How seasonal forcing influences the complexity of a predator-prey system

Almost all population communities are strongly influenced by their seasonally varying living environments. We investigate the influence of seasons on populations via a periodically forced predator-prey system with a nonmonotonic functional response. We study four seasonality mechanisms via a continuation technique. When the natural death rate is periodically varied, we get six different bifurcation diagrams corresponding to different bifurcation cases of the unforced system. If the carrying capacity is periodic, two different bifurcation diagrams are obtained. Here we cannot get a 'universal diagram' like the one in the periodically forced system with monotonic Holling type Ⅱ functional response; that is, the two elementary seasonality mechanisms have different effects on the population. When both the natural death rate and the carrying capacity are forced with two different seasonality mechanisms, the phenomena that arise are to some extent different. The bifurcation results also show that each seasonality mechanism can display complex dynamics such as multiple attractors including stable cycles of different periods, quasi-periodic solutions, chaos, switching between these attractors and catastrophic transitions. In addition, we give some orbits in phase space and corresponding Poincaré sections to illustrate different attractors.