The predator-prey model (or the Lotka-Volterra equations) are a system of ODEs that can be used to predict the population size of prey and predators through time.
The schematic below represents our model in the form of a flow diagram. The prey population grows in size due to births, and declines due to death by natural causes and death by predation.
The predator population size grows due to births as a result of predation, i.e. the predators eat the prey to survive. The predators can only die by natural causes – nothing eats the predators!
We can write down our assumptions from our flow diagram using words. We will write an ODE for each compartment. Therefore we will have two equations, one for prey population size and one for predator population size,
\[
\begin{aligned}
\mbox{prey population rate of change} & = \mbox{births} – \mbox{deaths}-\mbox{predation deaths} \\
\mbox{predator population rate of change} &= + \mbox{births via predation} -\mbox{deaths} \\
\end{aligned}
\]
We denote $X$ as the number of prey and $Y$ as the number of predators. Therefore the prey population rate of change with respect to time is written as $\frac{dX}{dt}$ and the the predator population rate of change is $\frac{dY}{dt}$.
We assume that the prey population gives birth at a rate of $b$ and dies from natural causes at a rate of $m$. So we can write $\alpha = b – m$ as the overall growth rate of the prey population.
The prey population also suffer from deaths via predation. The number of prey removed due to predation will depend on the number of contacts between the prey and predators $XY$, the rate of contact and the probability of successful predation given contact. We denote $\beta$ as the removal rate of prey due to predation, this parameter combines both the rate of contact and the probability of successful predation.
Putting all this together we can write the rate of removal (deaths) of prey due to predation as $\beta X Y.$
As the prey population declines form predation, the predator population will grow. The number of predators which are added (via births) will depend on the number of contacts between the prey and predators and the rate of contact as before, but now the rate of addition will not be the same as the rate of removal i.e. we cannot assume that the same number of prey that are removed are also added to the predators.
For example, if 5 prey are eaten, that does not mean that 5 predators are then born, the rate of births due to predation occurs at a different rate but due to the same number of contacts. Therefore we introduce another parameter $\delta$ which represents the growth rate of the predator population as a result of predation. Therefore our growth (births) via predation can be written as $\delta X Y.$
The predator population also declines from deaths from natural causes, which we assume occurs at a rate of $\gamma$.
We can now write down our system of ODEs representing the rate of change of he prey and predator populations:
\[
\begin{aligned}
\frac{dX}{dt} & = \alpha X – \beta XY \\
\frac{dY}{dt} &= \delta XY – \gamma Y
\end{aligned}
\]