Simple population models

Change in a population with continuous growth

$N_{t+1}=N_{t}+B-D+I-E$
${\Delta}N=r=B-D+I-E$

• $N_{t} =$ No. individuals at time $t$
• $B =$ No. births
• $D =$ No. deaths
• $I =$ No. immigrants
• $E =$ No. emigrants

Exponential population growth

$$\frac{dN}{dt} = rN$$ (1)

$$N_{t} = N_{0}e^{rt}$$ (2)

After Gotelli (1998)

Exponential population growth

Environmental resistance

Density dependence in limited populations

Density-dependent factors

Vary in significance with population density--e.g. predation, disease, competition

Density-independent factors

Do not vary in response to population density--e.g. weather events, volcanic eruptions, collision with an extra-terrestrial body

Logistic population growth

$$\frac{dN}{dt} = rN(1-\frac{N}{K})$$ (3)

$$N_{t}=\frac{K}{1+[\frac{(K-N_{0})}{N_{0}}]e^{-rt}}$$ (4)

After Gotelli (1998)

Logistic population growth

Oscillating population

Oscillating population

Age-structured population

$$\begin{pmatrix}n_{1}\\ n_{2}\\ n_{3}\\ {\vdots}\\ n_{x}\end{pmatr... ...trix}\begin{pmatrix}n_{1}\\ n_{2}\\ n_{3}\\ {\vdots}\\ n_{x}\end{pmatrix}_{(t)}$$ (5)

Called the Leslie matrix after Leslie (1945), see also Coulson and Godfray (2007)