Stochastic model

• Per capita birth rate (p) and death rate (q)
  • Model 1: Density-dependent birth, constant death.
    $$\begin{eqnarray*} p & = & b ( 1 - \frac{N}{k}) \\ q & = & d \\ \end{eqnarray*}$$
    
    

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

• Per capita rate near the carrying capacity K.
  
  $$K=\frac{b_1 - d_1}{b_2+d_2} = \frac{b -d}{b/k} = \frac{b-d}{b} \cdot k$$
  
  
  • Model 1

    current N		per capita birth rate		per capita death rate
    N = K		$p = b ( 1 - \frac{1}{k} \cdot \frac{b-d}{b} \cdot k) = d$		q = d
    N=K+1		$p = b ( 1 - \frac{1}{k} \cdot (\frac{b-d}{b} \cdot k + 1)) = d - \frac{k}{b}$		q = d
    N=K-1		$p = d + \frac{b}{k}$		q = d

  • Model 2

    current N		per capita birth rate		per capita death rate
    N = K		p = b		q = b
    N=K+1		p = b		q = b + b / k
    N=K-1		p = b		q = b - b/k

• Graphical representation