Role of seasonality on predator–prey–subsidy population dynamics

Abstract

The role of seasonality on predator-prey interactions in the presence of a resource subsidy is examined using a system of non-autonomous ordinary differential equations (ODEs). The problem is motivated by the Arctic, inhabited by the ecological system of arctic foxes (predator), lemmings (prey), and seal carrion (subsidy). We construct two nonlinear, nonautonomous systems of ODEs named the Primary Model, and the n-Patch Model. The Primary Model considers spatial factors implicitly, and the n-Patch Model considers space explicitly as a "Stepping Stone" system. We establish the boundedness of the dynamics, as well as the necessity of sufficiently nutritional food for the survival of the predator. We investigate the importance of including the resource subsidy explicitly in the model, and the importance of accounting for predator mortality during migration. We find a variety of non-equilibrium dynamics for both systems, obtaining both limit cycles and chaotic oscillations. We were then able to discuss relevant implications for biologically interesting predator-prey systems including subsidy under seasonal effects. Notably, we can observe the extinction or persistence of a species when the corresponding autonomous system might predict the opposite.

Dynamics from a predator-prey-quarry-resource-scavenger model

“…We compare our results with the ones as found by Nevai and Van Gorder for their autonomous PPS model (Nevai and Van Gorder 2012) and with the ones as found by Levy et al (2016) for their PPS model with oscillating subsidy input rate. We will also discuss the difference between the PPQRS model and apparent competition models, which we will find to look very similar to our system.…”

“…These standard values are chosen because they are similar to the values as used in Nevai and Van Gorder (2012) and Levy et al (2016), so we can make easy comparisons with their PPS models. When applied to a real biological system, either an Arctic system as considered throughout this paper, or any other system where allochthonous resources are important, their values should be adjusted accordingly.…”

“…Therefore, these dynamics might easily have been overlooked. Regarding less regular non-equilibrium dynamics, note that Rinaldi et al (1993) and Levy et al (2016) found chaotic behaviour in non-autonomous predator prey models. , μ = 0.9 and ρ = 0.5 (bottom left), μ = 0.9 and ρ = 1 (top right) and μ = 0.9 and ρ = 2 (bottom right).…”

“…We use the same approach as Levy et al (2016) (who studied a non-autonomous generalization of the PPS model in order to understand the role of seasonality in modifying predator-prey-subsidy dynamics) and replace the constants δ, ρ, β, r and γ by the following time-dependent parameters:…”

“…Note that if we take ρ 1 = 0, we get back our autonomous rate of growth (Turchin and Hanski 1997). We have chosen for a simple sinusoidal form of our oscillations, following (Levy et al 2016) and (Turchin and Hanski 1997). We see that ρ(t) has a minimum at t = 1 4 τ .…”