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 = population at time . Then is the rate of population growth.
Remember that both
Malthus (Doomsday) Model
In the doomsday model, the rate of growth is directly proportional to a a scaling value , with no other affecting factors:
For an initial population , the solution is:
Derivation
If the initial population is , it implies that is an initial condition (Separation Of Variables) ()
The doomsday model has different behaviours for different values of :
- : unbounded exponential growth
- : population dies out
- : population stays constant
It’s pretty unrealistic, as you can tell.
With harvesting
Remove some of the population at a constant rate.
Using Qualitative Analysis, we can see that:
- The equilibrium solution (i.e. when ) is
- The solution is unstable
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.
- is the birth rate of the population, like the Malthus model
- is the competition term, with being the carrying capacity, which is the maximum population an area can sustainably hold.
Using Qualitative Analysis:
- The equilibria solutions are
- is an unstable solution, while 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
Solutions for this one aren’t as generalised, it’s better with an example.