ)>0, 因此要想系统出现Turing不稳定现象, 必须限定扩散系数-d21>-1.885 8(即d21<1.885 8). 参数c的变化对Turing不稳定的影响如图3所示, 其中参数b=1.7, d21=1, d11=0.02, d22=6, d12=0.1. 由图3可见：当c=4.3时, 由于Re(λ)<0, 故不可能出现Turing不稳定现象; 当c<4.087 3时, 存在Re(λ)>0, 故当c=3.5, c=2时, 出现Turing不稳定现象, 此时临界值cT=4.087 3.下面利用MATLAB软件数值模拟Holling-Tanner系统的Turing斑图选择. 所有的数值模拟均采用齐次Neumann边界条件, 即在边界上种群的进出流量为0. 所有的数值模拟均在离散格子200×200内进行, 两个格子之间的距离由晶格常数Δ h确定, 取Δ h=1, 时间间隔Δ t=0.01. 设参数值b=1.7, d11=0.02, d22=6,d12=0.1, d21=1, 变化c的值, 则可得参数值μ1=-0.001 5, μ2=0, μ3=0.026 3, μ4=0.107 3. 种群u在不同时刻的空间斑图分别如图4～图7所示. 图4中c=3.5, 4个子图迭代步数分别为0,200 000,100 000,1 400 000. 由图4可见: (A)中颜色条数值基本不变, 初值选取为平衡解加上一个随机扰动; (B)中出现了类点状斑图; (C)中出现条状斑图; (D)中条状斑图占据整个区域, 且系统的动力学行为不再发生变化. 图5中c=3.89, 4个子图迭代步数分别为0,50 000,250 000,950 000. 由图5可见, 其中关于食饵的分布条状斑图与Hπ斑图(也称点状斑图)最终共存, 且条状斑图优于Hπ斑图. 此外, μ满足μ3<μ=0.048 3<μ4, 因此数值模拟的结果与理论分析相符. 两种斑图同时存在的双稳现象也称为别针效应[7]. 图6中c=3.94, 4个子图迭代步数分别为0,200 000,500 000,1 800 000. 由图6可见, 条状斑图与Hπ斑图最终共存, 但Hπ斑图优于条状斑图. 同理, μ满足μ3<μ=0.048 3<μ4, 数值模拟的结果与理论分析相符. 由图7可见, 当c增大为4.06接近于临界值cT=4.087 3时, Hπ斑图最终占据整个区域, 此时μ2<μ=0.006 7<μ3, 数值模拟结果与理论分析相符.当b=1.7, c=2, d11=0.02, d22=6, d12=0.1, 变化d21时得到的Turing斑图分别如图8～图10所示. 图8和图9分别为当d21=1.4,1.6时, 食饵u的时间演化图. 由图8和图9可见: 随着时间的演化, 最终点状斑图与条状斑图同时存在, 但图8中条状斑图占优; 而当d21的值增大至1.6时, 点状斑图会占优(图9(D)). 由图10可见, 当d21的值增大1.85趋近于临界值1.885 8时, 最终规则的点状斑图占满整个空间. 该结果与自然界的生态现象相符, 这是因为当d21越趋于临界值, 捕食者扩散至食饵过程越快, 捕食者追捕食饵也越主动, 对应于自然界凶猛的捕食者如老虎, 该类动物一般不喜欢群居, 在空间分布上显示为点状.综上, 本文通过研究自然界中猛兽群体的追捕现象, 给出了一类带有负交叉扩散系数的一般二维模型, 并对一类具比率依赖的Holling-Tanner捕食-食饵模型进行了理论和数值研究, 所得结果表明:1) 负交叉扩散项系数-d21影响了Turing斑图的生成与选择, 在其他参数固定的情况下, -d21必须大于某个临界值时, 系统才会出现Turing不稳定现象.2) 理论和数值结果表明, 带比率依赖的Holling-Tanner系统具有丰富的动力学行为. 随着负交叉扩散系数的变化, 系统展现了不同类型的斑图结构, 如点状、条状以及二者共存的斑图.【相关文献】[1] TANG Xiaosong, SONG Yongli. 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一类具有交叉扩散项的捕食-食饵模型的局部分歧容跃堂;董苗娜;何堤;王晓丽【摘要】研究一类带有交叉扩散项的捕食-食饵模型在齐次Dirichlet边界条件下分歧解的存在性.利用极大值原理和上下解法得到正解的先验估计,并借助Crandall-Rabinowitz分歧理论,得出局部分歧正解存在的充分条件.%The existence of bifurcation solutions for a predator-prey model with cross-diffusion under homogeneous Dirichlet boundary conditions is concerned.By the maximum principle,a priori estimate of positive solutions are obtained.Then by Crandall-Rabinowitz bifurcation theory,the sufficient conditions for the existence of positive solutions to a local bifurcation is proved.【期刊名称】《纺织高校基础科学学报》【年(卷),期】2016(029)004【总页数】7页(P443-449)【关键词】捕食-食饵;自扩散;交叉扩散;先验估计;局部分歧【作者】容跃堂;董苗娜;何堤;王晓丽【作者单位】西安工程大学理学院,陕西西安710048;西安工程大学理学院,陕西西安710048;西安工程大学理学院,陕西西安710048;西安工程大学理学院,陕西西安710048【正文语种】中文【中图分类】O175.26近年来,关于生物数学领域的捕食食饵模型的研究已经成为热点,尤其是对于种群扩散影响下的捕食模型,国内外学者均已取得了一些符合实际的研究成果.文献[1]研究了一类捕食模型的正常数平衡态解的稳定性及分歧;文献[2-3]利用极大值原理和分歧定理研究了一类捕食模型局部解的延拓;文献[4-7]利用分歧定理研究了模型在交叉扩散影响下的正解的存在性问题.在文献[8]中,作者提出了一类具有扩散项的捕食食饵模型,通过给出正解的先验估计及局部分歧解存在条件,进而得到该系统平衡态的全局分歧解及其走向;文献[9]则在上述基础上研究了该类模型在交叉扩散项影响下的分歧.在同时考虑交叉扩散和自扩散项时，本文将继续研究如下捕食-食饵模型在齐次Dirichlet边界条件下正解的存在性，即其中:Ω为RN中具有光滑边界∂Ω上的有界区域;u,v分别表示食饵和捕食者的种群密度;a,b,c,d,α,β都是正常数,m1,m3表示自扩散系数;m2,m4表示交叉扩散系数,反应函数是Bazykin研究捕食者的饱和不稳定性与食饵的稳定性时建立的功能反应函数,生物背景参见文献[10].本文将针对模型(1)的如下平衡态方程展开讨论.注:对于问题(2)的解(u,v),如果在Ω中,(u,v)中只有一个分量为0,则称其为半平凡解. 记}.定义中的范数为通常的Banach空间C1 (Ω)中的范数,令,则X是Banach空间. 首先,考虑特征值问题引理1[11] 假设为常数,则问题(3)的所有特征值满足λ1(p,q)<λ2(p,q)≤λ3(p,q)≤…→∞,相应的特征函数为φ1,φ2,….由文献[11]知λ1(p,q)是简单的且关于q(x)严格单调递增.为方便起见,简记λ1=λ1(0),相应的主特征函数φ1>0.再考虑边值问题引理2[11] (1) 如果a≤λ1,则u=0是问题(4)的唯一非负解;若a>λ1,则问题(4)的唯一正解为θa.(2) 如果c≤λ1,则v=0是问题(5)的唯一非负解;当c>λ1时,其存在唯一正解θc.因此,当a>λ1,问题(2)存在半平凡解(θa,0);当c>λ1,问题(2)存在半平凡解(0,θc).定义Z=(U,V),其中U=(1+m1u+m2v)u,V=(1+m3v+m4u)v,则即(u,v)≥0与(U,V)≥0之间存在一一对应的关系.现在,引入和问题(2)等价的半线性椭圆系统易知,当a,c>λ1时,问题(6)的两个半平凡解分别为,其中.引理3[12] 假设a>λ1,令,则L(a)的特征值均大于0.引理4 设c>λ1,则当cm4>d时,存在唯一的a=a*(c)∈(λ1,∞),满足,且a=a*(c)关于c严格单调递增.此外,∃ψ*≥0满足证明取.显然A(λ1,c)=λ1(-c)=λ1-c<0.由于当a→∞时θa→∞,故有.经计算得又因为与均严格单调递增,可知A(a,c)关于a严格单调递增.从而存在唯一的a=a*(c)>λ1,使得A(a*(c),c)=0.再对A(a*(c),c)=0两边关于c求导,得Aa(a*(c),c)·a*′(c)+Ac(a*(c),c)=0.由于Ac(a,c)<0,结合Aa(a,c)>0得知a*′(c)>0,即a=a*(c)关于c严格单调递增.类似可以证明以下引理.引理5 假设c>λ1,则当aβ>b时,就存在唯一的a=a*(c)∈(λ1,∞),满足,且a=a*(c)关于c严格单调递增.此外,∃φ*≥0满足现在,结合文献[12-13]中的方法给出系统(6)的正解存在的必要条件及先验估计.定理1 当a≤λ1,或者,则问题(6)没有正解.证明若问题(6)存在正解(U,V),由问题(6)中的第2个方程得两边同乘以V,分部积分得由Poincare不等式‖‖，可得,同理可证a>λ1.与已知条件矛盾,则定理1得证. 定理2 设且b-βa(1+αa)>0.若(U,V)是问题(6)的任意正解,则∀x∈Ω,有证明设∃x0∈Ω,使得(x).由于故有,则同理可得由(u,v)与(U,V)之间的关系知定理2成立.现在以a为分歧参数,参考文献[14-19],利用Crandall-Rabinowitz局部分歧定理,给出问题(6)发自半平凡解与的局部分歧正解的存在性.定理3 设且cm4>d,则为问题(6)的分歧点,且的领域内存在正解其中a*由唯一确定,ψ*>0满足证明令其中u,v均为(U,V)的函数.将问题(6)在(U,V)处Taylor展开为这里,偏导数为处的导数值,满足.同时对(U,V)求导,得令,则有显然T(a;0,0)=0.记关于在(a*;0,0)处的Frechlet导数是L(a*;0,0).经计算,L(a*;0,0)·(φ,ψ)=0等价于如果ψ≡0,那么由算子La*可逆知φ≡0,矛盾,所以ψ不恒为零.又,故有因此,算子L(a*;0,0)的核空间N(L(a*;0,0))=span{U0},U0=(φ*,ψ*)T,其中又令L*(a*;0,0)为L(a*;0,0)的自伴算子,类似可得由Fredholm选择公理知因此可得dimN(L(a*;0,0))=1,codimR(L(a*;0,0))=1.令,下面采用反证法证明假设∃(h,k)∈X,使得L1(a*;0,0)·(φ*,ψ*)=L(a*;0,0)·(h,k).经计算得那么有两边同时乘以ψ*,分部积分得由于cm4-d>0,且θa关于a严格单调递增,则上式左端大于0,矛盾.由Crandall-Rabinowitz局部分歧定理知,存在充分小的δ>0及C1连续曲线(a(s):Φ1(s),Ψ1(s)):(-δ,δ)→R×X满足a(0)=a*,Φ1(0)=0,Ψ1(0)=0,Φ1(s),Ψ1(s)∈Z 使得(a(s):(φ*+Φ1(s)),s(ψ*+Ψ1(s)))是T(a(s):的零点,其中X=Z⨁N(L(a*;0,0)),由于,因此可得到发自的局部分歧正解Γ*.同理可得到发自半平凡分支的局部分歧正解.定理4 设且aβ>b,则为问题(5)的分歧点,且的领域内存在正解a*由唯一确定,φ*>0满足充分小.这里(Φ2(s),Ψ2(s);a(s))是连续函数,满足a(0)=a*,Φ2(0)=0,Ψ2(0)=0,∫Ωψ2φ*dx=0,且RONG Yuetang,DONG Miaona,HE Di,et al.The local bifurcation for a kind of prey-predator model with cross-diffusion[J].Basic Sciences Journal of Textile Universities,2016,29(4):443-449.【相关文献】[1] 周冬梅,李艳玲.一类捕食模型正常数平衡态解的稳定性及分歧[J].科学技术与工程,2010,10 (23):5615-5619.ZHOU Dongmei,LI Yanling.Stability and bifurcation of positive constant steady-state solution for predator-prey model[J].Science Technology andEngineering,2010,10(23):5615-5619.[2] 李海侠,李艳玲.一类捕食模型正平衡解的整体分歧[J].西北师范大学学报:自然科学版,2006,42(2):8-12.LI Haixia,LI Yanling.Bifurcation of positive steady-state solutions for a king of predator-prey model[J].Journal of Northwest Normal University:Natural Science,2006,42(2):8-12. [3] 王妮娅,李艳玲.一类带收获率的的捕食模型的全局分歧和稳定性[J].安徽师范大学学报:自然科学版,2015,38(1):25-30.WANG Niya,LI Yanling.Global bifurcation and stability of a class of predator-prey models with prey harvesting [J].Journal of Anhui University:Natural Science Edition,2015,38(1):25-30.[4] KUTO K,YAMADA Y.Multiple coexistence states for a prey-predator system with cross-diffusion[J].J Differential Equations,2004,197(2):315-348.[5] 张晓晶,容跃堂,何堤,等.一类带有交叉扩散的捕食-食饵模型的分歧性[J].纺织高校基础科学学报,2014,27(3):322-326.ZHANG Xiaojing,RONG Yuetang,HE Di.Bifurcation for a prey-predator model with cross-diffusion[J].Basic Sciences Journal of Textile Universities,2014,27(3):322-326.