Weak Predation and Competition Dynamics
in a Food Webchain Model
We model the food chain between a single prey, two competing predators, and one top-predator that is at the top of the food chain. The food chain is carefully constructed out of the standard Rosenzweig-MacArthur model for a tritrophic food chain. Alternatively, one could consider our full system a construction from a prey-2 predator food web.
The actions and behaviors of the system depend heavily on different initial parameters of the system, parameters which model the behaviors and attributes of the various species. We distinguish mathematically a weak predator from a stronger predator, because the weak predator produces a certain qualitative behavior in the system, which is totally different from the behavior a strong predator produces. Additionally, we can define a dominant and non-dominant predator.
We scale the system conveniently, and using singular orbit / singular perturbation analysis, we are able to prove the existence of certain orbits for the limiting condition (and slightly perturbed condition) of the full system. We demonstrate that the Competition Exclusion Principle holds for the food web, but not for the full system, as well as analyzing other facts about the food chain and food web.
Finally, for the more perturbed full system, we show some chaotic orbits and chaotic structures possible. We do not analyze all the behaviors of the full system, as it would be a massive undertaking. However, we take a sub case of the food chain called the “slow flow case”, and analyze the various possible behaviors that can occur when a predator is added to the food chain. Even from the simple case in the food chain, where there are no complex structures present, the generalized food chain provides chaos and many complex orbits.