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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 $\hat{x}=1$ , $\hat{y}=1$ ?

Thank you in advance.

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In such nonlinear systems, the behaviour of trajectories around a fixed (or equilibrium) point is determined by a process called Linear Stability Analysis (LS). In it, we compute the Jacobian matrix for the system at each fixed point, and then determine its eigenvalues. These tell us the nature of the fixed point.

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.

Note however one technicality. As LS essentially involves a linear approximation to the nonlinear equations centered at each fixed point (which is what computing the Jacobian matrix is equivalent to), if the fixed point is not structurally stable, the effect of the nonlinear terms may destroy the fixed point predicted by LS. (Note the usage of 'may'. It may, or may not happen. We need more information to be sure.)

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).

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