[6] DUBEY B,DAS B,HASSAIN J.A prey-predator interaction model with self and cross-diffusion[J].Ecol Modelling,2002,141:67-76.[7] ZHANG Cunhua,YAN Xiangping.Positive solutions bifurcating from zero solution in a Lotka-Volterra competitive system with cross-diffusion effects[J].Appl Math J China Univ,2011,26(3):342-352.[8] 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 Xiaozhou,WU Jianhua.Global bifurcation and stability for predator-prey model with predator saturation and competition[J].Journal of System Science and Mathematical Sciences,2010,30(7):979-989.[9] 何堤,容跃堂,张晓晶.一类具有交叉扩散的捕食-食饵模型的分歧[J].纺织高校基础科学学报,2015,28(4):426-430.HE Di,RONG Yuetang,ZHANG Xiaojing.Bifurcation for a prey-predator model with cross-diffusion[J].Basic Sciences Journal of Textile Universities,2015,28(4):426-430.[10] BAZYKIN A D.Nonlinear dynamics of interacting population[M].Singapore:World Scientific,1998.[11] 叶其孝,李正元,王明新.反应扩散方程引论[M].北京：科学出版社,2011:40-56.YE Qixiao,LI Zhengyuan,WANG Mingxin.Introduction of reaction-diffusionequations[M].Beijing:Science Press,2011:40-56.[12] 何堤,容跃堂,王晓丽,等.一类具有交叉扩散的捕食-食饵模型的局部分歧[J].西安工业大学学报,2015,35(11):872-876.HE Di,RONG Yuetang,WANG Xiaoli,et al.Local bifurcation for a prey-predator model with cross-diffusion[J].Journal of Xi′an Technological University,2015,35(11):872-876.[13] 容跃堂,何堤,张晓晶.带交叉扩散项的Holling Ⅳ捕食-食饵模型的全局分歧[J].纺织高校基础科学学报,2015,26(3):287-293.RONG Yuetang,HE Di,ZHANG Xiaojing.The global bifurcation for a prey-predator model with cross-diffusion and Holling Ⅳ [J].Basic Sciences Journal of TextileUniversities,2015,26(3):287-293.[14] 马晓丽,冯孝周.一类具有交叉扩散的捕食模型的正解的存在性[J].安徽大学学报:自然科学版,2011,35(5):26-31.MA Xiaoli,FENG Xiaozhou.The existence of positive solutions for a predator-prey model with cross-diffusion[J].Journal of Anhui University:Natural Science Edition,2011,35(5):26-31.[15] 马晓丽.一类具有交叉扩散的捕食模型的整体分歧[J].西安工业大学学报,2010,30(5):506-510. 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Pollution•Dose-response analysis: expose to a toxin at different concentrations High LD50 means low toxicityPoison: LD50 of 50mg or less per kg of body weightED50: 50% shows negative effectThreshold dose•Acute effectChronic effecto Infection: result of a pathogen病菌invading bodyPathogensViruses, bacteria, fungi, protozoa原生动物, parasitic wormsVector: ticks spread spotted feverPollutantsPrimary: COSecondary: acid rain (SO2+SO3+water vapor), PANsCriteria pollutantsCO: binds irreversibly to hemoglobinPb: smeltingO3: good at highNO2: combustion engines -- smog, acid precipitationSO2: combustion of coal, smeltingParticulates: soot, sulfate aerosolsIndoor pollutantsVOC:dry cleaning, carpet, furniture; form O3 and smogTobacco smokeRadon: lung cancer; emitted by uraniumSmogAided by air inversions (trap pollutants) and fog (hold pollutants)Industrial smog (gray) and photochemical smog (brown)CFC: propellants, fire extinguishers, hairspray --depletes ozoneMontreal ProtocolOzone loss: greatest in spring as Cl breaks down O3into O2Acid rainCalcium acts as buffer to acid precipitationClean Air Act: established cap-and-trade program for SO2 in 1990National Ambient Air Quality StandardsCatalytic converter: control emissions in carsCorporate Average Fuel Economy (CAFE)Anthropogenic greenhouse gas: CO2, CH4, N2OTemperature inversionHypoxic zone (depletion of O2) caused by eutrophicationJudge water qualitypHHardness: concentrations of calcium and magnesiumDissolved oxygen: warm water = less DOTurbidityBOD:rate bacteria absorb O from waterDeal w wastewaterSludge processor: tank filled w aerobic bacteria; sludge further processed w anaerobic bacteria -- produces CH4 that can be used as fuel to run the treatment plant; sludge cake used as fertilizerTertiary treatment: pass secondary treated water thru sand and carbon filters and then chlorinationRecyclingPrimary: same productsSecondary: new productsHazardous wasteo Corrosiveo Ignitableo Reactiveo Toxico Disposed thru•Injection wells, surface impoundments (for liquid waste that evaporates), landfills Transuranic waste: left over from nuclear weaponsResource Conservation and Recovery ActHigh-level radioactive wastes: bury underground in remote areasSuper fund program in Rocky FlatsLove Canal: landfillo Cap-and-trade policy: economic incentives for limiting emissionso Market permitso Law of Conservation of MatterTrace elements (zinc, copper, iron) are required by living things that cycle, along w major elementsWater cycleEvaporation: from earth's surface and from living organismsTranspiration: plantsCarbon cycleRespiration: animals and plantsPhoto synthesisWays to release carbon▪Burn of fossil fuels▪VolcanoReservoirs: ocean, earth's rocks, fossil fuelsNitrogen cycleNitrogen fixation: nitrogen become ammonia/nitrates thru lightning storms or soilbacteria (rhizobium in legumes)Nitrification: soil bacteria converts ammonium (NH4+) into nitrate (NO3)Assimilation: plants absorb NH3, NH4+, NO3-.Heterotrophs (organisms receive energy by consuming other organisms) obtain nitrogenAmmonification: decomposing bacteria convert dead organisms to NH3 or NH4+ that can be reusedDenitrification: ammonia converted to nitrites and nitrates and to N2/N2O that rise to atmospherePhosphorus cycleFound in soil, rock, sediments; released thru chemical weathering in the form ofphosphate; limiting factor for plant growthSulfurNeed in plants' and animals' dietsIn rocks and sands and deep in the oceanWays to enter atmosphere▪Volcano, bacterial functions, decay of once-living organisms▪Industrial:SO2 and H2S gaseso Autotrophs/ heterotrophsProducersChemotrophs: chemosynthesis -- make food frominorganic chemicals in anaerobicenvironmentsNet Primary Productivity (NPP) -- amount of energy plants pass to herbivores, calculated by [Gross Primary Productivity (amount of sugar plants produce in photosynthesis) - amount of energy plants need for growth...]NPP: limiting factor for the number of consumersConsumers▪Primary▪Secondary▪Tertiary▪Detritivores: consume nonliving organic matter such as dead animals/ fallen leaves▪Decomposers: bacteria and fungi absorb nutrients from nonliving organic matter as plant, wastes of living organisms, and corpsesBiomagnification: increasing concentration of