Predator Prey Model

Question

Consider the following system of equations, and assume that population of prey is measured in thousands, and that the population of predators is measured in hundreds.

$$\frac{dx}{dt} = 5x(1-\frac{x}{3}) - xy$$ $$\frac{dy}{dt} = -9y + 4xy$$

(a) Explain which is the predator species and which is the prey species. Does the model assume that the predator species has anything else to eat other than the prey? Explain. (b) Determine the equilibrium points of the system, i.e. the points at which neither population will change. Explain what each equilibrium point represents physically. (c) Determine the equilibrium points of this system and classify the each one as a stable node, unstable node, or saddle. Use Polking’s software to create a phase diagram for the system.

Answer 1

(a) Explain which is the predator species and which is the prey species. Does the model assume that the predator species has anything else to eat other than the prey? Explain.

Well, in Volterra-type models predators and prey can be distinguished by how their population changes alone, without any other species. From point of view of dynamical systems the absence of one of species is ruled by setting one of the initial conditions to zero. So, let's check:

• If we set $x$ to zero, the equation for $y$ becomes $\frac{dy}{dt} = -9y$. Everybody dies.
• If we set $y$ to zero, the equation for $x$ becomes $\frac{dx}{dt} = 5x(1-\frac{x}{3})$. It's a logistic equation and behaviour of its solutions are well known: everything (except $x \equiv 0$) approaches solution $x \equiv \frac{1}{3}$.

For predators it's natural to die when there's nothing to hunt and to eat. So, $y$ corresponds to predator and $x$ to prey.

You could deduce this by writing equations in canonical form as they've written in Wikipedia (that was an advice from @AlexeyBurdin and @h-solatges).

Is this done by solving the systems of equations and finding the values of x and y, and thats the equilibrium? Given that I did that already (if its correct) what would each equilibrium mean?

Yes, if you have a system of differential equations of form $\dot{x} = F(x)$ (it's a system of autonomous differential equations, written in vector form), then all equilibrium points are found by solving the system of equations $F(x) = 0$. Basically, each equilibrium means that systems posesses a solution that doesn't change in time (that's why they are called equilibrium or steady state). In your case the coordinates of steady state tell you how many predators and prey should be presented such that their count doesn't change in time.

Usually the meaning of each equilibrium is subject of interpretation of specific model. If you have a system of ODEs, that comes from electric engineering, you should interpret the behaviour of quantities in terms of electric engineering.

From point of view of general dynamical systems, neighbourhood of steady state (and behaviour of solutions in that neighbourhod) is much more interesting and important subject to study than steady state itself. Here comes the last part of your question.

Determine the equilibrium points of this system and classify the each one as a stable node, unstable node, or saddle.

Start from here. I believe that everything here is explained quite well.