Predator-prey problem

In the typical predator-prey problem, there is a prey that would multiply without predators after the logistic growth. This means that initially the population grows exponentially as more animals also reproduce more. However, since there is not an unlimited amount of habitat or food available, there is a limit to this growth. The reproduction therefore also depends on the limit population.

The increase in prey (hares) is therefore determined by the typical difference equation

RabbitIncrease:=(Limit-Rabbits)*Rabbits*Factor

described.

In the graphic model:

The predators (foxes) also reproduce after the logistic growth. However, their limit population depends on the food, i.e. on the hares present. So we only need to replace the limit population in the above equation:

FoxesIncrease:=(Factor*Rabbits-Foxes)*Foxes*Factor

In the graphical model, instead of the constant limit state, an action arrow of the state variable "Rabbits" linked with a constant factor must be set.

In the end, however, the prey also decreases, the more predators and the more prey there is. It is therefore valid:

RabbitsDecrease:=Rabbits*Foxes*Factor

Now the model is complete.

The choice of the initial values and especially the factors influences the behaviour of the system considerably.

Assume the following initial values:

Limit population of rabbits: 1000

Initial population of rabbits: 200

Growth factor of rabbits: 0.001

Initial population of foxes:10

Ratio of limit fox population and rabbits population: 0.25

Growth factor of foxes: 0.0025

Decrease factor of rabbits: 0.04

By changing the values, you can basically obtain different results.

Plot the rabbit and fox population over time (scale foxes if necessary) and also the fox population over the hare population (phase diagram).