toxin at higher trophic levelsEcosystemsBiomesEcotones: transitional area where two ecosystems meetEcozones/ecoregions: smaller regions that share similar physical features▪Aquatic life zones▪Freshwater▪Saltwater▪Law of Toleranceo Law of the Minimum: living organisms will continue to live until the supply of materials is exhaustedPhylogenetic tree to model evolutionSpeciationHow evolution works▪Natural selection▪Genetic drift: accumulation of changes in the frequency of alleles (versions of a gene) due to sampling errors▪Microevolution▪Macroevolution▪ExtinctionBiologicalEcological: so few that cannot perform ecological function-Commercial/economicCompetition▪Intraspecific: same species compete▪Interspecific: different species compete▪Gause's principle: no two species can occupy the same niche at the same time, one has to…• Realized niche• Fundamental niche -- no competitionPredationSymbiotic relationships▪Mutualistic: sea anemone and clown fish▪Commensalistc: trees and epiphytes▪Parasitism: one is harmed the other benefitsEcological succession▪Primary▪Secondary▪Climax community -- balance b/t abiotic and biotic▪Pioneer species: lichenEdge effect: greater species diversity and biological density at boundariesTheory of Island Biogeography: number of species on"island" is determined byimmigration and extinctionPopulationo Population dispersion▪Random▪Clumping: most commonUniform.e.g.forestsBiotic potential -- the amount that population would grow if unlimited resourcesunpractical!Carrying capacity (K)Exponential (unrestricted) growth -J curveLogistic (restricted) growth -S curveReproductive strategy▪R-selected: bacteria, algae, and protozoa; reproduce early and often▪K-selected: humans, lions, and cows; reproduce late and fewerPopulation cycles▪Boom-and-bust• Common among r-strategists▪Predator-prey: rabbits and coyotesFactors influence population growth▪Density-dependent: increased predation, competition, toxic materials▪Density-independent: fire, storms, earthquakesActual growth rate = (birth rate - death rate)/10▪Birth/death per 1000Total fertility rate: number of children a woman bears during lifetimeReplacement birth rate: 2.5 for developing countriesAge-structure pyramids/diagrams describe populations▪Pre-reproductive 0-14▪Reproductive 15-44▪Post-reproductive 45+Demographic transition model -- predict population trends based on birth/death rates ▪Pre-industrial state: high birth and death rates; environmental resistance -- harsh living conditions▪Transitional state: high birth and low death▪Industrial state: low birth and death; high growth▪Postindustrial state: zero growthMacronutrients are needed in large amounts: proteins, carbon hydrates, and fatsMicronutrients: vitamins, iron, and minerals such as CaHunger < malnutrition < undernourished5 mass extinctionsSecond Harvest in US: redistribute food that would otherwise go to wasteUrban sprawl: emigrates from city to suburbsI = P x A x T (Impact, population, affluence, technology)Threatened < endangeredBiodiversity hotspot: a highly diverse region that faces threats and has lost 70% of vegetationLaws▪Marine mammal protection act▪Endangered species act▪Convention on international trade …EnergyPotential energyKinetic energy.o.thermodynamics.energ.ca.onl.betran.formed/transferred.E.g.photosynthesisSecond law of thermodynamics: increasing entropy--> energy is lost to universe as heat Ways to produce electricityFossil fuels 64% • How fossil fuels produce electricity:▪Burning fuel▪Water heated to steam▪Steam pushes turbine blades▪Spinning turbines rotate coils thru magnetic fields in generators▪Current induced as coils spin, producing electrical energy• Coal found in seams (long continuous deposits)• Largest coal reserve: in US• Exploratory wells; proven reserve• Oil extraction: take advantage of the differences in boiling pointsPrimary: release of oil and gas -- gusherPressure: uses mud, saltwater, and CO2Uses steam, hot water/gases to melt very thick crude oilOil can also be found in rock (shale oil) and surface sands (tar sands)• CoalAnthracite: pure carbonBituminousSubbitumimousLignite: the least pureCO2, NO…, Hg,SO2Can be removed by Scrubbers: alkaline substances that precipitate out SO2; the neutral compound formed in the scrubber (calcium sulfate) is eliminated in waste sludgeFly ash and boiler residueIron sulfide can be removed by grinding coal into small lumps and washingOrganic sulfur is only released during combustionBurn coal w limestone -- w Ca in limestone form calcium sulfateSmoke-stack scrubbingNuclear energy 17%▪Nonrenewable▪Uranium-238; uranium 235 splits thru fission▪Breeder reactors generate new fissionable materials faster than they consume such material▪Nuclear fusion: fusing tritium-2 neutrons anddeuterium-1 neutron▪Nuclear reactors in US• Boiling water reactor: two water circulation systems• Pressurized water reactor: three…Renewable energy sources 19%▪Biomass: wood, charcoal, animal wastes▪Gasohol: 90% gasoline and 10% ethanol; has higher octane; expensive and energy-intensive to produce▪Hydroelectric power: thermal pollution; dams; sediments trapped; more evaporation and water loss▪Solar energy• Passive collection• Active collectionPV cells: produced using fossil fuels▪Wind energy: fastest growing• Nacelle: base of windmill▪Geothermal energy: dissolved salt corrode machine; gases such as methane, CO2, hydrogen sulfide, and ammonia are released▪Ocean tides▪Hydrogen cells: cleanest, safest• Obtained from fossil fuels by reforming• Hydrogen released thru electrolysis (from H2O): may use fossil fuel• Hydrogen obtained from organic molecules: may use fossil fuel• Only waste is water vaporLaws not ratified by US▪Basel Convention on the Control of the Tran boundary Movements of Hazardous Wastes ▪Kyoto Protocol: cut greenhouse gas emissions▪Traditional subsistence agriculture: enough food for one family's survival▪Slash and burn --> deforestationFederal Insecticide, Fungicide, and Rodenticide Act(FIFRA): to approve pesticides Integrated pest management: keep pest population to economically viable level▪Introduce natural insect predators, inter cropping, mulch to control weeds, diversify crops, crop rotation, release pheromone/hormone interrupters, use traps, construct barriersGenetically