Spatial Population Models for Animals - Environmental Modelling: Finding Simplicity in Complexity - page 217

Box 1 - Does space make a difference?
Building spatial models is expensive - spatial data are difficult
to collect and spatial models are likely to suffer more from
uncertainty in representation and in parameterization than
their non-spatial counterparts. Spatial models are also more
difficult to evaluate. 'Realism', while desirable on the face of it,
comes at a cost! The tradeoff between tractability and realism is
at the heart of most model-building activity. Here we will
explore how the inclusion of space in a simple model can
change the predictions it makes. Comparison of spatial and
non-spatial models is of growing interest as a way of
distinguishing the specific effects of space and spatial pattern
on ecological processes (see Dieckman et al ., 2000 and
references therein).
Two key assumptions of Levins' patch model
(Equations 13.3 and 13.4) are that new colonists are available
globally and are dispersed randomly. To examine the
implications of this assumption Tilman et al . (1997) built a
stochastic cellular automata representation of the model in
which dispersal occurs locally within a neighbourhood of
dimension d . Although the cellular automata representation is
not mathematically tractable, it is directly comparable to
Levins' model and allows us to explore what happens when we
explicitly include space in Levins' model. An interactive
version of this model is available at: www.creative-
current.com/george/wandm-ed2/spatialLevins.html (also
linked from www.kcl.ac.uk/envmod).

spatial is that the mean level of occupation is lower than is
predicted by the deterministic model (Bascompte, 2003). This
difference is because colonists arriving at occupied sites are
lost, which does not occur in Levin's spatially implicit model.
We can also build a spatial representation of Levins' model
as extended by Tilman et al . (1994) to include habitat loss. A
version of this model is available at: www.creative-current.
com/George/wandm-ed2/spatialLevinsRestoration.html (also
linked from www.kcl.ac.uk/envmod).
In this form of the model we consider (following Huxel and
Hastings, 1999) the loss of habitat followed by four different
restoration strategies. In the absence of habitat loss the model
behaves as outlined above (see Figures B13.1 and B13.2).
However, as more and more habitat is lost patch occupancy
drops commensurately. On the restoration of habitat there is a
recovery in patch occupancy at a rate which varies with which
of the four restoration strategies is used:

random - degraded patches are selected at random to be
restored to the empty state;

•

random reintroduction - degraded patches are selected

at random to be restored to the occupied state;

+

•

spatial - degraded patches are selected at random to be
restored to the empty, with the constraint that at least one
patch in their dispersal neighbourhood is occupied;

•

reverse - degraded patches are restored to the empty state
in the reverse order from which they were degraded.

•

0.6

1.0

Equilibrium solution ( p * = 1 − m/c)

0.5 Analytical solution

0.8

r = 3

0.4

r = 2 Rev

Rnd
Adj
Rnd + Re

0.6

0.3

0.4

0.2

0.1

0.2

r = 1

0.0

0.0

0 50 100 150 200 250

0 100 200 300 400 500

Time ( t )

Time ( t )

Figure B13.1 Comparison of the analytical spatially implicit
(black) form of Levins' metapopulation model with Tilman
et al .'s spatially-explicit (grey) representation ( r

Figure B13.2 The outcome of four different restoration
strategies (10 replicates of each) following habitat loss, from
most to least effective (grey-scaled dark to light): random +
reintroduction, adjacent, random and reverse. Note that: (i) the
'adjacent' strategy is only slightly slower to restore patch
occupancy to pre-habitat loss levels than the 'random

distance
colonists can disperse from their parent patch); in all cases, c

=

=

0.16 and m 5) and the spatial simulation was

run 10 times for each level of r . Note the dramatic reduction in
patch occupancy that occurs when dispersal is strongly
constrained.

= 0 . 08 (so p * = 0 .

+
reintroduction' strategy and (ii) patch occupancy continues to
decline after habitat loss ceases (the 'extinction debt' - Tilman
et al ., 1994). In all cases, c

0.1 and habitat is lost at a
rate of 1% per year for a period of 75 years.

= 0.5, m =

Making Levins' classic metapopulation model spatially
explicit has two important results. First, sites occupied by
individuals are not randomly distributed in space but are
aggregated; this is an outcome of localized dispersal occurring
around occupied sites (a result more thoroughly described by
Durrett and Levin, 1994). A second effect of making the model

Unsurprisingly the metapopulation recovers most quickly
when the 'random + reintroduction' strategy is used.
However, the spatially sensitive approach of restoring sites