= Solution
Let $X,Y$ be <prey> and <predator> abundances. With constant positive demographic and <predation> coefficients, the <Lotka-Volterra equations> are
$$
\dot X=X(\alpha-\beta Y),\qquad
\dot Y=Y(\delta X-\gamma).
$$
Their positive <equilibrium> is $X_* =\gamma/\delta$, $Y_* =\alpha/\beta$. Its <Jacobian matrix> and <eigenvalues> are
$$
\boxed{J_*=
\begin{pmatrix}0&-\beta\gamma/\delta\\\delta\alpha/\beta&0\end{pmatrix},
\qquad \lambda_\pm=\pm i\sqrt{\alpha\gamma}.}
$$
The linearized populations oscillate with period $2\pi/\sqrt{\alpha\gamma}$, with the <predator> peak following the <prey> peak. This is a <center equilibrium>, not an attracting <equilibrium>. The conclusion is supported beyond <linearization> by the <Logarithmic first integral of the Lotka-Volterra equations>: in these units one may use
$$
V=\delta X-\gamma\log X+\beta Y-\alpha\log Y.
$$
Direct differentiation gives $\dot V=0$. Its <Hessian matrix> is a <positive-definite matrix> in the <positive quadrant>, and its minimum is at $(X_*,Y_*)$. Nearby level curves are closed <periodic orbits>, so there is no damping toward a preferred oscillation amplitude.
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