# Solution of the Lotka Volterra system of differential equations in $\mathbb{R^2}$ is cyclic

I'm currently reading an abstract about the Lotka Volterra equations and I've some questions. Please be aware I'm an amateur.

First I will give you the system of differential equations:

$$x^{'} = x - xy \mbox{ } \mbox{ } ; \mbox{ } x(0) = \hat{x}$$ $$y^{'} = -y + xy \mbox{ } \mbox{ } ; \mbox{ } y(0) = \hat{y}$$

where $$y$$ pradator species , $$x$$ prey.

Now I've read that the solution of this is cyclic for all inital values $$\hat{x}, \hat{y}$$ in the first quarter plane and I don't know why. Here I need your help. Moreover I've read that the cycles are around the equilibrium point $$\hat{x}=1$$ , $$\hat{y}=1$$. I know that this has something has to do with setting the time derivates on the left side of the equations above equal to zero. But how do I find out now that the cycles are around $$\hat{x}=1$$ , $$\hat{y}=1$$ ?

In our case, the eigenvalues of $$J$$ at $$(1,1)$$ are $$\pm i$$. As they are purely imaginary, the fixed point is a centre - that is, all trajectories in its neighborhood are closed i.e. orbits or cycles exist about $$(1,1)$$. Thus, LS predicts $$(1,1)$$ to be a centre fixed point.
Now, fixed points like saddles, nodes and spirals are structurally stable - the predictions made by LS is what is actually shown by the nonlinear system. However, unluckily for us, centres are not structurally stable. Thus, we cannot be sure that $$(1,1)$$ is indeed a centre. We need more information.
This information is got by the following fact about this system of equations - there exists a conserved quantity! This is significant, as a centre predicted by LS in a system with a conserved quantity is indeed a centre even on adding the nonlinear terms. (The conserved quantity is $$E(x,y) = \ln x - x + \ln y - y$$) (A derivation is here - Determine a conserved quantity in a dynamical system Lotka-Volterra)
Hence, the trajectories about $$(1,1)$$ are closed (or cyclic).