# Stable systems

The values we used in the examples above are all examples of a stable system. We get a nice oscillation in numbers of both predators and prey. Neither species dies out. We can plot the numbers in another way called a phase portrait. We plot the numbers of prey on the -axis with the numbers of predators on the -axis. This gives us the following phase portrait.

Each of our previous examples (coloured by the initial number of prey) is included, along with the equilibrium point for the stable system. You can see that the individual phase plots are ordered by the initial value of N but they all oscillate around the equilibrium point, which in this case is N=20, P=5.

This equilibrium value is dependent not on the values of N, or P but the values of a, b, c and d. For the system to be stable we must have $\frac{\text{dN}}{\text{dN}}=0$ and $\frac{\text{dP}}{\text{dt}}=0$. This gives us two equations to solve.

$0=\text{aN}-\text{bNP}$
$0=\text{cNP}-\text{dP}$

This has two solutions for (N,P):(0,0) or $\left(\frac{\text{d}}{\text{c}},\frac{\text{a}}{\text{b}}\right)$. Our equilbrium comes from the second of these and you can check the values for yourself.

### Exercise

Attempt the implement the Lotka-Volterra system numerically. It may help to modify some of the code we used last class.

In class we will see how to implement the system numerically (and we will compare methods with those you produce) and extend the model.

## Information

This resource was created as part of the Educational Development Unit's ASP2 project, developing resources for the Applied Science degree at the University of the Highlands and Islands.

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