Logistics人口模型

1.  function Population_1  
2.  %t:时间（年）  
3.  %p:各年人口总数（万人）  
4.  %pm:最大人口容量  
5.  %r:人口增长率  
6.  t = [1961:1:2020];  
7.  p =[65859,67296,69172,70499,72538,74542,76368,78534,80671,82992,85229,87177,89211,90859,92420,93717,94974,96259,97542,98705,100072,101654,103008,104357,105851,107507,109300,111026,112704,114333,115823,117171,118517,119850,121121,122389,123626,124761,125786,126743,127627,128453,129227,129988,130756,131448,132129,132802,133450,134091,134916,135922,136726,137646,138326,139232,140011,140541,141008,141178];  
8.  p0=p(1);  
9.    
10. %方程（5）的拟合  
11. r=polyfit(p(2:end),diff(p)./(p(2:end)),1);  
12. pm=-r(2)/r(1);  
13. r0=r(2);  
14.   
15. %拟合数据与实际数据的误差  
16. f=@(t)pm./(1+(pm/p0-1)*exp(-r0*(t-1961)));  
17. err=norm(f(t)-p);  
18.   
19. %绘图  
20. clf  
21. fplot(f,[1961,2050],'b')  
22. hold on  
23. plot(t,p,'*')  
24. axis([1961,2050,60000,pm])  
25. xlabel("时间T");ylabel("人口总数（万人）"); 

26. function Population_2  
27. %t:时间（年）  
28. %p:各年人口总数（万人）  
29. %pm:最大人口容量  
30. t = [1961:1:2020];  
31. p =[65859,67296,69172,70499,72538,74542,76368,78534,80671,82992,85229,87177,89211,90859,92420,93717,94974,96259,97542,98705,100072,101654,103008,104357,105851,107507,109300,111026,112704,114333,115823,117171,118517,119850,121121,122389,123626,124761,125786,126743,127627,128453,129227,129988,130756,131448,132129,132802,133450,134091,134916,135922,136726,137646,138326,139232,140011,140541,141008,141178];  
32. p0 = p(1);  
33.   
34. % φ(x)的拟合  
35. r_x=polyfit(p(2:end),diff(p)./p(2:end),1);  
36. pm = -r_x(2)/r_x(1);  
37.   
38. %方程（10）的拟合  
39. y=-log((pm*p0-p*p0)./(pm*p-p*p0));  
40. k=polyfit(t-1961,y,3);  %以3次多项式拟合
41.   
42. %拟合数据与实际数据的误差  
43. f=@(t)pm./(1+exp(-polyval(k,t-1961))*(pm/p0-1));  
44. err=norm(f(t)-p);  
45.   
46. %绘图  
47. clf  
48. fplot(f,[1961,2050],'b')  
49. hold on  
50. plot(t,p,'*')  
51. axis([1961,2050,60000,pm])  
52. xlabel("时间T");ylabel("人口总数（万人）");