Population Growth Models

Only four processes change population size: births and immigration (individuals moving in) add; deaths and emigration (individuals moving out) subtract. A population that exchanges individuals with others is open, and all four terms apply. A closed population has no migration, such as bacteria in a sealed flask or animals on an isolated island, so only births and deaths matter: $\Delta N=B-D$. The raw change $\Delta N$ is hard to compare between populations of different size, so ecologists use the per-capita growth rate (cgr), the change per individual over the time interval. A gain of 60 in a population of 2400 is a per-capita rate of $\frac{60}{2400}=0.025$, or 2.5% per year. The growth rate without the per-capita correction is simply $\frac{\Delta N}{\Delta t}$, the number added per unit time.

When resources are unlimited, each individual keeps reproducing at the same per-capita rate, so a bigger population adds more individuals each time. There are two versions of this idea. Geometric growth describes species that breed in discrete seasons, such as many insects, annual plants and seals that pup once a year. Ecologists compare the size in one breeding season with the previous one: the geometric rate of increase $\lambda$ (lambda) $=\frac{N_{t+1}}{N_t}$. If $\lambda >1$ the population grows, if $\lambda =1$ it is stable, and if $\lambda <1$ it declines. Exponential growth describes species that reproduce continuously, such as bacteria and humans. Its rate is the intrinsic rate of increase $r$ (per-capita births minus deaths per unit time), and growth follows $\frac{dN}{dt}=rN$. Both give the same J-shaped curve. A discrete version written with a percentage rate is also common: $N_t=N_0{(1+r)}^t$, where $\lambda =1+r$. Always check which form a question uses.

A population growing exponentially has a constant doubling time $t_d$. Setting $N_t=2N_0$ in $N_t=N_0{e}^{rt}$ gives $e^{r{t}_d}=2$, so $t_d=\frac{\ln 2}{r}\approx \frac{0.693}{r}$. Multiplying top and bottom by 100 turns $r$ into a percentage and gives the rule of 70: $t_d\approx \frac{70}{\text{growth rate in \%}}$. A population growing at 2% per year doubles in about 35 years; one growing at 7% per year doubles in about 10 years. The rule is an approximation that works best for small rates, and it shows why even modest steady growth becomes enormous: ten doublings multiply a population by $2^{10}=1024$.

No environment can support exponential growth for long. As a population grows, each individual gets less food, water, space and shelter, waste builds up and disease spreads more easily. The carrying capacity ($K$) is the maximum population size an environment can sustain indefinitely with its available resources. The logistic model adds a braking term to exponential growth: the factor $\frac{K-N}{K}$ is the fraction of carrying capacity still unused. When $N$ is tiny, that fraction is close to 1 and growth is nearly exponential at the maximum rate $r_{max}$. When $N$ approaches $K$, the fraction approaches 0 and growth stops. The result is an S-shaped (sigmoid) curve with a lag phase (few breeders, slow increase), a log or exponential phase (rapid increase), a deceleration phase as resistance builds, and a stationary phase where births balance deaths at about $K$. Because $\frac{dN}{dt}$ is the product of $N$ (growing) and $\frac{K-N}{K}$ (shrinking), the population adds individuals fastest at $N=\frac{K}{2}$.

Real populations rarely settle exactly on $K$. Resource limits act with a time lag: animals conceived in a year of plenty are born after food has already run short. A population may therefore overshoot $K$ and then suffer a dieback, as the St. Matthew Island reindeer did. If the overshoot damages the habitat, $K$ itself falls and the population may never recover to its former size. Other populations oscillate above and below $K$ in regular population cycles, giving a sinusoidal (wave-like) curve; predator–prey interactions are a common cause. So the three curve shapes to recognize are exponential (J), sigmoid (S) and sinusoidal (cycling about $K$). What stops growth falls into two groups. Density-dependent factors have a stronger effect per individual as density rises, so they are the forces that hold a population near $K$. Density-independent factors kill the same proportion regardless of density; they cause sudden crashes but do not regulate population size.