On Liénard's equation and the uniqueness of limit cycles in predator-prey systems
AbstractWe study a system of ODE'S modelling the interaction of one predator and one prey 
$$\begin{gathered} \frac{{dx}}{{dt}} = xg\left( x \right) - yp\left( x \right), \hfill \\ \frac{{dy}}{{dt}} = \gamma y\left[ { - \delta - \eta y - \alpha y^2 + h\left( x \right)} \right]. \hfill \\ \end{gathered}$$
 This system defines a two-species community which incorporates competition among prey in the absence of any predators as well as a density-dependent predator specific death rate. This system is investigated under ecologically natural regularity conditions and assumptions on g, p and h to ensure the existence and uniqueness of limit cycles. The proof uses the standard Hopf-Andronov bifurcation theory and the technique of Liénard's equation.