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具有Allee效应捕食−竞争系统的时空动态分析刘陈霖; 郑三强; 韩晓卓【期刊名称】《《广东工业大学学报》》【年(卷),期】2019(036)006【总页数】7页(P38-44)【关键词】Allee效应; 捕食−竞争系统; 稳定性分析; 数值模拟; 空间分布模式【作者】刘陈霖; 郑三强; 韩晓卓【作者单位】广东工业大学应用数学学院广东广州 510520【正文语种】中文【中图分类】Q141当一个种群的密度减少到一定程度时，种群数量将会维持在一个很低的水平或趋于灭绝，这种现象称之为Allee效应[1-4]. Allee效应的产生机制在于，当种群密度过低时，种群个体将面临近亲繁殖和难以抵御天敌等问题，从而导致出生率减少，且死亡率增加[5-6]. 近几十年来，大量的野外观测和实验研究表明，Allee效应出现在生态系统各个尺度或水平的物种上[7-8]，这极大地增加了物种局部甚至全球灭绝的风险[7,9]. 具有Allee效应的局域种群动态及实证研究已因此成为种群生态学关注的热点之一[10]. 已有研究表明Allee效应对捕食系统、竞争系统，甚至集合种群都产生重要影响[11-13]. 例如，Allee效应可以破坏捕食系统原有的稳定性[10,14]，并引起种群数量随时间震荡；对于竞争系统，Allee效应也导致其出现明显的不稳定性[12-13]. 所研究的捕食-竞争系统多以Lotka-Volterra模型和Levins模型的集合种群模型为主. 在集合种群尺度上，当斑块的占有率低于一定的阈值时，集合种群则呈现负增长趋势，进而导致集合种群灭绝，该现象被称之为似Allee效应[15]. 似Allee效应直接诱导集合种群的全局密度降低[16]或集合种群内部各局部种群同步灭绝.尽管Allee效应对局部种群动态的研究为种群生态学关注的热点之一，鉴于之前的研究发现，因为产生Allee效应的先决条件为种群密度过低[5-6]，Allee效应产生的作用强度有可能受到种群初始密度的影响，也即初始种群密度与Allee效应之间的关系鲜有报道. 另外，上述研究过多涉及Allee效应指数本身对系统稳定性，特别是种群数量或种群密度的作用关系，而对于该效应是否对物种或种群的空间模式具有一定的调节作用，也未见报道. 因此，基于两点，本文以Kylafis和Loreau[17]提出的捕食-竞争系统为基础，考虑到捕食者的Allee效应，建立具有Allee效应的捕食-竞争系统，并利用相平面分析和耦合映像粒子模型[18-19]分别给出系统的稳定性分析和空间分布模式. 通过MATLAB软件进行数值模拟，探讨Allee效应和捕食者初值的共同作用对系统稳定性及各物种种群空间分布模式的影响.1 模型Kylafis和Loreau基于生态位构建理论提出了一个反映生态作用关系与环境之间具有正向反馈作用的捕食-竞争系统模型[17]. 该模型含有一个捕食者P、两个作为食饵的竞争物种A和B，以及固有自然资源R，其中两个竞争物种通过摄取自然资源R生存，捕食者P则以捕食两个竞争物种为生存前提，该模型可表示为其中NP，NA和NB分别表示捕食者P、物种A和物种B的种群密度，NR为资源浓度；常数mA和mB分别表示捕食者P对物种A和对物种B的捕食率，fA和fB则表示物种A和物种B对资源R的摄取率；dP，dA和dB分别为物种P、物种A和物种B的自然死亡率. K表示资源R的供给总量，S表示资源R的转化再生率. 由方程(1)～(4)组成的系统记为系统(I)，该模型详细地阐述了捕食者、被捕食竞争者与资源的关系：捕食者的初值变化不改变系统最终的平衡点，而被捕食竞争者的初值变化只影响其自身的平衡点.在该捕食-竞争系统中，捕食者作为初始值较小的物种，其受到的Allee效应的影响最大，于是在捕食者上引入Allee效应，则由方程(1)变为如式(5)的捕食者动态方程：其中参数b为Allee效应强度. 由方程(2)～(5)组成的系统记为系统(II). 显然，当b=0时，系统(II)则退化为系统(I)，其他参数含义同上.本文将采用两种方法讨论捕食者具有Allee效应的捕食-竞争系统的时空动态. 首先，利用数值模拟和分岔图[20]检验Allee效应对于整个系统平衡动态的影响. 然后，利用耦合映像粒子模型对二维异质生境(n×n个斑块)上的具有空间结构的三物种进行空间分布模式的演化仿真. 为此，假设 At(i,j) 、 Bt(i,j)和Pt(i,j)分别表示在斑块( i,j)上t时刻时被物种A、物种B和捕食者P占据的概率，根据上述系统方程和概率转移模型[21-22]提出如下概率转移规则：其中，∑ At(i,j) ， Bt(i,j) 和∑ Pt(i,j)分别为焦点斑块在t时刻时的邻体被物种A、物种B和捕食者P占据的概率之和. 同理，斑块 (i,j) 在t时刻的资源量Rt(i,j)演化动态为需要说明的是在模拟过程中，三物种初始分布均为随机产生，演化规则采用同步更新、冯诺依曼的临体结构和周期边界，以消除边界效应对结果的扰动.2 结果2.1 系统稳定性分析根据稳定性理论，系统(I)存在一个非平凡平衡点为根据系统的生物学意义，当满足条件NP*＞0，NA*＞0，NB*＞0且NR*＞0时，意味着资源在不枯竭的前提下使得三物种持续共存. 此时，该系统中的参数关系应同时满足如下条件：即参数的选取遵循捕食者P对物种A的捕食率与物种A对资源R的摄取率的比值大于捕食者P对物种B的捕食率与物种B对资源R的摄取率的比值，捕食者P对物种A的捕食率与物种A的自然死亡率的比值大于捕食者P对物种B的捕食率与物种B的自然死亡率的比值，物种A的自然死亡率与物种A对资源R的摄取率的比值小于物种B的自然死亡率与物种B对资源R的摄取率的比值. 而对于系统(II)，根据稳定性理论可得参数关系与系统(I)的参数关系相同.根据如上参数选取范围，通过MATLAB软件给出系统(I)和系统(II)的三物种相平面图，如图1所示.初值取NP=0.2, NA=0.5, NB=0.5, NR=1. 其他参数取值为mA=0.4, mB=0.35, fA=0.40873, fB=0.36, dP=0.05,dA=0.018, dB=0.018,b=1, S=1, K=1. 由图1可知：两个系统中的3个物种的相空间曲线最终螺旋汇聚于内部平衡点，两个系统最终达到稳定. 在汇聚的过程中，系统(II)(见图1(b))的螺旋圈数远远多于系统(I)(见图1(a))，说明Allee效应导致系统(II)产生持续性阻尼震荡，从而延长了系统达到稳定的时间. 结果表明Allee效应可较大程度上改变系统的时间动态.图 1 系统(I)和系统(II)的相空间图Fig.1 The phase space diagram of system (I) and system (II)2.2 两系统中物种种群密度的时间动态分析事实上，具有Allee效应的捕食者的初始密度也将对系统产生较大的影响. 在模拟中，固定其他参数不变，观察当捕食者初始密度NP分别取0.1，0.182 和0.25时系统(I)和系统(II)的三物种时间动态图(见图2).其中参数为mA=0.4, mB=0.35,fA=0.40873, fB=0.36,dP=0.05, dA=0.018, dB=0.018, b=1, S=1, K=1,NA=0.5,NB=0.5, NR=1. 显然，捕食者P初值的变化对于无Allee效应的系统(I)无明显影响，捕食者P和物种A、B的种群密度都先呈现阻尼振动，约500时间步长后均达到稳定状态，且均收敛到相同的数值，如图2(a)、图2(c) 、图2(e)所示. 