Open Collections

Hewa Thudallage, Sanuri

University of British Columbia

Bylot Island, in the Canadian Arctic, hosts a predator-prey system that includes snowy owls, arctic foxes, and lemmings. The latter are prey for both the snowy owls and foxes. Snowy owls reside temporarily on the island, their presence or absence depending on the availability of the prey only. Hence, unlike many other predator-prey systems, the composition of the food web (one or two predators) depends on the internal system dynamics. We begin with an ordinary differential equation model, that we reduce to a simple onespecies discrete map. This simplification is possible because we assume that the fox and owl population on Bylot Island remains constant because of a fixed number of territories and nests. The simplified model allows us to prove several key aspects of solution behaviour. We investigate the asymptotic behaviour of solutions. We also fully characterise the cyclic and pseudo-cyclic dynamics in the lemming-arctic fox-snowy owl (LFO) system. Next, since arctic regions exhibit drastic seasonal changes, we extend the model to account for some behavioural changes. Specifically, sufficient lemmings are also a necessity for arctic foxes to breed. As in the previous case, we now obtain a model with two thresholds. The second threshold represents the lemming density above which arctic foxes breed. We observe and characterise more complex behaviour in pseudo-cyclic dynamics relative to our first model. We also discover bistable behaviour: long-term dynamics vary between two- or three-cycle solutions depending on the initial conditions. The availability of lemmings determines the presence or absence of snowy owls and the breeding behaviour of arctic foxes in our model. We implement these seasonal changes as thresholds. Using these threshold values and lemming growth rates, we explain the necessary conditions to observe cyclic dynamics in this predator prey system. Moreover, we demonstrate that our simple mathematical model can exhibit complex behaviour that nonetheless contains defined structure. Specifically, we present upper bounds on the number of consecutive increases or decreases in a positive solution of the LFO model based on parameter values. This qualitative behaviour differs from both overcompensation cycles and consumer-resource cycles.

Text

2024-09-04

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/89206

Mathematics

University of British Columbia

UBCO

http://creativecommons.org/licenses/by-nc-nd/4.0/