Population Dynamics ODE

Wells et al. (1998) give a DE model of species population size (count) as a function of time. The authors define the following six parameters:

1.

$f \in (0,\: 1)$: instantaneous birth rate.

2.

$r \in (0, \: 1)$: instantaneous mortality rate (death rate).

3.

$c \in (0, \: 1)$: proportion of the N animals that meet over a short period of time. This implies that cN is the number of meetings over a short period of time.

4.

P: the probability that any one meeting does not result in a litter.

5.

k: carrying capacity of the environment in terms of maximum number of animals that can be supported.

6.

N₀: initial count of the species within the region. The initial time at which this count exists is t₀.

The model is  

$$\frac{dN}{dt} = f(1 - P^{cN})N - rN - (f - r)\frac{N^2}{k}.$$ (2)

If all of the parameters in (9) were known and constant for a region, that region's species count at time t could be computed exactly by solving (9).