1、微分方程模型 应用和计算,华南理工大学理学院 刘深泉教授,典型微分方程模型,Malthus人口模型 Logistic模型 新产品推广模型 物种竞争模型 正规战-游击战模型 Lotka-Volterra Models 海洋种群生态学 相互作用和变化问题,世界人口增长概况,中国人口增长概况,年代 1860 1870 1880 1960 1970 1980 1990人口 31.4 38.6 50.2 179.3 204.0 226.5 251.4,美国人口的增长概况,马尔萨斯（Malthus）指数人口模型,假设人口增长率r是常数,特点：种群数量翻一番的时间固定,Logisitic模型,模型检验Log
2、istic模型效果如何呢？ 1945年克朗皮克（Crombic）人工饲养小谷虫的实验，数学生物学家高斯（EFGauss）做原生物草履虫实验，都和Logistic曲线吻合。,Logistic模型描述种群增长高斯把5只草履虫放进盛有0.5cm3营养液的小试管，开始时草履虫以每天230.9%的速率增长，此后增长速度不断减慢，到第五天达到最大量375个，实验数据与r=2.309，a=0.006157，N(0)=5的Logistic曲线：,The Logistic Model,where K is capacity. In General Situation:,Application of Logist
3、ic model modeling population growth modeling of growth of tumors In chemistry: reaction models In physics: Fermi distribution In linguistics: language change Double logistic function,Malthus and Logistic模型,Malthus模型和Logistic模型。前一模型假设了种群增长率r为一常数。后一模型则引入了一个竞争项。,Logistic 模型的应用推广,新产品的推广模型,需求量上界K，销售数量x(t
4、)，未使用人数Kx(t). 统计筹算律,记比例系数为k，则x(t)满足：,此方程Logistic模型，解为：,Logistic 模型示例,新技术的传播和商业品牌的S形传播 谣言或网络信息的传播-选举应用 计算机病毒或传染病的扩散模型 城市房地产价格的logistic曲线 细胞分泌胰蛋白酶原和污染浓度扩散 公司财务危机的Logistic回归模型 自治系统和非自治系统,Predator-prey model,Predator-prey models are argubly the building blocks of the bio- and ecosystems as biomasses are
5、 grown out of their resource masses. Species compete, evolve and disperse simply for the purpose of seeking resources to sustain their struggle for their very existence. Depending on their specific settings of applications, they can take the forms of resource-consumer, plant-herbivore, parasite-host
6、, tumor cells (virus)-immune system, susceptible-infectious interactions, etc. They deal with the general loss-win interactions and hence may have applications outside of ecosystems. When seemingly competitive interactions are carefully examined, they are often in fact some forms of predator-prey in
7、teraction in disguise.,Types of predators,Carnivores 食肉动物 kill the prey during attackHerbivores 食草动物 remove parts of many prey, rarely lethal.Parasites寄生生物 consume parts of one or few prey, rarely lethal.Parasitoids拟寄生类 kill one prey during prolonged attack.,Predator-prey Model,x = amount of prey, y
8、 = amount of predatordx/dt = xg(x) yp(x) dy/dt = y-s + cp(x)g(x) is a growth function, g(x), monotonic non-increasing, p(x) is predation function p(x), monotonic increasing,Lotka -Volterra Models,Simplest model of predator-prey = Lotka-Volterra Lotka and Volterra independently propose a pair of diff
9、erential equations that model the relationship between a single predator and a single prey in a given environment:,Variable and Parameter definitions x prey species population y predator species population r Intrinsic rate of prey population Increase a Predation coefficient b Reproduction rate per 1
10、 prey eaten c Predator mortality rate,Ratio-Dependent Predator-Prey Model,Parameter/Variable Definitions x prey population y predator population a capture rate of prey d natural death rate of predator b predator conversion rate,Prey growth term,Predation term,Predator death term,Predator growth term
11、,两种群竞争模型 - Lotka-Volterra模型应用,竞争结局有三种结果 （1） 种1胜而种2被排除； （2） 种2胜而种1被排除； （3） 两种共存。,植物与食植动物的食物链模型,其中：V为植物密度；H为食草动物密度；,r1植物内禀增长率；K未放牧时植物最大密度；,d1在植被稀少时，动物的牧食效率（寻觅效率）；,a当草场被啃平时，动物的下降率；,C1每头食草动物最大取食率；,C2当草地高密度时对动物下降状况的改善率；,d2在植被变稀时的动物繁殖能力；,意识 动物 植物 地球 环境 社会,Lotka-Volterra-正规战与游击战,战争建模的格式 正规战争：正规部队与正规部队作战2)
