matlab酵母培養(yǎng)物的離散阻滯增長模型

1、遼寧工程技術(shù)大學(xué)上機(jī)實(shí)驗(yàn)報(bào)告實(shí)驗(yàn)名稱酵母培養(yǎng)物的離散阻滯增長模型院系理學(xué)院專業(yè)理科實(shí)驗(yàn)班級姓名學(xué)號日期2015.4.20實(shí)驗(yàn)?zāi)康暮喪霰敬螌?shí)驗(yàn)?zāi)康模赫莆战湍概囵B(yǎng)物的阻滯增長差分方程模型的分析與建模方法，并會利用Matlab作數(shù)值計(jì)算與誤差分析.實(shí)驗(yàn)準(zhǔn)備你為本次實(shí)驗(yàn)做了哪些準(zhǔn)備：上課聽講，下課復(fù)習(xí)課本，不明白之處查閱資料。實(shí)驗(yàn)進(jìn)度本次共啟1個(gè)練習(xí)，完成1個(gè)。實(shí)驗(yàn)總結(jié)本次實(shí)驗(yàn)的收獲、體會、經(jīng)驗(yàn)、問題和教訓(xùn)：教師評語具體處理過程和相應(yīng)實(shí)驗(yàn)結(jié)果:已知從測量酵母培養(yǎng)物增長的實(shí)驗(yàn)收集的數(shù)據(jù)如表:/h0123456789生物量/g9.618.329.047.271.1119.174.6257.350.441.

2、1370時(shí)刻/h101112131415161718生物量/g513.559.594.629.4640.8651.655.9659.661.378168實(shí)驗(yàn)要求：1、作圖分析酵母培養(yǎng)物的增長數(shù)據(jù)、增長率、與相對增長率.2、建立酵母培養(yǎng)物的增長模型.3、利用線性擬合估計(jì)模型參數(shù)，并進(jìn)行模型檢驗(yàn)，展示模型擬合與預(yù)測效果圖4、利用非線性擬合估計(jì)模型參數(shù)，并進(jìn)行模型檢驗(yàn)，展示模型擬合與預(yù)測效果圖5、請分析兩個(gè)模型的區(qū)別，作出模型的評價(jià).實(shí)驗(yàn)內(nèi)容：1、作圖分析酵母培養(yǎng)物的增長數(shù)據(jù)、增長率、與相對增長率.(1)增長數(shù)據(jù)：繪制x關(guān)于k的散點(diǎn)圖：x=9.6,18.3,29.0,47.2,71.1,119.1,

3、174.6,257.3,350.7,441.0,513.3,559.7,594.8,629.4,640.8,651.1,655.9,659.6,661.8;k=0:1:18;plot(k,x,'k+')xlabel('時(shí)間k(小時(shí))')ylabel('生物量x(克)')title('x關(guān)于k的散點(diǎn)圖')x關(guān)于k的散點(diǎn)圖700,1F,1-1F,4600+-4500-4-)+克400rrx(+S300-生4200-r+100，一-+440-1-11111111024681012141618時(shí)間k(小時(shí))即x關(guān)于k的散點(diǎn)沿s型曲線分布，

4、x隨著k單調(diào)增加，x可能趨于穩(wěn)定值，極限可能存在(2)增長率：繪制x差值關(guān)于k的散點(diǎn)圖以及繪制x差值關(guān)于x的散點(diǎn)圖:xk=8.7,10.7,18.2,23.9,48,55.5,82.7,93.4,90.3,72.3,46.4,35.1,34.6,11.4,10.3,4.8,3.7,2.2,0;k=0:1:18;subplot(1,2,1);plot(k,xk,'k+')xlabel('時(shí)間k(小時(shí))')ylabel('一階差分xk(克)')title('xk關(guān)于k的散點(diǎn)圖')xk=8.7,10.7,18.2,23.9,48,55.