engineered plants▪Golden rice contains vitamin A and ironOld growth forest: never been cut or disturbed for hundreds of yrs; incrediblebiodiversitySecond growth forest95% of forests are naturally occurring; the rest are plantations/tree farms -- same age;for commercialSilvi culture: forest for harvesting timber▪Clear-cutting: fast-growing plants (pine)▪Uneven-aged management• Selective cutting: for trees that take longer to grow/ interested in specific types• Shelter-woodcutting: mature trees cut over 10-20 yrs; leaves some mature trees to reseedLaws▪Wilderness Act: road-free areas▪Wild and Scenic Rivers ActForest fires▪Surface fires: protect forests from more harmful fires by removing underbrush and dead materials▪Crown fires: huge threat; spread quickly and high temperature▪Ground fires: in bogs/swamps; originating from surface fires; difficult to detect and extinguishOcean resources▪12-mile limit from shore --> 200-mile from shoreCoal-mining: deposition of iron pyrite and sulfur--> acid mine drainageExtracting gold: cyanideGangue: waste material; Tailings: piles of ganguesStrip mining▪Strip overburden to expose a seam of mineral ore; practical when ore is close to the surface --used for coal-mining▪Least expensive/dangerous but large impact on environment▪Mountaintop removal: transforms summits and destroys ecosystems Shaft mining: vertical tunnelsMine waste used for: concrete for buildings, fill for road gradingEarthEarthCore▪Solid inner (nickel and iron, solid due to pressures)▪Molten outer (iron and sulfur, semi-solid due to lower-pressure)Mantle - solid rock▪Asthenosphere: slowly flowing rockLithosphere▪Rigid upper mantle▪CrustTectonic platesConsists only of ocean floor: Nazca plateContain both continental and oceanic: NA platePacific plate is the largest platePlate boundaries▪Convergent: two plates pushed toward each other; one of them pushed deep into the mantle▪Divergent: move away from each other--> gap filled with magma(molten rock), new crust formed when it cools▪Transform fault: slide from side to sideSoilAbiotic/bioticClay < silt < sandAcidity▪most 4--8 (solubility of nutrients)▪More acidic --> Hg and Al can leach into ground water(Al damages the gills of fish and causes suffocationFormation▪Physical/mechanical weathering: by wind and water▪Chemica.weathering.interaction.bt.wate.an.gases.an.th.bedrock.E.g.rus.forme.b.iron/m eta.interactin..waterBiologica.weathering.e.g.tre.root.growing/expandin.thr.rocksLayers▪O: organic; Humus for med by decomposition of organic material▪A: topsoil; plant growth; zone of leaching▪B: zone of illuviation (dissolved material moves from higher soil to lower soil due to gravity of water)▪C: larger rock; not much weathering▪R: bedrockLoamy: same amount of three textures; best for plant growthMost fertile soil are aggregatesMonoculture: can be prevented by crop rotationGreen Revolution▪Salinization in over-irrigated soil▪Land degradation: prevented by drip irrigationLaws▪Soil and water conservation act▪Food security act: prevented conversion of wetlands to non wetlands AtmosphereTroposphere: majority of water vapor and clouds; greenhouse gasTropopause: temperature increases w altitudeStratosphere: o zoneMesosphere:Thermosphere/lonosphere: thinnest; aurora; ionization; reflects radio waves▪Earth evenly heated b/c▪Motion of air as the result of solar heating▪Rotation of earth▪Properties of air, water, and landPrevailing winds: winds moving north are deflected to right/east -- Coriolis effect Dew point: water vapor condenses to liquid -- clouds, precipitationWinds▪Trade winds▪Quickly propel trade ships▪Northeast trade winds (blow from NE)▪Southeast trade windsWesterly -- result of Coriolis effect▪Ferrel cell (reverse of Hadley cell) accounts for westerlies▪Polar westerlies (60 degrees)Horse latitudes (30-35 degrees)▪Dry air & high pressure --> weak windsDoldrum (5 degrees)▪Intertropical convergence zone (ITCZ): heaviest precipitationJet stream: high-speed in upper troposphere▪Monsoon w heavy rainfall: land heats/cools more quickly than waterRain shadow effect: Olympic rainforest on the Washington State coast▪Windward: moisture▪Leeward: dryHurricanes contains more energy than nuclear explosion▪Hurricanes in Atlantic▪Typhoons/cyclones in PacificEl Nino▪Southern oscillation▪Fish population declinesEl nina -- Corilolis effectENSO events -- nino & ninaHydrosphereFreshest sea water: Gulf of Finland, part of Baltic SeaMost saline: Red SeaFreshwater▪Lake Baikal: 20% of world's freshwater▪Delta: deposited sediments▪Estuaries• Saltwater marshes, mangrove forests, inlets, bays, and river mouths▪Wetlands: marshes, swamps, bogs, prairie potholes, and flood plains• Defining characteristics: soil type, hydrology, species composition▪Epilimnion, thermos cline, hypolimnion▪Littoral (rooted, emergent plants), limnetic,profundal (aphotic -- light cannot reach), benthicBarrier islands as bufferOcean▪Coastal▪Euphotic▪Bathyal▪AbyssalUpwelling provides nutrients --> toxic algal bloom (red tide - caused by prolife ration of din of lagellates)Interbasin transfer to deal w water shortagesGroundwater -- water from wells or aquifers▪Unconfined aquifer▪Confined aquiferWater-stressed/water-scarce▪US: not water-scarce; certain regions are water-stressed • Irrigation> thermoelectric power > public supply > …Riparian河边rightPrior appropriationLargest area of old-growth forest in US is in Alaska。

