A Lyapunov functional for a stage-structured predator-prey model with nonlinear predation rate (Q708521)

The dynamics of the following general stage-structured predator-prey model is studied \[ x' = n(x) - \omega(x, y_2),\quad y_1' = k\omega(x, y_2) - c_1m(y_1),\quad y_2' = c_2m(y_1) - d(y_2), \] where \(x(t)\) is the density of the prey at time \(t\), \(y_1(t)\) and \(y_2(t)\) are the population densities of the predator at immature and mature stages. This model includes some well studied models when the functions \(n\), \(\omega\), \(m\) and \(d\) are given particular explicit definitions. Using the Lyapunov function method, sufficient conditions are derived for the local stability of the equilibria together with estimates of their respective domains of attraction. It is observed that these conditions also yield global stability results in some important particular situations. The biological significance of these conditions is also discussed and the positive steady state is also investigated.