然而，上述结果与系统(II)的三物种时间动态图作对比则发现：捕食者P初值较小时，系统出现无规则的振动状态(见图2(b))，系统无法达到稳定；当捕食者P初值增加时，Allee效应则使系统各物种种群密度产生剧烈的阻尼振动，且震荡的振幅和持续时间都大幅增加(见图2(d))；捕食者P初值取值为0.25时，Allee效应仍然引起系统的阻尼震荡(见图2(f))，但相比图2(d)，振幅明显减小，且快速收敛到稳定状态. 该结果表明Allee效应可增加系统的复杂性，所产生的结果在一定程度上依赖于初值的选择，这一结果与图1相吻合.图 2 系统(I)与系统(II)的时间动态Fig.2 The temporal dynamics of system (I) and system (II)2.3 捕食者初值及Allee阈值对系统稳定性影响为了更清晰地表明捕食者P初值大小与Allee效应常数对系统稳定性的影响，笔者给出两组不同初值下Allee效应对系统分岔图的分析，如图3所示. 其他参数的取值为mA=0.4, mB=0.35, fA=0.40873,fB=0.36, dP=0.05, dA=0.018, dB=0.018, S=1, K=1. 对于捕食者P初值较大时(NP=0.2)，Allee效应常数并未使系统产生分岔现象，并且随Allee阈值b的增加，捕食者P种群密度大幅减少(见图3(a))，而两食饵的种群密度都显著提高(见图3(b)和图3(c)). 对于捕食者P初值较小的情况(NP=0.1)，Allee效应则使系统产生明显的混沌现象，特别是随Allee阈值b的逐渐增加，系统越发不稳定(见图3(e)和3(f)). 因此，在下文如无特别说明，模拟中取较大的捕食者初值.图 3 物种P、A和B的种群密度关于Allee阈值b的分岔图Fig.3 The bifurcation diagram of system (II) regarding Allee threshold b2.4 系统三物种空间动态分析图4为系统(I)和系统(II)的空间分布图，参考取值同图3. 从图4可以发现，3个系统处于稳定时的各个物种的空间分布呈现出聚集性[23]分布. 同时发现，在系统(I)和系统(II)中，物种A和物种B的分布呈现出斑块互补的现象. 在空间分布图上，首先以资源层为基础底层，然后在此基础上是以资源为生的被捕食者层，最后是以被捕食者为生的捕食者层. 在被捕食者层中，在物种A的密度较高的地方，物种B 的密度较低，这便出现互补的现象. 图5为系统(I)和系统(II)捕食者P第1 200步到第2 000步(间隔为100步)的空间分布图，参考取值同图3. 同一时刻，系统(I)与系统(II)比较，可以清楚地得到具有Allee效应的系统在空间的分布动态上比起不具有Allee效应的系统，呈现出的聚集性分布更加明显.3 讨论图 4 系统(I)和系统(II)空间分布图Fig.4 Spatial dynamics of systems (I) and (II) 本文以一个捕食-竞争系统为基础，构建了具有Allee效应的捕食-竞争系统，之后对此系统进行稳定性分析和时空动态分析. 结果表明，(1) Allee效应可以使稳定的系统变得不稳定，这具体表现在：当初始时刻捕食者丰富度较高时，Allee效应导致系统各物种种群数量阻尼振动的振幅和持续时间都大幅增加；当初始时刻捕食者丰富度较低时，Allee效应导致系统的某些物种种群数量无规则变动且不会趋于稳定状态，另一些物种甚至面临灭绝风险. (2) Allee效应降低了个体的空间聚集程度，这主要体现在较短的时间尺度上；若时间很长，分散分布的个体会逐渐形成聚集状态.图 5 系统(I)和系统(II)从第1 200步到第2 000步的空间动态(从左到右, 相隔100步), 分别见(a)和(b)Fig.5 Spatial dynamics of system (I) and system (II) from the 1 200 th step to the 2 000 th step are respectively shown in (a) and (b) 对于Kylafis和Loreau提出的捕食-竞争系统[17]，韩晓卓与莫昌健[24]的研究表明，生态位构建不会影响系统的稳定性，反而增强了物种的竞争能力. 本文则探讨了Allee效应对系统稳定性和空间分布模式的影响，因此本文的研究有助于进一步了解捕食-竞争系统稳定性的演化机制. 之前关于Allee效应对捕食和竞争系统影响的研究表明，Allee效应破坏了系统原有的稳定性[25-26]；本研究虽然也得到了相同的结论，但还表明Allee效应对系统稳定性的影响也受制于种群初始数量：若初始时刻种群数量较高，系统在经过长时间的阻尼振动后仍可趋于稳定状态；但若初始时刻种群数量较低，系统无法达到稳定状态.本文建立具有Allee效应的捕食-竞争系统，探讨Allee效应和捕食者初值的共同作用的变化对其影响. 研究结果可为保护生物多样性提供理论基础，有助于人们加深对生态系统稳定性的了解，提高人们保护物种的意识. 本研究仅在捕食者上加入Allee效应，但食饵也可能具有Allee效应[11]，因此将Allee效应加到食饵上会对捕食-竞争系统造成何种影响有待于进一步解决.参考文献：【相关文献】[1]ALLEE W C. Animal aggregations: a request for information [J]. The Condor, 1923, 25(4): 129-131.[2]ETIENNE R S, HEMERIK L. The founder and Allee effects in the patch occupancymetapopulation model[M]//REYDON T A. C, HEMERIK L. Current themes in theoretical biology. New York: Springer, 2005: 203-232.[3]GREENE C M. Allee effects [J]. Encyclopedia of Ecology,2008, 35(3): 123-127.[4]LIU Z G, ZHAO X, ZHANG F P. Allee effects of local populations and the synchrony of metapopulation [J]. Acta Ecologica Sinica, 2012, 32(1): 1-6.[5]GASCOIGNE J C, LIPCIUS R N. Allee effects driven by predation [J]. Journal of Applied Ecology, 2004, 41(5): 801-810.[6]GASCOIGNE J, BEREC L, GREGORY S, et al. Dangerously few liaisons: a review of mate-finding Allee effects[J]. Population Ecology, 2009, 51(3): 355-372.[7]AMARASEKARE P. Allee effects in metapopulation dynamics [J]. American Naturalist, 1998, 152(2): 298-302.[8]WANG W X, ZHANG Y B, LIU C Z. Analysis of a discrete-time predator-prey system with Allee effect [J]. Ecological Complexity, 2011, 8(1): 81-85.[9]HANSKI I. Uniting two general patterns in the distribution of species [J]. Science, 1997, 275(5298): 397-400.