时间测度上具有时滞的互惠系统的周期解鲁红英【摘要】互惠相互作用关系是生物种群之间相互作用的基本关系之一,是生态学、生物数学的研究热点.2种群互惠系统是指每一种群的存在对另一种群的增长都会起促进作用的系统.由于时滞对一系统所带来的影响,在自然现象中是屡见不鲜的.因此,生态系统中,为了更真实的反应自然,时滞是一种不应忽略的因素.时标理论的提出,整合和统一了连续与离散的分析.因此时标上的动力系统更为一般,包含微分方程与差分方程作为它的特例.在时间测度上研究了具有时滞的两种群互惠系统,利用重合度理论中的延拓定理讨论此系统周期解的存在性问题,从而使这一类系统的连续时间情形与离散时间情形的周期解存在性问题得到了统一研究.并且所获得的周期解存在性定理,推广了文献[15]的主要结果.%Mutual interaction, which is one of the basic relationships between populations, has long been dominant themes in both ecology and mathematical ecology. Mutualism, an interaction of two-species of organisms that benefits both, is found in many type of communities. Since time delays occur so often in nature, a number of models in ecology can be formulated as systems of differential equations with time delays. So, time delay is a factor that should not be ignored. The theory on time scales unified analysis of continuous process and discrete process. Therefore, the dynamic equations on time scales are more general, including differential equations and difference equations as special cases. In this paper, the existence of periodic solutions for a delayed mutualism system on time scales is considered. By using the continuation theorem of coincidence degree,a set of sufficient conditionswhich ensure the existence of periodic solutions of the system are obtained. So, the study of existence of periodic solutions for the continuous differential equations and discrete difference equations are unified. In addition,The existence theorem of periodic solutions generalized the main results in [15].【期刊名称】《沈阳师范大学学报（自然科学版）》【年(卷),期】2011(029)002【总页数】5页(P165-169)【关键词】时间测度;互惠;时滞;周期解【作者】鲁红英【作者单位】东北财经大学,数学与数量经济学院,辽宁,大连,116025【正文语种】中文【中图分类】O175.1生态系统的动力学行为一直是生态数学的一个重要的研究问题,许多现象可以表达为微分方程或差分方程,近年来有许多作者对此进行了深入的研究,获得了很好的结果[1-18]。

Predator-Prey CyclesHow do predators affect populations of the prey animals? The answer is not as simple as might be thought. Moose reached Isle Royale in Lake Superior by crossing over winter ice and multiplied freely there in isolation without predators. When wolves later reached the island, naturalists widely assumed that the wolves would play a key role in controlling the moose population. Careful studies have demonstrated, however, that this is not the case. The wolves eat mostly old or diseased animals that would not survive long anyway. In general, the moose population is controlled by food availability, disease and other factors rather than by wolves.When experimental populations are set up under simple laboratory conditions, the predator often exterminates its pre and then becomes extinct itself, having nothing left to eat. However, if safe areas like those prey animals have in the wild are provided, the prey population drops to low level but not extinction. Low prey population levels then provide inadequate food for the predators, causing the predator population to decrease. When this occurs, the prey population can rebound. In this situation the predator and prey population may continue in this cyclical pattern for some time.Population cycles are characteristic of small mammals, and they sometimes appear to be brought about by predators. Ecologists studying hare populations have found that the North American snow shoe hare follows a roughly ten-year cycle. Its numbers fall tenfold to thirty in a typical cycle, and a hundredfold change can occur. Two factors appear to be generating the cycle: food plants and predators.The preferred foods of snowshoe hares are willow and birch twigs. As hare density increases, the quantity of these twigs decreases, forcing the hares to feed on low-quality high-fiber food. Lower birth rates, low juvenile survivorship, and low growth rates follow, so there is a corresponding decline in hare abundance. Once the hare population has declined, it takes two to three year for the quantity of twigs to recover.A key predator of the snowshoe hare is the Canada lynx. The Canada lynx shows a ten-year cycle of abundance that parallels the abundance cycle of hares. As hare numbers fall, so do lynx numbers, as their food supply depleted.What causes the predator-prey oscillations? Do increasing number of hares lead to overharvesting of plants, which in turn results in reduced hare populations, or do increasing numbers of lynx lead to overharvesting hares? Field experiments carried out by Charles Krebs and coworkers in 1992 provide an answer. Krebs investigated experimentalplots in Canada’s Yukon territory that contained hare populations. When food was added to those plots (no food effect) and predators were excluded (no predator effect) from an experimental area, hare numbers increased tenfold and stayed there—the cycle was lost. However, the cycle was retained if either of the factors was allowed to operate alone: if predators were excluded but food was not added (food effect alone), or if food was added in the presence of predators (predator effect alone). Thus both factors can affect the cycle, which, in practice, seems to be generated by conjunction of the two factors.Predators are an essential factor in maintaining communities that are rich and diverse in species. Without predators, the species that is the best competitor for food, shelter, nesting sites, and other environmental resources tends to dominate and exclude the species with which it competes. This phenomenon is known as “competitor exclusion”. However, if the community contains a predator of the strongest competitor species, then the population of that competitor is controlled. Thus even the less competitive species are able to survive. For example, sea stars prey on a variety of bivalve mollusks and prevent these bivalves from monopolizing habitats on the sea floor. This opens up space for many other organisms. When sea stars are removed, species diversity falls sharply. Therefore, from the stand point of diversity, it is usually a mistake to eliminate a major predator from a community.Paragraph 1: How do predators affect populations of the prey animals? The answer is not as simple as might be thought. Moose reached Isle Royale in Lake Superior by crossing over winter ice and multiplied freely there in isolation without predators. When wolves later reached the island, naturalists widely assumed that the wolves would play a key role in controlling the moose population. Careful studies have demonstrated, however, that this is not the case. The wolves eat mostly old or diseased animals that would not survive long anyway. In general, the moose population is controlled by food availability, disease and other factors rather than by wolves.1.In paragraph 1, why does the author discuss the moose and wolves on Isle Royale?O To provide an example of predators moving to new habitats by following migrating preyO To show that the interactions between predator populations and prey populations are not always might be expectedO To suggest that prey populations are more influenced by predation than food availability and diseaseO To argue that studies of geographically isolated populations tend not to be useful to naturalistsParagraph 2: When experimental populations are