### Consequences of population models on the dynamics of food chains

Kooi, B.W., Boer, M.P. and Kooijman, S.A.L.M. 1998.
Consequences of population models on the dynamics of food chains.
*Math. Biosci.* **1533**: 99 - 124
#### Abstract

A class of bioenergetic ecological models is studied for the dynamics
of food chains with a nutrient at the base. A constant influx rate of
the nutrient and a constant e!ux rate for all trophic levels is
assumed. Starting point is a simple model where prey is converted into
predator with a fixed e ciency. This model is extended by the
introduction of maintenance and energy reserves at all trophic levels,
with two state variables for each trophic level, biomass and reserve
energy. Then the dynamics of each population are described by two
ordinary differential equations. For all models the bifurcation
diagram for the bi-trophic food chain is simple. There are three
important regions; a region where the predator goes to extinction, a
region where there is a stable equilibrium and a region where a stable
limit cycle exists. Bifurcation diagrams for tri-trophic food chains
are more complicated. Flip bifurcation curves mark regions where
complex dynamic behaviour (higher periodic limit cycles as well as
chaotic attractors) can occur. We show numerically that Shil'nikov
homoclinic orbits to saddle-focus equilibria exists. The codimension 1
continuations of these orbits form a `skeleton' for a cascade of flip
and tangent bifurcations. The bifurcation analysis facilitates the
study of the consequences of the population model for the dynamic
behaviour of a food chain. Although the predicted transient dynamics
of a food chain may depend sensitively on the underlying model for the
populations, the global picture of the bifurcation diagram for the
dfferent models is about the same.