[10]ZHOU S R, LIU Y F, WANG G. The stability of predatorprey systems subject to the Allee effects [J]. Theoretical Population Biology, 2005, 67(1): 23-23.[11]MOROZOV A, ANDREW Y, PETROVSKII S, et al. Bifurcations and chaos in a predator-prey system with the Allee effect [J]. Proceedings Biological Sciences, 2004,271(1546): 1407-1414.[12]WANG G, LIANG X G, WANG F Z. The competitive dynamics of populations subject to an Allee effect [J]. Ecological Modelling, 1999, 124(2-3): 183-192.[13]SINGH M K, BHADAURIA B S, SINGH B K. Bifurcation analysis of modified Leslie-Gower predator-prey model with double Allee effect [J]. Ain Shams Engineering Journal,2016, 9(4): 1263-1277.[14]BOUKAL D S, BEREC L. Single-species models of the Allee effect: extinction boundaries, sex ratios and mate encounters [J]. Journal of Theoretical Biology, 2002, 218(3):375-394. [15]周淑荣, 王刚. 集合种群的似Allee效应[J]. 生物数学学报,2002, 17(3): 324-328.ZHOU S R, WANG G. Allee-like effects in metapopulation[J]. Journal of Biomathematics, 2002, 17(3): 324-328.[16]惠苍. 集合种群空间混沌的模拟研究以及Allee效应、拥挤效应与捕食效应对空间模式的影响[J]. 西北植物学报,2004, 24(3): 370-383.HUI C. Spatial chaos of metapopulation incurred by Allee effect, overcrowding effect and predation effect [J]. Acta Botanica Boreali-Occidentalia Sinica, 2004, 24(3): 370-383.[17]KYLAFIS G, LOREAU M. Niche construction in the light of niche theory [J]. Ecology Letters, 2011, 14(2): 82-90.[18]KANEKO K. Coupled map lattice[M]. Saarbrücken: Alphascript Publishing, 1991: 237-247.[19]王志斌, 张旭峰. 均匀耦合映像格子中的时空周期图案[J]. 中北大学学报(自然科学版), 2000,21(3): 202-205.WANG Z B, ZHANG X F. Spatiotemporal periodic patterns in symmetrically coupled map lattices [J]. Journal of North University of China, 2000, 21(3): 202-205. [20]BOCCARA N. Modeling complex systems [J]. Agricultural Economics, 2005, 58(2): 65-66.[21]ZHANG F, LI Z Z, HUI C. Spatiotemporal dynamics and distribution patterns of cyclic competition in metapopulation [J]. Ecological Modelling, 2006, 193(3-4): 721-735. [22]HUI C, LI Z. Distribution patterns of metapopulation determined by Allee effects [J]. Population Ecology, 2004,46(1): 55-63.[23]王鑫厅, 王炜, 刘佳慧, 等. 植物种群空间分布格局测定的新方法: 摄影定位法[J]. 植物生态学报, 2006, 30(4):571-575.WANG X T, WANG W, LIU J H, et al. A new method measuring plant population spatial patterns: photography orientation [J]. Chinese Journal of Plant Ecology, 2006,30(4): 571-575.[24]莫昌健, 韩晓卓. 生态位构建作用对捕食-竞争系统的影响[J]. 生态科学, 2016, 35(5): 73-81.MOC J, HAN X Z. The effects of niche construction on prey-competition system [J]. Ecological Science, 2016,35(5): 73-81.[25]黄亚玲. Allee效应对具有生态位构建作用的捕食系统的影响[D]. 广州: 广东工业大学, 2016.[26]HUI C, LI Z. Dynamical complexity and metapopulation persistence [J]. Ecological Modelling, 2003, 164(2-3): 201-209.。
常微分方程与动力系统英文版Differential equations play a crucial role in various scientific and engineering disciplines, providing a mathematical framework to describe the dynamics of systems over time. 常微分方程在各种科学和工程学科中发挥着至关重要的作用，提供了描述系统随时间演化动态的数学框架。
From modeling population growth to analyzing electrical circuits, differential equations offer powerful tools for understanding and predicting the behavior of complex systems. 从模拟人口增长到分析电路，常微分方程为理解和预测复杂系统的行为提供了强大的工具。
One particular area where differential equations are extensively used is in the study of dynamical systems. 动力系统的研究是差分方程广泛应用的一个特定领域。