5、5,82.7,93.4,90.3,72.3,46.4,35.1,34.6,11.4,10.3,4.8,3.7,2.2,0;x=9.6,18.3,29.0,47.2,71.1,119.1,174.6,257.3,350.7,441.0,513.3,559.7,594.8,629.4,640.8,651.1,655.9,659.6,661.8;subplot(1,2,2);plot(x,xk,'k+')xlabel('生物量xk(克)')ylabel('一階差分xk(克)')100克k(x分差階xk關(guān)于k的散點(diǎn)圖908070605040302010時(shí)

6、間k(小時(shí))克k(x分差階xk關(guān)于x的散點(diǎn)圖title('xk關(guān)于x的散點(diǎn)圖')1009080706050403020100200400600800生物量xk(克)0觀察x差值關(guān)于k的散點(diǎn)圖，難以發(fā)現(xiàn)二者的近似而簡單的函數(shù)關(guān)系。觀察x差值關(guān)于x的散點(diǎn)圖，發(fā)現(xiàn)二者近似二次函數(shù)關(guān)系xk=-a1*xkA2+a2*xk;,實(shí)質(zhì)就是離散阻滯增長模型。(3)相對增長率：繪制rk差值關(guān)于k的散點(diǎn)圖以及繪制rk關(guān)于xk的散點(diǎn)圖:rk=0.90625,0.5847,0.62759,0.50636,0.67511,0.46599,0.47365,0.363,0.25749,0.16395,0.09

7、0395,0.62712,0.058171,0.018112,0.016074,0.0073721,0.0056411,0.0033354,0;k=0:1:18;subplot(1,2,1);plot(k,rk,'k+')xlabel('時(shí)間k(小時(shí))')ylabel('增長率rk(%)')title('rk關(guān)于k的散點(diǎn)圖')rk=0.90625,0.5847,0.62759,0.50636,0.67511,0.46599,0.47365,0.363,0.25749,0.16395,0.090395,0.62712,0.05817

8、1,0.018112,0.016074,0.0073721,0.0056411,0.0033354,0;x=9.6,18.3,29.0,47.2,71.1,119.1,174.6,257.3,350.7,441.0,513.3,559.7,594.8,629.4,640.8,651.1,655.9,659.6,661.8;subplot(1,2,2);plot(x,rk,'k+')xlabel('生物量xk(克)')ylabel('增長率rk(%)')title('rk關(guān)于x的散點(diǎn)圖')0.21+0.1+-+。1105101520

9、時(shí)間k(小時(shí))rk關(guān)于k的散點(diǎn)圖10.9個(gè)rk關(guān)于x的散點(diǎn)圖1r1r-0.20.1200400600800生物量xk(克)876000%/率長增0.94觀察rk關(guān)于k的散點(diǎn)圖，難以發(fā)現(xiàn)二者的近似而簡單的函數(shù)關(guān)系。觀察rk關(guān)于xk的散點(diǎn)圖，發(fā)現(xiàn)二者近似線性遞減關(guān)系rk=r*(1-xk/N);由rk=(x(k+1)-x(k)/x(k),代入上式，建立離散阻滯增長模型。2、建立酵母培養(yǎng)物的增長模型.在營養(yǎng)有限的環(huán)境下，假設(shè)用前差公式計(jì)算的增長率rk隨著生物量xk的增加而線性遞減，即rk=rk=(x(k+1)-x(k)/x(k)=r*(1-x(k)/N),k=0,1,2根據(jù)模型假設(shè)，即可建立離散阻滯增

10、長模型x(k+1)=x(k)+r*x(k)*(1-x(k)/N),k=0,1,2,3、利用線性擬合估計(jì)模型參數(shù)，并進(jìn)行模型檢驗(yàn)，展示模型擬合與預(yù)測效果圖首先，根據(jù)rk和xk的數(shù)據(jù)擬合出r(k)=r*(1-x(k)/N)的參數(shù)r和N,擬合效果圖如1.(3)-1圖所示。然后根據(jù)觀測數(shù)據(jù)直接取x0=9.6,用循環(huán)語句按照x(k+1)=x(k)+r*x(k)*(1-x(k)/N),k=0,1,2,進(jìn)行迭代計(jì)算，算出第018小時(shí)酵母生物量的模擬值，并計(jì)算誤差平方和，繪制模擬效果圖和模擬誤差圖。用matlab編程如下：t=0:18;x=9.6,18.3,29.0,47.2,71.1,119.1,174.6

