2.1.2 Alternate equilibria
Evidence is accumulating that certain large-scale
complex systems may have alternate equilibria
and critical tipping points. Examples are the well-
known trophic cascades in freshwater lakes (Car-
penter and Kitchell 1993; Schefferet al.1993).
Again, the criticism of studies that suggest the
existence of alternate equilibria deals with the
length of the study relative to that of the genera-
tion time of the organisms involved, and the phys-
ical characteristics of the different sites at which
the species were studied (Connell and Sousa 1983).
Also more recently the possible multiple stable
states of a system have been stated to be difficult
to be proven experimentally (Scheffer and Carpen-
ter 2003; Schro ̈deret al.2005), despite several
recent new suggestions (Kefiet al.2007; van der
Heideet al.2007; Carpenteret al.2008). The transi-
tion from one state to an alternate state is nowa-
days called a ‘catastrophic regime shift’, indicating
the serious ecosystem-level implications of this
phenomenon. The major problem is that ecological
resilience cannot be measured in practice. Models
can be used as indicators of ecological resilience
(Carpenteret al. 2001), but the mechanisms of these
transitions are often poorly known (Scheffer and
Carpenter 2003). Recently, van Nes and Scheffer
(2007) and Carpenteret al. (2008) have successfully
explored various indicators of an upcoming cata-
strophic shift, such as the ‘critical slowing down’
known from physics.
2.1.3 Stable limit cycles
Apart from the predator–prey oscillations based on
the Lotka–Volterra equations which are only neu-
trally stable, the Holling–Tanner model (Holling
1965; Tanner 1975) produces a range of dynamics.
This model shows no tendency to return to the
equilibrium point, but displays another form of
predator–prey oscillations: stable limit cycles. In
the above-mentioned microbial food web study,
obvious stable limit cycles were established at low
dilution rates of the chemostat systems (Beckset al.
2005). Maxima and minima for the predator and the
two preys recurred during the whole period of the
study.
2.1.4 Chaotic dynamics
The long-term persistence of complex food webs is
not automatically linked to stability, and many
mathematical models predict that species inter-
actions can create chaos and species extinctions.
Despite receiving an overwhelming amount of the-
oretical attention, experimental demonstrations of
chaos are rare. Only a few single species systems
(Costantinoet al.1997; Ellner and Turchin 2005), the
already mentioned microbial food web study, in
which at intermediate dilution rates of the chemo-
stat systems chaotic dynamics were observed
(Beckset al.2005) as well as nitrifying bacteria in a
wastewater bioreactor (Grahamet al.2007), show
compelling evidence for chaos. Recently, it has been
shown that in a long-term experiment with a plank-
ton community, consisting of bacteria, several phy-
toplankton species, herbivorous and predatory
zooplankton species, and detritivores, chaotic dy-
namics also appear (Benincaet al.2008). The food
web showed strong fluctuations in species abun-
dances, attributed to different species interactions.
We refer to this study later. As both the structure
and the dynamics of a closed, local community are
the result of the interactions among the constituting
species, we needpopulation dynamic modelsto repre-
sent them. Such interactions may be of different
kinds, such as between predators and prey,
between competitors for the same resource, and
non-trophic interactions, e.g. through environmen-
tal modification (Olffet al.2009). In the present
chapter we will discuss the dynamics ofsmall food
web modulesand those ofcomplex interactions.2.2 Dynamics of food web modules
Insights from specific ‘few-species-interaction-
configurations’ or modules (Menge 1995; Holt
1997; Bascompte and Melian 2005) of consumer–
resource interactions have much increased over
the last few decades. For example, we know much
more now about resource competition (Schoener
1974; Tilman 1982), mutualism (Oksanen 1988), ap-
parent competition (Holt 1977), indirect mutualism
(Vandermeer 1980; Ulanowicz 1997), intra-guild
predation (Poliset al.1989), positive interactions
such as facilitation (Callaway 2007), positive26 DYNAMICS