Dynamical systems involve the study of how systems evolve over time, often exhibiting behaviors such as stability, chaos, and bifurcations. 动力系统涉及研究系统随时间演化的方式，通常表现出稳定性、混沌和分支等行为。
Through the use of differential equations, researchers can model and analyze the behavior of dynamical systems in a quantitative and systematic way. 通过使用常微分方程，研究人员可以以定量和系统的方式对动力系统的行为进行建模和分析。
精品课程申报表
申请学校同济大学是否部属□√
是否已获市级精品课程□否获奖年度
课程名称理论力学
课程层次（本/专）本科
课程类型□理论课（不含实践）□√理论课（含实践）□实践(验)课所属一级学科名称工学
所属二级学科名称力学类
课程负责人徐鉴
申报日期2008.5.16
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课程负责人情况
课程类别：公共课、基础课、专业基础课、专业课课程负责人：主持本门课程的主讲教师
2. 主讲教师情况⑴
课程类别：公共课、基础课、专业基础课、专业课
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课程类别：公共课、基础课、专业基础课、专业课2. 主讲教师情况⑷
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The Chinese diet is renowned for its rich variety and deep cultural roots.It is characterized by a balance of flavors,textures,and colors,reflecting the harmony and diversity of Chinese culture.Here are some key aspects of the Chinese diet that make it unique and appealing:1.Diversity of Ingredients:Chinese cuisine uses a wide range of ingredients,including various types of grains,vegetables,fruits,meats,and seafood.This diversity ensures a balanced intake of nutrients and a variety of flavors in every meal.2.Balance of Flavors:Chinese dishes often strive for a balance of the five basic tastes: sweet,sour,bitter,spicy,and salty.This balance is achieved through the careful selection and combination of ingredients and seasonings.e of Spices and Herbs:Chinese cooking employs a variety of spices and herbs,such as ginger,garlic,star anise,and Sichuan peppercorns,which not only add flavor but also have health benefits.4.Cooking Techniques:The Chinese diet features a multitude of cooking methods, including stirfrying,steaming,boiling,braising,and deepfrying.Each technique is chosen to best preserve the natural flavors and nutritional value of the ingredients.5.Emphasis on Whole Grains:Whole grains like rice,wheat,and millet are staples in Chinese meals.They provide a good source of energy and are rich in fiber,vitamins,and minerals.6.Plenty of Vegetables:Chinese meals typically include a generous serving of vegetables, which are often stirfried,steamed,or used in soups.This ensures a high intake of vitamins,minerals,and antioxidants.7.Moderation in Meat Consumption:While meat is a part of the Chinese diet,it is often consumed in moderation,with more emphasis on plantbased foods.This approach aligns with modern dietary recommendations for a healthy lifestyle.8.Inclusion of Tofu and Soy Products:Tofu and other soy products are common in Chinese cuisine,providing a good source of protein for vegetarians and a healthy alternative to meat.9.Tea Culture:Tea is an integral part of the Chinese diet and is consumed throughout the day.It is not only a beverage but also a part of social and medicinal practices.10.Seasonal Eating:The Chinese diet often follows the principle of eating according to the seasons,believing that certain foods are more beneficial when consumed at their peak growing times.11.Health and Wellness:Traditional Chinese Medicine TCM principles often guide dietary choices,with an emphasis on foods that are believed to promote health and prevent disease.12.Culinary Traditions:Each region in China has its own unique dishes and culinary traditions,which are passed down through generations and reflect the local culture and climate.The Chinese diet is a testament to the countrys rich history and cultural heritage.It offers a wealth of flavors and health benefits,making it an attractive and nutritious choice for people around the world.。
