Numerical bifurcation analysis of a tri-trophic food web with omnivory
Kooi, B.W., Kuijper, L.D.J., Boer, M.P. and Kooijman, S.A.L.M. 2002.
Numerical bifurcation analysis of a tri-trophic food web with omnivory
Math. Biosci., 177 - 178: 201 - 228.
Abstract
We study the consequences of omnivory on the dynamic
behaviour of a three species food web under chemostat conditions.
The food web consists of a prey consuming a nutrient, a predator
consuming a prey and an omnivore which preys on the predator and the
prey. For each trophic level an ordinary differential equation
describes the biomass density in the reactor. The hyperbolic
functional response for single and multi prey species figures in the
description of the trophic interactions. There are two limiting
cases where the omnivore is a specialist; a food chain where the
omnivore does not consume the prey and competition where the
omnivore does not prey on the predator. We use bifurcation analysis
to study the long-term dynamic behaviour for various degrees of
omnivory. Attractors can be equilibria, limit cycles or chaotic
behaviour depending on the control parameters of the chemostat.
Often multiple attractor occur. In this paper we will discuss
community assembly. That is, we analyze how the trophic structure
of the food web evolves following invasion where a new invader is
introduced one at the time. Generally, with an invasion, the invader
settles itself and persists with all other species, however, the
invader may also replace another species. We will show that the food
web model has a global bifurcation, being a heteroclinic connection
from a saddle equilibrium to a limit cycle of saddle type. This
global bifurcation separates regions in the bifurcation diagram with
different attractors to which the system evolves after invasion. To
investigate the consequences of omnivory we will focus on invasion
of the omnivore. This simplifies the analysis considerably, for the
end-point of the assembly sequence is then unique. A weak
interaction of the omnivore with the prey combined
with a stronger interaction with the predator seems advantageous.