set up under simple laboratory conditions, the predator often exterminates its pre and then becomes extinct itself, having nothing left to eat. However, if safe areas like those prey animals have in the wild are provided, the prey population drops to low level but not extinction. Low prey population levels then provide inadequate food for the predators, causing the predator population to decrease. When this occurs, the prey population can rebound. In this situation the predator and prey population may continue in this cyclical pattern for some time.Paragraph 3: Population cycles are characteristic of small mammals, and they sometimes appear to be brought about by predators. Ecologists studying hare populations have found that the North American snow shoe hare follows a roughly ten-year cycle. Its numbers fall tenfold to thirty in a typical cycle, and a hundredfold change can occur. Two factors appear to be generating the cycle: food plants and predators.2. The word “rebound” in the passage is closest in meaning toO escapeO recoverO surviveO resist3.Paragraph 2 implies which of the following about experimental environments in which predators become extinct?O They may yield results that do not accurate predict changes of populations in the wild.O In these environments, the prey species is better adapted than the predator species.O These environments are appropriate only for studying small populations of predators and prey.O They are unrealistic because some predators are also the prey of other predators.4.Which of the following can be inferred from paragraphs 2 and 3 about the small mammals that experience population cycles?O Their population cycles are not affected by predators.O Their predators’ populations periodically disappear.O They typically undergo ten-year cycles.O They have access to places safe from predators.5. The word “roughly” in the passage is closest in meaning toO usuallyO repeatingO approximatelyO observable6. The word “generating” in the passage is closest in meaning toO producingO changingO speeding upO smoothing outParagraph 4: The preferred foods of snowshoe hares are willow and birch twigs. As hare density increases, the quantity of these twigs decreases, forcing the hares to feed on low-quality high-fiber food. Lower birth rates, low juvenile survivorship, and low growth rates follow, so there is a corresponding decline in hare abundance. Once the hare population has declined, it takes two to three year for the quantity of twigs to recover.7.According to paragraph 4, all of the following are true of the food of snowshoe hares EXCEPTO The preferred food fore hares consists of willow and birch twigs.O High fiber food is the most nutritious for hares.O Depletion of the supply of willow and birch twigs cause low birth and growth rates.O The food supply takes two or three years to recover after a peak in hare population density.8. The word “conjunction” in the passage is closest in meaning toO determinationO combinationO alternationO transformationParagraph 5: A key predator of the snowshoe hare is the Canada lynx. The Canada lynx shows a ten-year cycle of abundance that parallels the abundance cycle of hares. As hare numbers fall, so do lynx numbers, as their food supply depleted.9.According to paragraph 5, which of the following statements best characterizes the abundance cycle of the Canada lynx?O It closely follows the cycle the snowshoe hare.O When the numbers of lynx fall, the numbers of snowshoe hares soon decrease.O When hare numbers decrease, lynx numbers increase.O It is not clearly related to the availability of lynx food.Paragraph 6: What causes the predator-prey oscillations? Do increasing number of hares lead to overharvesting of plants, which in turn results in reduced hare populations, or do increasing numbers of lynx lead to overharvesting hares? Field experiments carried out by Charles Krebs and coworkers in 1992 provide an answer. Krebs investigated experimental plots in Canada’s Yukon territory that contained hare populations. When food was added to those plots (no food effect) and predators were excluded (no predator effect) from an experimental area, hare numbers increased tenfold and stayed there—the cycle was lost. However, the cycle was retained if either of the factors was allowed to operate alone: if predators were excluded but food was not added (food effect alone), or if food was added in the presence of predators (predator effect alone). Thus both factors can affect the cycle, which, in practice, seems to be generated by conjunction of the two factors.10.According to paragraph 6, which of the following was true of the hare population cycle in Krebs’s experiment?O The effects of providing food while at the same time introducing predators cancelled each other, so there was no cycle.O The cycle existed when either the food supply was limited or there were predators.O There was a cycle when there were no predators and food was supplied.O If the hares had places to hide from the lynx, the hare population increased tenfold and then remained at that level.Paragraph 7: Predators are an essential factor in maintaining communities that are rich and diverse in species. Without predators, the species that is the best competitor for food, shelter, nesting sites, and other environmental resources tends to dominate and exclude the species with which it competes. ■This phenomenon is known as “competitor exclusion”. ■However, if the community contains a predator of the strongest competitor species, then the population of that competitor is controlled. ■Thus even the less competitive species are able to survive. ■For example, sea stars prey on a variety of bivalve mollusks and prevent these bivalves from monopolizing habitats on the sea floor. This opens up space for many other organisms. When sea stars are removed, species diversity falls sharply. Therefore, from the stand point of diversity, it is usually a mistake to eliminate a major predator from a community.11.According to paragraph 7, which of the following statements correctly characterizes the effect of sea stars on the ecosystem in which they are predators of bivalves?O Bivalve population are kept low, allowing species that compete with bivalves to