Western cuisine is a diverse and rich culinary tradition that has evolved over centuries,influenced by various cultures and regions.It encompasses a wide range of dishes,flavors,and cooking techniques,making it a fascinating subject to explore in an English essay.Introduction to Western CuisineWestern cuisine primarily refers to the food traditions of Europe and the Americas,but it also includes the culinary practices of other Western countries,such as Australia and New Zealand.The introduction of your essay could provide an overview of this culinary diversity,highlighting the unique characteristics of Western food.Historical InfluencesYour essay could delve into the historical influences that have shaped Western cuisine. For example,the Roman Empires impact on European food,the introduction of new ingredients from the New World to Europe during the Age of Exploration,and the influence of immigration on American food culture.Key Ingredients and FlavorsDiscuss the staple ingredients that are common in Western cooking,such as wheat,dairy, meat,and various herbs and spices.You could also explore the regional variations in flavors,from the rich and hearty dishes of Northern Europe to the lighter,fresher fare of Mediterranean countries.Cooking TechniquesWestern cuisine is known for its variety of cooking techniques,from baking and roasting to grilling and sautéing.Your essay could provide examples of how these techniques are used in different dishes,such as the slowcooked stews of France or the quick stirfries of American Chinese cuisine.Dining EtiquetteAn interesting aspect of Western food culture is the dining etiquette that accompanies it. Your essay could touch on the importance of table manners,the use of cutlery,and the structure of a formal meal,which often includes appetizers,main courses,and desserts.Popular Dishes and Their OriginsWestern cuisine is home to many iconic dishes that have become popular worldwide. You could write about the origins and characteristics of dishes such as pizza from Italy, hamburgers from the United States,and fish and chips from the United Kingdom.Health and SustainabilityIn recent years,there has been a growing focus on health and sustainability in Western cuisine.Your essay could discuss the rise of organic farming,the popularity of vegetarian and vegan diets,and the efforts to reduce food waste.ConclusionConclude your essay by summarizing the key points discussed and reflecting on the importance of understanding and appreciating the diversity of Western cuisine.You might also consider the future of Western food,including trends such as fusion cuisine and the influence of global food movements.Sample Paragraph:The Evolution of Western CuisineOver time,Western cuisine has undergone significant transformations,reflecting the dynamic nature of culinary arts.From the medieval banquets of Europe,characterized by lavish spreads and intricate presentations,to the modernday emphasis on simplicity and fresh ingredients,Western food has continuously