Spatial Population Models for Animals - Environmental Modelling: Finding Simplicity in Complexity - page 214

climate and land-use change has become more acute
(Thomas et al ., 2004). Many empirical modelling tools
have been used to predict species distribution (species
distribution models - SDMs); amongst the commonest
have been climate envelope models or generalized linear
models (GLMs) often with stepwise model selection
(although stepwise model selection is now much
criticized - Whittingham et al ., 2006). The 2000s saw
ecologists adopt new statistical tools for predicting species
distributions, including Bayesian methods, techniques
adopted from machine-learning, ensemble forecasting
methods, and information theoretic methods of model
selection (Elith and Leathwick, 2009). Recently new
methods have been developed that allow SDMs to be
built solely on the basis of presence/occurrence (without
absence) data such as those commonly available from
taxonomic collections and museums (Elith et al ., 2006).
In a management context, the results from SDMs are
increasingly being coupled with conservation planning
tools that optimize habitat conservation and restoration
programs in both terrestrial and aquatic ecosystems (e.g.
Moilanen et al ., 2008; Thomson et al ., 2009). Of course,
static models of distribution can only tell us where a
given species may be found either now or at some other
point in time. Exactly how individuals will get there is
often as important a question, especially in fragmented
landscapes where there may be significant spatial barriers
to range shifts. Although spatial autocorrelation and
landscape context can be factored into SDMs (e.g.
see Thomson et al ., 2009), it is often seen as more
appropriate to adopt dynamic modelling approaches in
cases where dispersal is expected to play a critical role in
species responses to landscape change.

to computer simulation). Here we will outline three
broad approaches (loosely following Hastings, 1994)
adopted by ecologists seeking tomodel the spatial dynam-
ics of ecological systems: (i) reaction-diffusion systems,
(ii) metapopulation (patch-occupancy) approaches, and
(iii) individual-based models.

1. Reaction-diffusion systems
Reaction-diffusion systems model the growth and inter-
action of populations (the 'reaction' component) cou-
pled with a representation of dispersal (the 'diffusion'
component - Kareiva, 1990; Hastings, 1994). Models of
this type have a long tradition of use in population ecol-
ogy (Skellam, 1951), and originate in attempts to model
the spread of mutations through populations in genetics
(Fisher, 1937). In continuous space-time, they are of the
general form (for one species in one-dimension):
∂

D ( x ) ∂

N ( x , t )
∂ t = ∂

N

+ N · f ( N , x ) (13.1)

∂ x ∂ x

where: N ( x , t ) is the density function for the population
size at location x at time t , D ( x ) is the diffusion coefficient
(dispersal rate), and the function f ( N , x )isthe per capita
rate of increase of the population at x .
A widely used example of a continuous form of a
reaction-diffusion model is the Fisher equation, which
has been frequently used to model invasion processes
(Hastings et al ., 2005b):

rN ( x , t ) 1

2 N ( x , t )
∂ x 2

∂ N ( x , t )
∂ t =

D ∂

N ( x , t )
K

+ − (13.2)

where: r the rate of population increase, and K is the

environment's carrying capacity.
In the Fisher equation, the reaction component is the
logistic (Verhulst) model (the rightmost term). The pop-
ulation will spread in a wave at a rate governed by r
and D . Reaction-diffusion systems are mathematically
tractable and provide a rich framework in which the
implication of changes in dispersal dynamics and popu-
lation growth and interactions may be analysed. Andow
et al . (1990) used the example of muskrats ( Ondatra
zibethicus ) spreading after their release in Prague in 1905
to highlight some of the strengths of the reaction-diffusion
approach. Using data for location and timing of first sight-
ings after release, Andow et al . created a time series of
maps showing the muskrats' range and compared it with
the theoretical mean squared displacement of the wave

=

13.2.3 Dynamicapproaches

Dynamic models are those in which temporal trajecto-
ries of change, rather than simply the long-term average
outcomes are modelled, frequently in a way that incor-
porates spatial and temporal feedbacks (autocorrelation).
The models used by ecologists range from very abstract
'strategic' models, through to highly-detailed site- and
scale-specific 'tactical' models ( sensu May, 1974). Within
this broad spectrum, modelling occurs with different
purposes (predictive vs. exploratory aims), at different
scales (from the dynamics of individuals in populations
to ecosystem fluxes across entire continents) and using
different methods (from classical analytical approaches