survive.O The numbers of most species of bivalves are greatly reduced, leaving the bivalve species that is the strongest competitor to dominate among the survivors.O Biological diversity begins to decrease because many bivalve species disappear.O Sea stars dominate at first but then die off because of the depleted food supply.12.According to paragraph 7, which of the following is true of the phenomenon of competitor exclusion?O It results in more diverse communities.O It requires the presence of predators.O It affects all competitions equally.O It happens only when there is a dominant competitor.13. Look at the four squares [■] that indicate where the following sentence could be added to the passage.As a result, there are not enough of the strong competitions to monopolize the environment’s resources.Where would the sentence best fit?14 Directions: An introductory sentence for a brief summary of the passage is provided below. Complete the summary by selecting the THREE answer choices that express the most important ideas in the passage. Some sentences do not belong in the summary because they express ideas that are not presented in the passage or are minor ideas in the passage. This question is worth 2 points.The relationships between predators and prey are complex.●●●Answer ChoicesO Studies of the interactions between wolves and moose on Isle Royale in Lake Superior reveal that wolf predation is not the primary factor controlling the moose population.O Predators help maintain biological diversity by limiting populations of a dominant competitor species, thereby preventing that species from excluding others.O A species’ population tends to rise and falls in a cycle pattern if the food supply for the population is limited, or if the population has a major predator.O Ecologists are interested in studying predator-prey population cycles because understanding how predators and prey interact will allow better wildlife management programs.O In predator-prey population cycles, predator populations increase or decrease following similar population changes in the species they prey on.O The removal of sea stars reduces the diversity of the community in which they are predators, and is therefore a bad idea.参考答案1.○22.○23.○14.○45.○36.○17.○28.○19.○210.○211.○112.○413.○314. A species’ population tends… Ecologists are interested in…In predator-prey population cycles…。

Lotka-Volterra Predator-Prey ModelsCreated by Jeff A. Tracey, PhD.Background:The interaction between predators and prey is of great interest to ecologists. In the 1920s, Alfred Lotka and Vito Volterra independently derived a pair of equations, called the Lotka-Volterra predatory-prey model, that have since been used by ecologists to describe the interactions of predators and prey. These equations, as you will see, can produce cyclical rise and fall in the abundance of predators and prey. These cycles have been observed in many predator-prey systems, including the famous example of Canadian lynx and snowshoe hares (Figure [PHOTO]). In lynx and snowshoe hare population trajectories reconstructed from trapping records (REF, Figure X), both lynx and hare numbers appear to rise and fall (cycle), with the lynx numbers beginning to decline after the number hares (their primary food source) declines.[ --- PHOTO (MAYBE) OF LYNX CAPTURING HARE --- ]This area of inquiry is even more important today as ecologists and conservation biologists consider the role of predators in ecosystems and the consequences of their extirpation. Recent research has emphasized the importance of predators in many ecosystems (for example, the reintroduction of wolves in Yellowstone National Park).The purpose of this learning module is to explore deterministic and stochastic versions of the Lotka-Volterra predator-prey equations. For a given set of initial conditions (say, the number of predators and the number of prey at the time the model begins) and model parameters, deterministic models yield the same results each time the model is run. However, we do not understand all of the processes that affect the behavior of predators and prey (or other systems), so we often include some randomness in the models to account for this uncertainty. Models that contain such randomness are called “stochastic models,” and yield different results each time they are run even if the initial conditions and parameters are the same.In this learning module, you will be able to explore deterministic and stochastic versions of four variations of the Lotka-Volterra predator-prey model using a computer program designed for this purpose.The Model:With the Lotka-Volterra predator-prey model, we model the change in the number of predators (P) and number prey (V) in continuous time via a system of two ordinary differential equations. In the equations, d V/dt represents the change in the number of prey at an instant in time, and d P/dt represents the change in the number of predators at an instant in time. Each term in the equations has a biological interpretation which I will give.The predators have a death rate (q) which controls the rate of exponential decline of the predator population (P) in the absence of prey. Prey on the other hand, have a growth rate (r) which controls the rate at which the prey population (V) grows in the absence of predators (for the exponential Models 1 and 2). With respect to prey, models 1 and 2 are “exponential models” for population growth. In models 3 and 4 we include another term, (-r/K)V2, which causes the prey growth rate to decline as the number of prey increases. This is called a “logistic model” for population growth. The logistic models include a second parameter, the carrying capacity (K), which limits the size the prey population can attain.In order for the predator population to grow, predators must consume prey. This interaction is modeled by the last term in the dV/dt equation and the first term in the d P/dt equation. This term involves a functional response, which models the rate at which predators capture and consume prey. For models 1 and 3, the functional response is aV. This is a Type I functional response in which the rate of prey capture and consumption by each predator increases linearly with the number of