11、,257.3,350.7,441.0,513.3,559.7,594.8,629.4,640.8,651.1,655.9,659.6,661.8;r=(x(2:19)-x(1:18)./x(1:18);a1=polyfit(x(1:18),r,1);r1=a1(2),N1=-a1(2)/a1(1)x1=x；fork=1:18x1(k+1)=x1(k)+r1*x1(k)*(1-x1(k)/N1);endresd1=x-x1;sse1=sum(resd1.A2)subplot(2,1,1),plot(t,x,'k*',t,x1,ks,)axis(-1,19,0,670),legen

12、d(觀測值','模擬值',4)xlabel('時(shí)間k(小時(shí)),),ylabel('生物量x_k(克)')title(,(1)離散阻滯增長模型的模擬效果圖，線性擬合,)subplot(2,1,2),plot(t,resd1,'k.',-1,19,0,0,'k')axis(-1,19,-40,40),legend('觀測值','模擬值',4)xlabel('時(shí)間k(小時(shí)),),ylabel('模擬誤差')title('(2)離散阻滯增長模型的模擬誤差，線性

13、擬合,)結(jié)果：r1=0.6693N1=635.7055sse1=6.2932e+03(1)離散阻滯增長模型的模擬效果圖，線性擬合克kx量物生6004002000'r'口0由由古古丁古口印斗觀測值田田口模擬值由用田bIIIIYII024681012141618時(shí)間k(小時(shí))差誤擬模(2)離散阻滯增長模型的模擬誤差，線性擬合4、利用非線性擬合估計(jì)模型參數(shù)，并進(jìn)行模型檢驗(yàn)，展示模型擬合與預(yù)測效果圖用matlab編程如下：函數(shù)文件fun_3_4_2.m：functiony=fun_3_4_2(b,x)y=zeros(size(x);y(1)=b(3);fork=2:length(x)y

14、(k)=y(k-1)+b(1).*y(k-1)*(1-y(k-1)./b(2);end腳本：t=0:18;x=9.6,18.3,29.0,47.2,71.1,119.1,174.6,257.3,350.7,441.0,513.3,559.7,594.8,629.4,640.8,651.1,655.9,659.6,661.8;a2,resd2=nlinfit(t,x,fun_3_4_2,0.5,660,9.6)sse=sum(resd2.A2)subplot(2,1,1)plot(t,x,'k*',t,fun_3_4_2(a2,t),'ks')axis(-1,19

15、,0,670)legend(觀測值','模擬值',4)xlabel('時(shí)間k(小時(shí)),ylabel('生物量x_k(克)')title('(1)離散阻滯增長模型的模擬效果以，非線性擬合')subplot(2,1,2)plot(t,resd2,'k.',-1,19,0,0,'k')axis(-1,19,-40,40)xlabel('時(shí)間k(小時(shí)),ylabel('模擬誤差')title('(2)離散阻滯增長模型的模擬誤差，非線性擬合')結(jié)果如下:a2=0.5604652.463414.9997resd2=-5.39975.134111.84062.48075.1393-4.9118-6.756210.87271.03847.88769.6673-7.4948-11.6748-14.2442-17.8488-4.1746-4.7021-4.1982-3.0640sse=1.3535e+03(1)離散阻滯增長模型的模擬效果圖，非線性擬合克kx量物生600400200040困a曲曲國步觀測值囚m值口模擬值ooO22-差誤擬模-40024681012141618時(shí)間k（小時(shí)）024681012141618時(shí)間k(小時(shí))(2)離散阻滯