growth rate

[drifter god]

can someone give me an equation to find a populations growth rate(% per year) using these variables. 1) pop. size 2) life expectancy 3)births per person (ie: for every couple, 3 kids born or something) 4) parents age at time of birth. and anything else needed that im not thinking of

if anyone can come up with an equation i would be most grateful

[Cyborg Stan]

I think it would be around 100% * r^(1/g), where r is the ratio of kids per a single person, and g is the time between generations in years. This is just off the top of my head, so feel free to correct me.

[drifter god]

hey wow, i think that actually works, although i am guessing these numbers are right, but hey, thanks a lot

if someone can verify if this equation is accurate, that would be cool

[Kuroneko]

can someone give me an equation to find a populations growth rate(% per year) using these variables. 1) pop. size 2) life expectancy 3)births per person (ie: for every couple, 3 kids born or something) 4) parents age at time of birth. and anything else needed that im not thinking of

A model that depends on population size N and a constant growth rate r per unit time (which is a combination of birth rate and death rate) is simple: dN/dt = rN, so at time t, N(t) = N(0) e^(rt), N(0) being the initial population.

The main problem with the above is that the growth-rate rarely stays constant, so it's accuracy is very poor in the long-term. The population is usually limited by several factors... living space, food, whatever. If it depends linearly on a single constant factor (e.g. the population size is mainly driven by the food supply, which for some reason is essentially constant), so the growth rate is dependent on the current population, r(N) = a - bN. This leads to the logistic equation dN/dt = N(a - bN), which is separable and thus easily solvable: N(t) = a/(b + (a - bN(0))/N(0) e^(-at)).

The above could also be revised in the following way: if growing population consume more food than those not (even at the same population size), then the model becomes dN/dt = ( aN(F - bN) )/(1 + acN), where F is the food supply (or some other resource), and the food consumption is determined by bN + c dN/dt. This equation is also separable, so little difficulty (but I'm too lazy to do the algebra again).

If you want an age-dependent model... say, you have m+1 age groups of equal temporal length t, where m is some age beyond which the fraction of the population is negligible, and you want to compute the population after some time kt. Suppose you also have the birth rate contributions b0, b1, ..., bm and their surivorship rates s0, s1, ..., sm-1 per time step t.

For example: if t = 1 year, s0 is the fraction of newborns (age 0) that survive to their first year, and b0 = 0 since newborns don't breed. A larger time step would make it more tractable--if t = 13 years, s0 is the fraction of newborns that survive to 13 years, etc., but you lose some of the accuracy.

This has a relatively simple solution for the kth step: N(kt) = A^k N(0), where N is the m+1 column vector of the number in each age group, A is the matrix with b's for first row and s's for the subdiagonal (one below the diagonal). You can eigen-decompose A to make the matrix exponentiation simple.