# The Discrete Logistic

##### Model Description

The logistic is the simplest representation of population dynamics  that contains a minimum level of biological realism.  If we imagine a  population of individuals at time t, the number of individuals in the next period (periods can range from seconds to days to decades depending on the organism) is the number of individuals now plus the number of births minus the number of deaths.  If we assume in the simplest case there are no environmental constraints on births or deaths, then it is reasonable to assume that a constant proportion of living individuals give birth (i.e. fertile females) and have a total number of offspring per period.  For example, in the USA in 2010, there were 64.1 births per 1,000 women aged 15-44 years (typical fertile period).  When the proportion of women aged 15-44 years in the population is taken into account (about 20%), the birth rate per 1,000 population is 13.  The death rate is around 8 per 1,000. If we label the number of individuals in period t as x(t), we have

births = b*x(t), deaths=d*x(t)

where b and d are the per-capita birth and death rates (in the US, b = 13/1000 = 0.013, d = 8/1000 = 0.08). Thus, simply counting individuals, we have

x(t+1) = x(t) +  b*x(t) - d*x(t)

and factoring out x(t) yields

x(t+1) =  (1 +  b - d) x(t)

or, if we define the net intrinsic growth rate r'' as

r = (1 +  b - d)

then we have the simple population growth rule (computation):

x(t+1) =  r x(t).

Note, if births > deaths, then r > 1 and the population grows.  If the opposite is true, r < 1 and the population dies out (x(t) goes to 0).

In the real world, births and deaths are not constant as in the example above.  Namely, they change with population size. The simplest way to express this is that when the population goes up, individuals must compete for resources so births go down and deaths go up due to lower nutrition or outright conflict.  The absolute simplest what to capture this dynamic is it assume that the intrinsic growth rate is a decreasing function of x such that we replace r'' with r*(1-x).  It  is simple to see that as x increases, r decreases.  There are many ways to represent this mathematically, and each leads to the same qualitative behavior.

This leads to the famous logistic growth'' model presented here, i.e.

x(t+1) = rx(t)(1-x(t)).

discrete logistic phase diagram
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