hidden assumption of all of the foregoing models is that the environment is constant.  This shows up in the fact that the parameters of these models are fixed.  In reality, environments vary with the cycle of the seasons, none more drastically than those at the poles.  In the far North, the wobbling of the planet Earth as it moves about the sun has its most pronounced consequences.  Winters are long and very severe, and all the animals must adapt to them in some way.

To model the effects of a seasonally-varying environment, the simplest thing is to let one of the parameters vary periodically in time.  For example, we can replace the constant r in the Rosenzweig-MacArthur model with

a sinusioidally varying function of time.

Interestingly, the consequences of allowing seasonality are profound.  Essentially, what we have done is to couple two oscillators: one ecological, the other astrophysical.  The dynamics of coupled nonlinear oscillators has long been a subject of mathematical study and a great deal about their general properties is known.  Of special importance to our biological question are two phenomena: resonance and phase-locking.