人口增长模型

————计算物理第五次作业

未分类
Author:GUO Xiao

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摘要

继1.5题之后，又采用Euler法求解课本上的problem1.6

背景介绍

人口的增长速率有两项，一项是出生率，正比于总人口数N，另一项是死亡率，正比于总人口数的平方

$$\frac{dN}{dt}=aN-bN^2$$
解出此微分方程就可以得到总人口数N(t).

Numerical solution

Discretize this differential equation, we obtain

$$N(i+1)=N(i)+dt(aN(i)-bN(i)^2)$$
Then it can be numerical solved by iteration.

Exact solution

$$\frac{dN}{N(a-bN)}=dt$$
$$\frac{b dN}{a}(\frac{1}{bN}+\frac{1}{a-bN})=dt$$
$$ln\frac{bN}{bN-a}=at+C$$
When $t=0$,$N=N_0$, so
$$N=\frac{aN_0e^{at}}{a+bN_0(e^{at}-1)}$$
From this solution,we can find that if $b\neq0$, when $t\rightarrow\infty$,$N\rightarrow\frac{a}{b}$;if $b=0$,$N=N_0e^{at}$.

Results

Solution program code
To makes sure the accuracy of numerical solution,I let dt=0.001.

1.

When$a=1$, b has different values.

When $b=0$,$N=N_0e^{at}$ is exponenital growth;
When $b>0$,$N$ will decrease(or increase) till reach a stable state $N=\frac{a}{b}$.If $\frac{a}{b}=N_0$,$N$ will keep constant $N=N_0$.

2.

When $a=0.1$

3.

When $a=2$

Conclusions

-1.If $b=0$,N will exponenitally grow without limit.
-2.If $b\neq0$,N will gradually reach a stable state $N=\frac{a}{b}$.At the same time,if$\frac{a}{b}=N_0$,$N$ will keep constant $N=N_0$.