Nonlinear oscillators in Emden-Fowler form and as predator-prey systems

The nonlinear pseudo-oscillator recently tackled by Gadella and Lara is mapped to an Emden-Fowler (EF) equation that is written as a predator-prey system for which we provide the phase-space analysis and the parametric solution. Through an invariant transformation we find periodic solutions to a certain class of EF equations that pass an integrability condition. We show that this condition is necessary to have periodic solutions and via the predator-prey analysis we also find the sufficient condition for periodic orbits. EF equations that do not pass integrability conditions can be made integrable via an invariant transformation which also allows us to construct periodic solutions to them. Two other nonlinear equations, a zero-frequency Ermakov equation and a positive power Emden-Fowler equation are discussed in the same context