12、游击战争：游击队与游击队作战3) 混合战争：正规部队与游击部队作战,求微分方程的数值解,1, 常微分方程数值解的定义,2, 建立数值解法的一些途径,3, 用Matlab软件求常微分方程的数值解,返 回,输入命令 ：x,y,z=dsolve(Dx=2*x-3*y+3*z,Dy=4*x-5*y+3*z,Dz=4*x-4*y+2*z, t)；x=simple(x) % 将x化简y=simple(y)z=simple(z),结 果 ： x = (c1-c2+c3+c2e -3t-c3e-3t)e2ty = -c1e-4t+c2e-4t+c2e-3t-c3e-3t+c1-c2+c3)e2tz = (-c
13、1e-4t+c2e-4t+c1-c2+c3)e2t,用Matlab软件求常微分方程的数值解,t，x=solver（f,ts,x0,options）,解: 令 y1=x，y2=y1,1、建立m-文件vdp1000.m如下：function dy=vdp1000(t,y)dy=zeros(2,1);dy(1)=y(2);dy(2)=1000*(1-y(1)2)*y(2)-y(1);,2、取t0=0，tf=3000，输入命令：T,Y=ode15s(vdp1000,0 3000,2 0); plot(T,Y(:,1),-),3、结果如图,To Matlab（ff4）,解 1、建立m-文件rigid.m
14、如下：function dy=rigid(t,y)dy=zeros(3,1);dy(1)=y(2)*y(3);dy(2)=-y(1)*y(3);dy(3)=-0.51*y(1)*y(2);,2、取t0=0，tf=12，输入命令：T,Y=ode45(rigid,0 12,0 1 1);plot(T,Y(:,1),-,T,Y(:,2),*,T,Y(:,3),+),3、结果如图,To Matlab（ff5）,图中，y1的图形为实线，y2的图形为“*”线，y3的图形为“+”线.,返 回,地中海鲨鱼问题,鱼类种群相互制约关系，第一次世界大战1914年1918年 ,地中海各港口几种鱼类捕获量的资料，发现鲨
15、鱼等的比例有明显增，而供其捕食的食用鱼的百分比却明显下降.战争使捕鱼量下降，食用鱼增加，鲨鱼等也随之增加，但为何鲨鱼的比例大幅增加呢？,意大利数学家V.Volterra 建立一个食饵捕食系统的数学模型，定量地回答这个问题.,模型反映了在没有人工捕获的自然环境中食饵与捕食者之间的制约，没有考虑食饵和捕食者自身的阻滞作用，是Volterra提出的最简单的模型.,首先，建立m-文件shier.m如下：function dx=shier(t,x)dx=zeros(2,1); dx(1)=x(1)*(1-0.1*x(2);dx(2)=x(2)*(-0.5+0.02*x(1);,其次，建立主程序shark
16、.m如下：t,x=ode45(shier,0 15,25 2); plot(t,x(:,1),-,t,x(:,2),*)plot(x(:,1),x(:,2),To Matlab(shark),求解结果：,左图反映了x1（t）与x2（t）的关系。可以猜测： x1（t）与x2（t）都是周期函数。,考虑人工捕获,设表示捕获能力的系数为e，相当于食饵的自然增长率由r1 降为r1-e，捕食者的死亡率由r2 增为 r2+e,设战前捕获能力系数e=0.3, 战争中降为e=0.1, 则战前与战争中的模型分别为:,模型求解:,1、分别用m-文件shier1.m和shier2.m定义上述两个方程,2、建立主程序s
17、hark1.m, 求解两个方程，并画出两种情况下鲨鱼数在鱼类总数中所占比例 x2(t)/x1(t)+x2(t),To Matlab(shark1),实线为战前的鲨鱼比例，“*”线为战争中的鲨鱼比例,结论：战争中鲨鱼的比例比战前高！,种群生态学 Creating Food Systems,3-Level system predator-prey Lotka-Volterra Ecosystem modeling with 3-level system that describes a planktonic marine ecosystem. Franks and Chen coupled a N
18、utrient-Phytoplankton-Zooplankton (NPZ) model into a primitive equation model and applied it to examine the summertime plankton dynamics on GB. That was the first modeling effort to study the biological process under the “realistic” physical environment in the GoM/GB region.,2009 ICM China universit
19、y of mining and technology,Improved Alga spices model,Spices of population 2,Spices of population 3,Model of 3-Populations,Analytical Hierarchy Process,Develop a commercial polyculture to remediate Bolinao,A commercial polyculture scheme,BIOGEOCHEMISTRY,BIODIVERSITY “Terra Incognita”,In understandin
20、g lies the road to prediction (e.g.: if we want to understand the biogeochemical cycling of carbon &/or other elemental cycling, we need to know other half of ecosystem equation),Grand Challenge: Understanding this Variety, What Determines Species Diversity?, How Will Big Pictures Emerge From a Sea
21、of Biological Data?,Science, Vol 309, 1 July 2005,Two of Sciences Top 25 Questions,CREDIT: MICHAEL T. SHIPLEY,Systems approach. Circuit diagrams help clarify nerve cell functions.,“New institutions around the world are gathering interdisciplinary teams of biologists, mathematicians, and computer specialists to help promote systems biology approaches” Sounds a little like ESMF,多谢!,
