# Population growth equation

Mahlon G. Kelly mgk at darwin.clas.Virginia.EDU
Mon Sep 18 01:48:36 EST 1995

```jackf90000 at aol.com  writes:
> The "basic" exponential growth equation is the solution to the
> differential equation:
>      dx/dt = k x(t)
>  which says the rate of change of x(t) is proportional to x(t), where x(t)
> is taken to be the population size at time t.
>    The solution  is:
>      x(t) = k1 exp(-k2 t)

Sorry, but that is wrong. The integration gives
x(t) = x(0) exp(kt)

>      the constants k1 & k2 depend on K an the initial population size at
> time t=0.

x(0) IS the initial population size at t=0. No k1 or k2 exist.

The equation is more usually written
dN/dt = (b-d)N

where N is population density, b is natality (number born per
individual per time) and d is mortality (number dying per
individual per time). Of course this assumes no emigration or
immigration.

The integrated form is
N(t) = N(0) e^(b-d)t
Which, of course, is the same as the equation I gave above,
with b-d = k.

To get the doubling time, just let N(t) = 2N(0) and take the
log of both sides, thus
2=e^(b-d)D
where D is doubling time, and thus
ln(2) = (b-d)D
or
D = ln(2)/(b-d)

>  For more on population, get the book "[I can't remember the title,
> something like"Introduction to Population Genetics"] by Crow & Kimura.
>
> For the binary fission problem, Suppose that at the population size
> doubles every generation:
>     No
>     N(1) = 2* No
>     N(2) = 2* N(1) = 2*2*No
>     N(k) = 2**k * No
>
> Does this help?
>
> Jack
--
Associate Professor (Emeritus)
University of Virginia
mgk at darwin.clas.virginia.edu

```