Logistic population growth

In the previous immigration-emigration model, the population size changed due to migration. We assumed that there is no birth or death.

Now we consider a closed ecological system, where there is no migration. Dynamics of population size is due to internal birth and death.

• Exponential population growth
  

  The population growth is proportional to the current population size (N). Mathematically,

  
  

  $$\frac{dN}{dt} = r N$$
  
  

  r is per capita growth rate.

  r can be influenced by both birth and death rate.

  

  Fortunately, no organim grows to infinite population size.
  

  This is only realized when the population size is relatively small, so individuals do not interfare with each other.

• Logistic population growth
  
  • This model capture the fact that population size doesn't grow infinitely.

    As the population grows, the population growth become slower.

    
    

    $$\frac{dN}{dt} = f(N) = r N (1 - \frac{N}{K}) = r N - \frac{r}{K} N^2 = a_1 N - a_2 N^2$$
    
    

    K is the ``carrying capacity'' of the population.
    The 2nd order polynominal expression.
    As $a_2 \rightarrow 0$, it becomes closer to density-independent, exponential growth.

    

  • Why is there reduced population growth with high N?
    No implicit mechanism in this model.

  • More mechanistic representation:
    
    • Birth rate decrease with large N? Lack of nutrients?
      Birth rate as a function of N: $B(N) = b_1 N - b_2 N^2$
    • Death rate increase with large N? Higher transmission of disease in a dense population?
      Death rate as a function of N: $D(N) = d_1 N + d_2 N^2$
    • overall population growth
      $$\begin{eqnarray*} \frac{dN}{dt} & = & B(N) - D(N) \\ & = & (b_1 N - b_2 N^2) ... ...\ & = & (b_1 - d_1) N (1 - \frac{b_2 + d_2}{b_1 - d_1} N) \\ \end{eqnarray*}$$
      
      
      Comparison with the original logistic growth model suggests that
      $$\begin{eqnarray*} r & = & b_1 - d_1 \\ K & = & \frac{b_1 - d_1}{b_2 + d_2}\\ \end{eqnarray*}$$
      
      

    • This suggests that several different mechanisms of density dependence end up with the same logistic equation.

      Example:
      

      1. Death is density independent ( )
         r = 0.2 - 0.1 = 0.1
         K = (0.2 - 0.1)/ (0.001 + 0) = 100
      2. Birth is density independent ( $b_1 = 0.2, b_2 = 0, d_1 = 0.1, d_2 = 0.001$)
         r = 0.2 - 0.1 = 0.1
         K = (0.2 - 0.1)/ (0 + 0.001 ) = 100

  • Question: Does the mechanisms of density dependence influence the dynamics?
    • Model 1: Density-dependent birth, constant death.
      $$\begin{eqnarray*} B(N) & = & b N ( 1 - \frac{N}{k}) = b N - \frac{b}{k} N^2\\ D(N) & = & d \cdot N\\ \end{eqnarray*}$$
      
      

    • Model 2: Constant birth, density dependent death
      $$\begin{eqnarray*} B(N) &=& b N\\ D(N) &=& d \cdot N (1 + \frac{N}{\frac{d}{b} k}) = d \cdot N + \frac{b}{k} N^2 \\ \end{eqnarray*}$$
      
      

    • Relationship of the parameters

      	Birth		Death
      	density-indep.	density-dep.	density-indep.	density-dep.
      	$b_1$	$b_2$	$d_1$	$d_2$
      Model 1	b	$\frac{b}{k}$	d	0
      Model 2	b	0	d	$\frac{b}{k}$

• Implementation strategy.
  • Re-interpretation of Model 1 (per capita death and birth rate).
    • Each individual dies with the probability d after a short time
    • Each individual have a baby with the probability b ( 1 - N / k) after a short time.
  • Stochastic version