evolved.The industrial revolution brought about mass production and the availability of new ingredients,while globalization has introduced a plethora of flavors and cooking styles into the Western culinary landscape.Today,Western cuisine is a melting pot of flavors,techniques,and traditions,offering a rich tapestry of gastronomic experiences for food lovers around the world.Remember to use descriptive language and engaging examples to make your essay on Western cuisine both informative and enjoyable to read.。
带有乘法Allee效应的捕食-食饵模型的共存解李海侠【摘要】讨论了一类带有乘法Allee效应的捕食-食饵扩散模型正解的存在性和稳定性。
利用局部分歧理论研究了分歧正解的存在性，考察了分歧解的稳定性，运用全局分歧定理将局部分歧进行延拓从而得到了正解存在的充分条件。
结果表明当参数满足一定条件时，两物种能共存而且共存解稳定。
%The existence and stability of a predator-prey diffusive model with multiplicative Allee affect are discussed. The existence of bifurcating positive solutions is investigated by means of the local bifurcation theory. The stability of bifurcation solutions is determined. By using the global bifurcation theory, the local bifurcation is extended and the suffi-ciently conditions of the existence of positive solutions are obtained. The results indicate that the two species will coexist when the parameters satisfy certain conditions, furthermore the coexistence solutions are stable.【期刊名称】《计算机工程与应用》【年(卷),期】2015(000)010【总页数】4页(P16-19)【关键词】乘法Allee效应;分歧理论;稳定性【作者】李海侠【作者单位】宝鸡文理学院数学系，陕西宝鸡 721013【正文语种】中文【中图分类】O175.261 引言本文讨论如下带有乘法Allee效应的捕食-食饵模型：其中，u和v分别表示食饵和捕食者的浓度，r和e分别是u和v的最大增长率，k是承载能力，m表示没有捕食者时食饵物种u的Allee效应阀值或最小族群量。
ar X i v :n l i n /0012056v 1 [n l i n .C D ] 26 D e c 2000Rich Variety of Bifurcations and Chaos in a Variant of Murali-Lakshmanan-ChuaCircuitK.Thamilmaran and kshmanan ∗Centre for Nonlinear Dynamics,Department of Physics,Bharathidasan University,Tiruchirapalli 620024,IndiaK.MuraliDepartment of Physics,Anna University,Chennai 600025,India(Dated:)A very simple nonlinear parallel nonautonomous LCR circuit with Chua’s diode as its only non-linear element,exhibiting a rich variety of dynamical features,is proposed as a variant of the simplest nonlinear nonautonomous circuit introduced by Murali,Lakshmanan and Chua(MLC).By constructing a two-parameter phase diagram in the (F −ω)plane,corresponding to the forcing am-plitude (F )and frequency (ω),we identify,besides the familiar period-doubling scenario to chaos,intermittent and quasiperiodic routes to chaos as well as period-adding sequences,Farey sequences,and so on.The chaotic dynamics is verified by both experimental as well as computer simulation studies including PSPICE.I.INTRODUCTIONSometime ago,the simplest nonlinear dissipative nonautonomous electronic circuit consisting of a forced seriesLCR circuit instead of the forced series∗Electronicaddress:lakshman@bdu.ernet.inRNi FIG.1:Circuit realization of the simple non-autonomous MLC variant circuit.Here N is the Chua’s diode.nal periodic forcing voltage source and only one non-linear element(N ),namley,the Chua’s diode.In the dynamically interesting range,the v −i characteris-tic of the Chua’s diode is given by the usual three segment piecewise-linear function [Chua et al.,1987;Kennedy,M.P,1992;Cruz,J.M &Chua,L.O,1992].The nonlinear element is added to the familiar forced parallelLCR of MLC circuit[Murali et al.,1994].The resulting circuit can be consid-ered as another important very simple