Population Models

The use of differential equations allows mathematical models for population to be created. Here are a few common ones. Qualitative Analysis is very useful here, as it shows the state of the model with certain initial populations.

Let $p$ = population at time $t$. Then $\frac{dp}{dt}$ is the rate of population growth.

Remember that both $p,t\ge 0$

Malthus (Doomsday) Model §

In the doomsday model, the rate of growth is directly proportional to a a scaling value $k$, with no other affecting factors:

$\frac{dp}{dt}\propto p⟹\frac{dp}{dt}=kp,k>0$ (Separable ODEs)

For an initial population $p_0$, the solution is:

$$p(t)=p_0\times e^{kt}$$

Derivation

If the initial population is $p_0$, it implies that $p(0)=p_0$ is an initial condition $\frac{dp}{dt}=kp;p(0)=p_0$ $\frac{1}{p}\frac{dp}{dt}=k$ (Separation Of Variables) $\int \frac{1}{p}dp=\int kdt$ $\log (|p|)=kt+c$ $\log (p_0)=0+c$ ($p(0)=p_0$) $\log (|p|)=kt+\log (p_0)$ $\log \left(\frac{p}{p_0}\right)=kt$ $\frac{p}{p_0}=e^{kt}$ $p(t)=p_0\times e^{kt}$

The doomsday model has different behaviours for different values of $k$:

• $k>0$ : unbounded exponential growth
• $k<0$ : population dies out
• $k=0$ : population stays constant

It’s pretty unrealistic, as you can tell.

With harvesting §

Remove some of the population at a constant rate.

$$\frac{dp}{dt}=kp-h,h>0.$$

Using Qualitative Analysis, we can see that:

• The equilibrium solution (i.e. when $\frac{dp}{dt}=0$ ) is $\frac{h}{k}$
• The solution is unstable

500

Logistic Model §

Inspired by Pierre-François Verhulst, the model introduces a ”competition” term, because a larger population will result in more overcrowding, diseases, etc. This competition model helps keep the population semi-stable.

$$\frac{dp}{dt}=kp-\frac{k}{a}{p}^2=kp\left(1-\frac{p}{a}\right),a>0$$

• $kp$ is the birth rate of the population, like the Malthus model
• $-\frac{k}{a}{p}^2$ is the competition term, with $a$ being the carrying capacity, which is the maximum population an area can sustainably hold.

Using Qualitative Analysis:

• The equilibria solutions are $p=0,a$
• $p=0$ is an unstable solution, while $p=a$ is a stable solution, as can be seen in the phase plot: 1000

Fun fact, the logistic model accurately described the population of USA from 1790-1950.

With harvesting §

Again, removing population at a constant rate

$$\frac{dp}{dt}=kp-\frac{k}{a}{p}^2-h=kp\left(1-\frac{p}{a}\right)-h,a>0$$

Solutions for this one aren’t as generalised, it’s better with an example.