### Consequences of symbiosis for food web dynamics

Kooi, B.W., Kuijper, L.D.J. and Kooijman, S.A.L.M. 2004.
Consequences of symbiosis for food web dynamics.
*Journal of Mathematical Biology* **3**: 227 - 271
#### Abstract

Basic Lotka-Volterra type models in which mutualism (a
type of symbiosis where the two populations benefit both) is taken
into account, may give unbounded solutions. We exclude such
behaviour using explicit mass balances and study the consequences
of symbiosis for the long-term dynamic behaviour of a three
species system, two prey and one predator species in the
chemostat. We compose a theoretical food web where a predator
feeds on two prey species that have a symbiotic relationships. In
addition to a species-specific resource, the two prey populations
consume the products of the partner population as well. In turn,
a common predator forages on these prey populations. The temporal
change in the biomass and the nutrient densities in the reactor is
described by ordinary differential equations ({\sc ode}). Since
products are recycled, the dynamics of these abiotic materials
must be taken into account as well, and they are described by {\sc
ode}s in a similar way as the abiotic nutrients. We use
numerical bifurcation analysis to assess the long-term dynamic
behaviour for varying degrees of symbiosis. Attractors can be
equilibria, limit cycles and chaotic attractors depending on the
control parameters of the chemostat reactor. These control
parameters that can be experimentally manipulated are the nutrient
density of the inflow medium and the dilution rate. Bifurcation
diagrams for the three species web with a facultative symbiotic
association between the two prey populations, are similar to that
of a bi-trophic food chain; nutrient enrichment leads to
oscillatory behaviour. Predation combined with obligatory
symbiotic prey-interactions has a stabilizing effect, that is,
there is stable coexistence in a larger part of the parameter
space than for a bi-trophic food chain. However, combined with a
large growth rate of the predator, the food web can persist only
in a relatively small region of the parameter space. Then, two
zero-pair bifurcation points are the organizing centers. In each
of these points, in addition to a tangent, transcritical and Hopf
bifurcation a global heteroclinic bifurcation is emanating. This
heteroclinic cycle connects two saddle equilibria where the
predator is absent. Under parameter variation the period of the
stable limit cycle goes to infinity and the cycle tends to the
heteroclinic cycle. At this global bifurcation point this cycle
breaks and the boundary of the basin of attraction disappears
abruptly because the separatrix disappears together with the
cycle. As a result, it becomes possible that a stable
two-nutrient--two-prey population system becomes unstable by
invasion of a predator and eventually the predator goes extinct
together with the two prey populations, that is, the complete food
web is destroyed. This is a form of over-exploitation by the
predator population of the two symbiotic prey populations. When
obligatory symbiotic prey-interactions are modelled with Liebig's
minimum law, where growth is limited by the most limiting
resource, more complicated types of bifurcations are found. This
results from the fact that the Jacobian matrix changes
discontinuously with respect to a varying parameter when another
resource becomes most limiting.