CALCULUS FOR MODELLING 05
Population Models
Population models translate assumptions about births, deaths, migration and competition into equations for population change.
The balance principle
For a simple population, births and immigration are inflows; deaths and emigration are outflows.
1. Birth and death rates
If per-capita birth and death rates are \(b\) and \(d\), then
\[\frac{dP}{dt}=bP-dP=(b-d)P.\]Defining \(r=b-d\) gives the exponential model.
2. Density dependence
If crowding reduces net growth, a logistic form may be used:
\[\frac{dP}{dt}=rP\left(1-\frac{P}{K}\right).\]3. Immigration
A constant immigration rate \(m\) changes the equation to
\[\frac{dP}{dt}=rP+m.\]The additional term changes both the trajectory and possible equilibrium.
4. Harvesting
A constant removal rate \(h\) gives
\[\frac{dP}{dt}=rP\left(1-\frac{P}{K}\right)-h.\]If harvesting is too large, the population may fail to maintain a positive equilibrium.
5. Interacting populations
When one population affects another, a system of ODEs is needed:
\[\frac{dX}{dt}=f(X,Y),\qquad \frac{dY}{dt}=g(X,Y).\]Predator-prey and competing-species models have this form.
6. State variables and mechanisms
The modelling process is to identify the state variables, identify processes that change them, assign mathematical rates to those processes, and combine inflows and outflows.
7. Equilibria
An equilibrium population \(P^*\) satisfies
\[\boxed{\frac{dP}{dt}=0}.\]It represents a state where the model's gains and losses balance.
8. Model assumptions
Simple population models may assume homogeneous individuals, constant parameters, continuous population size and no time delays. These assumptions determine how the model should be interpreted.
9. Why calculus is useful
The derivative describes instantaneous population change, while solutions show how repeated local changes accumulate into long-term population behaviour.