Population Growth Model

Abstract

• I choose population growth model as this equition:

  $\frac{dN}{dt} = aN - bN^2$ (1)

• It describe how the number of individuals in a population, N, varies with time. Here the first term aN corresponds to the birth of new member, while the second term $-bN^2$ corresponds to deaths. The death term is proportional to $N^2$ to allow for the fact that food will becomeharder to find when the population N becomes larger.

Work

Eular method

• We could solving (1) with $b=0$ using the Eular method.
  And we can get :
  $N(t+dt) = N(t)+[aN(t) - bN^2(t)]dt$

• Then we could use the python code to get the figure of the model.

Exact solution

• Using some methodes of ordinary diffrential equation, we could get the exact solution of the equition(1) :
  $N = \frac{aN_{0}e^{at}}{a-bN_{0}+bN_{0}e^{at}}$
  This is the analytical expression of this model.

Analysis

I set the $N(0) = 1000$ , $a = 10$

• When $b = 0$ , we get this graph :
  
  It means the population grows like　 J 　model
• When $b = 0.01$ , we get a line :
  
  It means the population is stable
• When $b = 0.005$ , the figure is :
  
  The population grows rapidly first, and then begins flat
• When $b = 0.02$ and $b=0.05$ , it shows :
  
  
• When $b=0.1$ , it's :
  
  The line is nearly a 　L　 model
  Given fig4 , 5 , 6. When b is bigger, the population will reduce more rapidly.