dissipative second order nonautonomous nonlinear circuit and a variant of the MLC circuit.By applying Kirchhoff’s laws to this circuit,the governing equations for the voltage v across the capacitor C and the current i L through the inductor L are represented by the following set of two first-order2 non-autonomous diffrential equations:C dvR(f sin(Ωt)−v)−i L−g(v),(1a)Ldi Ldt (2b) whereβ=C/LG2,F=f/B p,G=1/R.Obviously g(x)=bx+0.5(a−b)(|x+1|−|x−1|).Here a= G a/G,b=G b/G.Note that the twocoupledfirst order ordinary differential equation given by Eqs.(2)can also be written as a single second order differential equation of the Lienard’s type in the form¨y+˙y+βg(˙y/β)+βy=βF sin(ωt).(3) We note at this point chaos via torus breakdown gen-erated in a piecewise-linear forced van der Pol equation of the form(3)with asymmetric nonlinearity has been studied by Inaba and Mori[Inaba,&Mori,1991]some-time ago.However the corresponding circuit uses more nonlinear elements than the present circuit.Now the dynamics of equation(2)or equivalently(3)depends on the parameters a,b,β,ωand F.Then for the above chosen experimental circuit parameter values,we have β=0.05,a=−1.121,b=−0.604andω=0.105.We use the amplitude F of the external periodic forcing as the control parameter and carry out a numerical simu-lation of Eqs.(2)either by integrating Eqs.(2)or by solving Eq.(3)analytically and numerically,for increas-ing values of F.For the above choice of parameters the numerical simulation of Eqs.(2)exhibits novel dynamical phenomena.FIG.3:(a)One parameter bifurcation diagram in the(ω−x) plane and(b)maximal Lyapunov spectrum(λmax).The value of the forcing amplitude has beenfixed at F=0.28To begin with we have carried out an experimental study of the dynamics of the circuit given by Fig.1.The driving amplitude(f)is slowly increased from0V and the response of the system is observed to progress through a series of transition from periodic motion to aperiodic motion.When the driving amplitude f=0(correspond-ing to the autonomous case),a period-1limit cycle is observed.By increasing the amplitude from zero up-wards,the circuit of Fig.1is found to exhibit a quasiperi-odic(torus)attractor which then transits to chaos via torus breakdown,followed by periodic windows,period-adding sequences etc.We have confirmed these results also by using the standard4th-order Rugne-Kutta inte-gration routine and carrying out a numerical analysis of Eqs.(2)or equivalently Eq.(3)with the rescaled circuit parameters of Fig.1as given above.We have observed a series of appearence and disappearence of the quasiperi-odic(torus)and phase locking(periodic)attractors,and period-adding sequences alternatively by varying the am-plitude F of the external source atfixed frequency.Illus-trated in Fig.2is the two paramater phase diagram which is plotted in the(F−ω)plane.In particular,quasiperi-odic motions and period-adding sequences besides the3FIG.4:(a)One parameter bifurcation diagram in the(F−x) plane and(b)maximal Lyapunov spectrum(λmax).The value of the forcing frequency has beenfixed atω=0.105 standard bifurcation sequences have been observed in re-gions of the lower drive amplitude(F)and frequency(ω) values.For example,typical quasiperiodic attractors ex-ist in the range F=(0.15,0.2),andω=(0.09,0.16).Sim-ilarly,a period adding