Over any given interval of time, the size of a population increases by the number of individuals added to the population by births and by immigration (the movement of individuals into the population from elsewhere) and decreases by the number of individuals lost from the population by deaths and by emigration (individuals leaving the population to go elsewhere). This relationship is expressed mathematically as
Nt = N0 + (B – D) + (I – E) (54.1)
where Nt = the population size at time t, N0 = the population size at time 0, B = the number of individuals born between time 0 and time t, D = the number that died between time 0 and time t, I = the number that immigrated between time 0 and time t, and E = the number that emigrated between time 0 and time t.
As you saw earlier, population size rarely remains the same but instead changes over time. We can use Equation 54.1 to estimate how population size changes over the interval of time 0 to t and get a population growth rate, or the rate of change in population size over time. This can be expressed mathematically as
ΔN = (B – D) + (I – E), or the change in N (ΔN) for the time interval 0 to t
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To understand population growth rate from births and deaths alone, ecologists often assume a “closed system” that does not include immigration and emigration. This simplifies the equation to
ΔN = (B – D) (54.2)
The change in population size can then be calculated if one knows the number of births (B) and deaths (D) that have occurred over a given period of time. The birth and death numbers will naturally depend on the size of the population and the time interval considered. As a result we need to convert B and D into rates that will allow us to estimate population growth for any change in population size over time. Therefore we can further express B and D as
B = bN0
where B is the product of the per capita birth rate (b) (i.e., the number of births per individual per unit of time) and N0 (population size at time 0). Likewise, the death rate can be calculated as
D = dN0
where D is the product of the per capita death rate (d) (i.e., the number of deaths per individual per unit of time) and N0 (population size at time 0).
The term “per capita” literally translates to “per head” and is meant to indicate births and deaths per individual in the population. The change in the total number of births and deaths can then be calculated by multiplying the per capita birth and death rates by the population size at N0. Here’s what that looks like mathematically:
ΔN = bN0 – dN0 or ΔN = (b – d)N0
The difference between the per capita birth and death rate (b – d) is the per capita growth rate and is symbolized by r. Substituting r for (b – d), we get
ΔN = rN0 (54.3)
This simple model reflects why populations change in size. A population will increase in size if the per capita birth rate exceeds its per capita death rate; that is, if b > d, then r > 0. Likewise, the population will decline in size if b < d, or r < 0. If b = d, then r = 0, and the population will not change in size.
Ecologists use differential calculus to express the change in population size (ΔN) over very short, instantaneous periods of time (dN/dt). In this way, they are assuming that population growth is continuous, meaning the time interval is infinitely small and the growth curve will be smooth. This is expressed by
dN/dt = rN0 (54.4)
Once we have a value for r, and we know population size, we can use it to characterize patterns of growth in populations through time. One pattern of growth that occurs when populations are not limited by resources is exponential growth, which we discuss next.