Population dynamics

• Let's think about an immigration-emigration process.

  

  Interested in the dynamics of population size (# of ducks in Smith Lake).

  • Immigration rate: $\alpha$ immigrants per unit time
  • Emigration rate: $\beta$ emigration per unit time.

  After a short time increment $\Delta t$, population size at time t ($N_t$) changes to:

  
  

  $$N_{t+\Delta t} = N_t + \alpha \Delta t - \beta \Delta t$$
  
  

  After simple algebra:

  
  

  $$\frac{N_{t+\Delta t} - N_t}{\Delta t} = \alpha - \beta$$
  
  

  This suggests that:

  
  

  $$\frac{dN}{dt} = \alpha - \beta$$
  
  

  Simple, intuitive description of the immigration-emigration process.

• Is this simplistic analytical model enough to answer all of your questions about how this system behaves?

  Let's take a look at a stochastic simulation.