sequence exists for F=(0.36,0.38) andω=(0.1,0.115),a period-doubling bifurcation se-quence for F=0.39,andω=(0.09,0.1),and also type I intermittency has been identified for F=0.38and ω=(0.10636).In addition for F=(0.24,0.27)and ω=(0.138,0.146),as well as for F=(0.26,0.29),ω= (0.128,0.138)and for F=(0.28,0.33),ω=(0.11,0.125), Farey sequences[Kaneko,1986]exist.The regions of chaos are also indicated in Fig.2.Finally Fig.3(a).represents the one-parameter bifurcation in the(ω−x) plane,for F=0.28which consists of quasiperiodicity, chaos,windows,period adding sequences and the familiar period doubling bifurcation sequence,intermittency and so on.Infig.3(b)its corresponding maximal Lyaponov spectrum in(ω−λmax)is plotted.Alsofig.4(a)repre-sents the one-parameter bifurcation in the(F−x)plane, forω=0.105,fig.4(b)depicts the corresponding maxi-mal Lyaponov spectrum(F−λmax).Further,the exper-imental results obtained for a choice of circuit parameters were also confirmed by using the PSPICE[Roberts,1997] deck available to simulate the behavior of the circuit in Fig.1.In Fig.5,we have included for comparison the chaotic attractor corresponding to F=0.389V in Eq.(1) for experimental and SPICE analysis and corresponding numerical simulation for amplitude f=0.383in Eq.(2).III.CONCLUSIONIn this letter,we have presented a very simple second order dissipative nonlinear nonautonomous circuit which is a variant of the MLC circuit and carried out.It is FIG.5:(a)Chaotic attractor corresponding to F=0.389V,Ω=1.116KHz in Eq.(1):(a)experimental(b)PSPICE and (c)corresponding numerical simulation of Eq.(2).shown to possess a very rich variety of dynamical phe-nomena.In view of the appearance of several ubiquitous routes to chaos such as quasiperiodicity,intermittency, period doubling,period-adding and Farey sequence in such a single but simple circuit,one can make use of the circuit in varied investigations on chaotic dynamics and applications,including spatio-temporal studies.A more detailed analysis of these aspects will be published else-where.AcknowledgmentsThe work of kshmanan forms part of a Depart-ment of Science and Technology,Government of India research project.We are thankful to Mr.A.Venkatesan for his assistance for numerical analysis.4REFERENCESChua,L.O.,Desoer,C.A.&Kuh,E.S.[1987]Linear and Nonlinear Circuits(McGraw-hill,New York). Cruz,J.M,&Chua,L.O.[1992]“A CMOS IC nonlinear reisitor for Chua’s circuit,”IEEE Trans.Circuits Syst-I.,vol.39,pp.985-995.Inaba,N.&Mori,S.[1991]“Chaos via torus break down in a piecwise-linear forced van der Pol os-cillator with a diode,”IEEE Trans.Circuits and Syst-I.,vol.38,pp.398-409.Kaneko,K.[1986]Collapse of Tori and Genesis of Chaos in Dissipative Systems(World Scientific, Singapore).Kennedy,M.P.[1992]“Robust op amp realization ofChua’s circuit,”Frequanz,vol.46,pp.66-80.Lakshmanan,M.&Murali K.[1996]Chaos in Non-linear Oscillators:Controlling and Synchronization (World Scientific,Singapore).Murali,K.,Lakshmanan,M.&Chua,L.O.[1994]“The simplest dissipative nonautonomous chaotic circuit,”IEEE Trans.Circuits Syst-I.,vol.41, pp.462-463.Murali,K.,Lakshmanan,M.&Chua,L.O.[1994]“Bifurcation and chaos in the simplest dissipa-tive nonautonomous circuit,”Int.J.Bifurcation and Chaos,vol.4,pp.1511-1524.Roberts,W.[1997]SPICE(Oxford University Press, New York).。
