Results (don't read this section until you have finished the modifications)

• The dynamics of population size for each simulation run.

  Results of 5 runs from modification A (alpha=0.1, beta=0.1, init. n = 10)
  

• Average dynamics (from modification C)

  Average of 10000 simulations.

  alpha = 0.1, beta = 0.1
  

  alpha = 0.1, beta = 0.12
  

• Does this results matches with the deterministic expectation?

• In this simple case, we can have an analytical stochastic model.
  • Consider imigration rate is smaller than emigration rate ( $\alpha < \beta$).
  • After a long time, the population size n can be anywhere between 0 and Infinity.
  • Probability that the population size is k is:
    
    $$Pr(n = k) = (1-\frac{\alpha}{\beta}) (\frac{\alpha}{\beta})^k$$
    
    

  • Steady state distribution of population size

    Population size after time = 3000
    (alpha = 0.1, beta = 0.12, 10000 simulations)

    

  • Geometric distribution,
    

    Expected (average) population size is:
    

    $$\bar{n} = \frac{\alpha}{\beta - \alpha}$$
    
    

• Discrepancy:
  

  Comapre the expectation from analytical stochastic model with simulation (modification C). Try additional runs with different values of alpha and beta to convince you.