Supplementary Information C. Density-dependence and recovery times
Comparing population recovery after insecticide exposure for four aquatic invertebrate
species using models of different complexity. J. M. (Hans) Baveco, Steve Norman, Ivo
Roessink, Nika Galic and Paul J. Van den Brink
We test whether our finding that the intensity of density dependence does not affect the time
to recovery in the logistic model also holds for our relatively simple IBM run for a pond
system with uniform exposure. In the pond, with its relatively small dimensions, spatial
processes are likely to meet the condition of perfect mixing.
To test the impact of intensity of density dependence, simulations are performed with
different values for DD of 0.001 (lowDD), 0.0025 (mediumDD) and 0.005 (highDD). Density
dependence acts through local density in each of the cells making up the landscape. The times
series for exposed population density divided by control population density are calculated. In
addition, the time series for the simulated dynamics are compared to the predicted dynamics
applying the logistic equation.
Figure SC1. Time series of simulated dynamics and predicted dynamics from the logistic
equation, for the 3 intensities of density dependence and local density dependence in the IBM.
Results
The dynamics of the Asellus, Gammarus and Chironomus IBM are similar to the dynamics
predicted from the logistic equation (Fig. SC1). For the mayfly, however, with more intense
density-dependent feedback, simulated population growth is slower and equilibrium level
lower than predicted. There are other small deviations, like the oscillations in Chironomus and
the mayfly, that are possibly caused by starting population not being in exact stable agestructure and the high reproductive capacity of both species (Table 1).
When comparing exposed and control dynamics (Fig. SC2) for Asellus, Gammarus and
Chironomus there is no impact of density dependence on recovery. For the mayfly, it shows
that there is not only a difference between IBM and logistic equation, but also between IBM
simulations with different density dependence. In other words, we find that for the mayfly
density dependence affects the time to recovery: with stronger density dependence recovery is
(much) slower.
An explanation may be the locally-defined density dependence in the IBM, interacting with
the spatial processes, leading to a relatively stronger impact of density dependence at low
densities (larger effective intensity of density dependence). In particular this might be related
to the clumped reproduction of the mayfly (all offspring of a female produced at the same
location), that, together with the immobility of the instar stages, result in persisting
heterogeneous spatial patterns – but only at the low density (high intensity) case. If this
explanation holds, we would expect 1) to observe similar dynamics with local density
dependence for Chironomus, but at higher intensity 2) to find the impact of density
dependence on recovery to disappear with density dependence acting through global density
instead of local density.
Figure SC2. Time series of (simulated) exposed population size divided by control population
size, for the 3 intensities of density dependence and local density dependence in the IBM.
To test this explanation, we ran the same simulations 1) with higher intensities (DD of 0.0075
and 0.01) of local density dependence and 2) with global density dependence: individual
mortality risk now does not depend on the density in an individual’s local cell, but on a
system-averaged density.
With higher intensities of local density dependence the outcome is unchanged for Asellus and
Gammarus (not shown). For Chironomus indeed the same phenomenon occurs as for the
mayfly: with higher intensity, density dependence starts to have an impact on recovery and
recovery times (Fig. SC3).
Figure SC3. Top: time series of simulated (IBM) and predicted (logistic equation) dynamics,
for high intensities of density dependence, local density dependence in the IBM, for
Chironomus and the mayfly. Bottom: Time series of (simulated) exposed population size
divided by control population size..
The new simulations with density dependence based on global density show that for all
species the logistic equation captures the density dynamics after stress (Fig. SC4). Also the
plots of exposed / control densities (Fig. SC5) show that for all species recovery time and
dynamics are completely independent of the strength of density dependence.
Figure SC4. Time series of simulated dynamics and predicted dynamics from the logistic
equation, for the 3 intensities of density dependence and with global density dependence in
the IBM.
Figure SC5. Time series of (simulated) exposed population size divided by control population
size, for the 3 intensities of density dependence and global density dependence in the IBM.
The main insights we obtain from these comparisons is that recovery dynamics in the IBM are
similar to the dynamics predicted from the non-spatial logistic model, when density
dependence in the IBM is defined as acting through global density. This is however violating
one of the main assumptions (and advantages) of IBM, that of local interactions.
When density dependence acts through local density, the recovery dynamics in the IBM can
still be identical to the dynamics predicted by the logistic equation, if there are no
mechanisms incorporated in the model that create persisting spatial patterns (heterogeneity) in
density (or if we remain outside the range of coefficient values for which these mechanisms
may create these patterns). In our model, clumped reproduction (all offspring in same cell)
combined with immobility of instars constituted such a mechanism. In general, it may not be
clear whether a model incorporates such a mechanism and coefficients are in the sensitive
range. Therefore, the safest approach seems to be to always test the model for impact of
density dependence on recovery dynamics, when density dependence acts through local
density. It may still be possible to use the intensity of density dependence to scale densities in
the IBM to (computationally) feasible levels; however, the validity of the underlying
assumption that this leaves recovery times unaffected should always be ascertained.