prey. In this model, prey are converted to predators by the first term in the dP/dt equation as bV, where b is called the “conversion efficiency.” However, it might be unrealistic to assume that there is no limit to the number of prey a predator may consume, so models 2 and 4 use a Type II functional response, in which the rate of prey capture and conversion to predators increases non-linearly with the number of prey. The formulation for each of the four Lotka-Volterra predator-prey models is:Deterministic Models: In the deterministic models, the number of prey (V) and predators (P) is treated as a continuous quantity. The program uses the equations above, initial values for V and P, and numerical methods to produce the deterministic population trajectories.Stochastic Models: In the stochastic models, the number of prey (V) and predators (P) is treated as a discrete (whole number) quantity. It is beyond the scope of this introduction to fully explain how the stochastic models work, so here I will give a brief explanation. In each of the two equations, the sum of the terms with positive signs can be thought of as the rate at which individuals are added to the population, and the sum of the absolute values of the terms with the negative signs can be thought of as the rate at which individuals are removed from the population. From these rates we determine a length of time to the next event, and which type of event occurs. The possible events are:●birth of a prey, in which case V increases by 1●death of a prey, in which case V decreases by 1●birth of a predator, in which case P increases by 1●death of a predator, in which case P decreases by 1From this process, we simulate the stochastic population trajectories. In the absence of extinction of predators or prey, the stochastic versions will, on average produce behavior similar to the deterministic models (but notice what happens if the predators become extinct).The Program:When the program is started, two windows are displayed. The first is a Control Panel (Figure 1) and the second is a Population Trajectory panel (Figure 2 and 3). The Control Panel allows you to select the version of the model to run, set all of the parameters and initial conditions for the model, the number of time steps, reset the model, run stochastic versions, and quit the program. The Population Trajectory panel displays the number of predators (in red) and prey (in blue) over time. If stochastic simulations are run (by clicking the “STOCHASTIC” button, which can be done repeatedly), the number of predators are displayed in light red and the number of prey in light blue.Figure 2: The Control Panel.Figure 3: The Population Trajectory panel displaying deterministic trajectories.Figure 4: The Population Trajectory panel displaying deterministic and stochastic trajectories.The program is started by double clicking the PredPreySim.jar file under Windows or calling the PredPreySim.sh BASH shell script from a command-line terminal in Linux or Mac OS. In order to run the program you must have the Java Virtual Machine (JVM) properly installed and on your computer's search path. If you do not have the JVM installed on your computer, you can download it for free from the Sun Microsystems website(/en/download/index.jsp).Once the program is running, deterministic predator and prey population trajectories will be displayed for the default parameters. If you change parameters, click the “RESET” button and then “RESUME.” To plot stochastic population trajectories, click the “STOCHASTIC” button.Note, however, that stochastic simulations can take a very long time for exponential models, especially for long time steps or if the prey intrinsic rate of increase is high.Exercises:Investigation 1: What happens if there are no predators? Set the predator slider to 0 and lower the time steps slider to about 20. Run each of the four models, and vary the prey rate of increase and carrying capacity.Investigation 2: Now add predators to the system. What happens to the number of prey? Explore the behavior of the model by varying the following parameters:●Model 1 – vary prey rate of increase, predator death rate, the encounter rate, andconversion efficiency.●Model 2 – vary prey rate of increase, predator death rate, the encounter rate,conversion efficiency, and prey handling time.●Model 3 – vary prey rate of increase, prey carrying capacity, predator death rate, theencounter rate, and conversion efficiency.●Model 4 – vary vary prey rate of increase, prey carrying capacity, predator death rate,the encounter rate, conversion efficiency, and prey handling time.The initial number of predators and prey can also be changed.What different behaviors are produced by the model? How do the deterministic and stochastic model results differ?Based on your explorations on one version of the model, write a 1 – 2 page description of (a) the model, (b) the basic kinds of behaviors it can produce, (c) how each of the model parameters affects the behavior of the model, and (d) the outcome of stochastic versions of the model and how they compare to the deterministic version.。

具有阶段结构及B-D功能反应的食饵-捕食模型宋燕;胥东方;李昂【摘要】Stability of a predator-prey model with Beddington-DeAngelis functional response and stage structure for the prey is investigated. By the method of characteristic, the local stability of a predator-extinction equilibrium and a coexistence equilibrium of the model is discussed. By introducing a new lemma and applying the comparison principle of differential equation and iterative method, respectively, sufficient conditions are obtained for the global stability of the predator-extinction equilibrium and the coexistence equilibrium. Finally, numerical simulations are carried out to illustrate the main results.%研究一类食饵具有阶段结构及Beddington-DeAngelis功能反应的食饵-捕食模型的稳定性，利用特征根法，讨论了捕食者灭绝平衡点及食饵和捕食者共存平衡点的局部稳定性；通过引入新的引理，利用微分方程比较定理及迭代法，得到了捕食者灭绝平衡点和食饵、捕食者共存平衡点全局稳定的充分条件；数值模拟验证了主要结果。

【期刊名称】《计算机工程与应用》【年(卷),期】2016(052)015【总页数】6页(P49-54)【关键词】阶段结构;时滞;Beddington-DeAngelis功能反应;食饵-捕食模型;稳定性【作者】宋燕;胥东方;李昂【作者单位】渤海大学数理学院，辽宁锦州 121000;渤海大学数理学院，辽宁锦州 121000;渤海大学数理学院，辽宁锦州 121000【正文语种】中文【中图分类】O175.1SONG Yan，XU Dongfang，LI Ang.Computer Engineering and Applications，2016，52（15）：49-54.在种群相互作用模型中，食饵-捕食模型引起人们广泛关注。