Population Growth
Overview
Population growth models are used to understand how the number of inididuals in a population grows and/or declines
and what rate that occurs. It is applied across population types. The mathematics of population growth is
similar to the mathematics of growth in situations that do no involve populations, such as
asset prices.
Non Stochastic Population Growth
The standard equation for modeling population growth, is the differential equation:
{% N(t+1) -N(t) = r N(t) %}
{% d N(t)/dt = r N(t) %}
It states that the rate of growth of a population is proportional to the current population size. Here,
the constant of proportionality, r, is some constant.
The growth equation is equivalent to the following:
{% d log N(t)/dt = r %}
The solution to this equation is given below:
{% N(t) = N_0 e^{rt} %}
typically, we assume r to be
{% r = \beta - \mu %}
where
- {% \beta %} is the fertility rate, i.e. each individual gives rise to approx {% \beta \Delta t %} new individuals
-
{% \mu %} is the death rate, i.e. the proportion of living individuals dying in a given time segment is {% \mu %}
Adding Noise
The above equations are unrealistic in that they are determistic. The differential equations can be
generalized within the ito calculus as a
geometric brownian motion
{% d log N(t) = r dt + \sigma dW(t) %}
