population : the more the better?

S.F.Guo

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Summary:

This is a problem we are familar, which we have learned in highschool biology. Actually this is a Markov process：the latter outcome is depend only on the former result. In this work, I not only found the outcome depending on the timestep, but also found that different parameters creat totally different result in this model.

The pseudocode:

I want to define a class to do with, but I failed because I don't really know about how to use the class fountion. Finally, I use the most simple way to write this code. that is inistialize, calculate, store and show. In the calculate function, I used the relation:

$$\frac{dN}{dt}=aN-bN^2$$
Actually this differential equation has an accurate solution. But as we woluld take advantage of the computer, we don't bother to solve it. Instead, I solved the linar equation when $b=0$. So we can see how the timestep determines the pewcision of the numerical solution.

The code and result:

To show the timestep-dependance of the final result, I put some curve together in one figure. This time I didn't use the read_file script. On the contrary, I wrote allt the command in one script, including the discussion on whether the constant $b=0$. And when the parameter $b=0$ , we the have the exponential solution. finally I plot three curve in on figure to illsutrate the error caused by the longer timestep. If $b\ne0$, I use different timestep to get the same curve and prove that the solution is good enough and inpendent on the timestep. THIS is the final codes.

• First figure is the $b=0$ condition and the accurate solution.
  
• Second figure is about the $b\ne0$ and the inistial population was not large, and $a\gg b$, so that the population bocome larger and attained the biggest level.
  
• Third figure is about the $b\ne0$ and the inistial population was not large, and b is not that small, so the nonlinar Quadratic Term influence the population and the population bocome smaller and attained the smallest level.

The analysis and conclusion

• We can easily found that the smaller the timestep is, the more acccurate the results are. And when the timestep is small enough, the solutions converged to the accurate solution and independant on the timestep.
• Finally, let us see the physical model itself. As expected, we gained two different results when we changed the parameters. Actually, We can infer that if the initial nunber and the parameters are match, we can gain the stationary configuraion. In this model we will finally go to an equlibrium population as long as there are enough time.