Linking Specialisation and Stability of Plant ... - OPUS Würzburg

4.3 the model 67
rium, measures the rate of return to the equilibrium following
an infinitely small perturbation (Otto & Day, 2007). The second
measure, which is likewise related to the concept of an equilibrium
state, quantifies the size of the "domain of attraction"
around an equilibrium. This is the collectivity of all initial conditions
from which a certain equilibrium state is reached after
an arbitrarily long time period. Hence, this second stability
measure does not only account for a system’s behaviour in the
immediate vicinity of the equilibrium state, but also considers
the effects of larger perturbations. On the other hand, unlike
the resilience criterion, the domain of attraction does not incorporate
information on the time required for recovery following
a disturbance. Both factors are accounted for by the third stability
measure ("persistence criterion") that considers the starting
conditions allowing all species to persist above a threshold
density within a certain ecologically relevant time span. In contrast
to the first two measures, this criterion does not only consider
the long-term (equilibrium) behaviour of a system, but
also its transient dynamics after a perturbation. For real ecological
systems that are subject to frequent disturbances from a
variety of sources, transients can often be more important than
the stability of an equilibrium state that may never be reached
(Hastings, 2004). However, many conservation projects are targeted
at preserving certain ecological functions rather than individual
species (e.g. Moonen & Barberi, 2008; Sutherland et al.,
2010). Therefore, the fourth stability measure ("ecosystem function
criterion") refers to the preservation of the ecological function
"pollination" (or another mutualistic service provided by
animals) for the plant community and thus quantifies the fraction
of starting conditions that allow persistence of both plants
and at least one animal species.
4.3 the model
Here, we briefly introduce the model equations and their main
underlying assumptions. See Benadi et al. (2012b) for a more
detailed description of this model and its stability properties.
Note that unlike in Benadi et al. (2012b), here we assume that
both animals and plants are obligate mutualists, i.e., they cannot
reproduce without the services offered by their mutualistic
partners.
Our model describes the dynamics of a plant community
comprising m species and a community of animals with n
species, where P i indicates the population density of the ith
plant species and A j the density of the jth animal species. Each