If organisms reproduce and die independently of other individuals (with the exception of the finite carrying capacity), then a simple birth/death process can be simulated with a transition probability given by P ( s → → s A | s N ) = [ r b δ δ n ( s A ) , n ( s ) + 1 + r d δ δ n ( s A ) , n ( s ) − − 1 ] Θ Θ [ n ( s A ) ] Θ Θ [ n ( K − − s A ) ] where r b , r d ∈ ∈ [ 0 , 1 ] , r b + r d ≤ ≤ 1 are constant birth and death probabilities, respectively, δ δ i , j is the Kronecker delta which ensures only one birth/death event happens every time step, and Θ Θ ( x ) is the Heaviside function, which makes sure particle numbers are positive and bounded by the carrying capacity K .