704

Geographic profiling as a novel spatial tool for targeting the control of invasive species

Mark D. Stevenson, D. Kim Rossmo, Robert J. Knell and Steven C. Le Comber

M. D. Stevenson (s.c.lecomber@qmul.ac.uk), R. J. Knell and S. C. Le Comber, School of Biological and Chemical Sciences, Queen Mary, Univ. of London, London E1 4NS, UK. – D. K. Rossmo, Center for Geospatial Intelligence and Investigation, Dept of Criminal Justice, Texas State Univ., 601 Univ. Drive, San Marcos, TX 78666, USA.

Geographic profiling (GP) was originally developed as a statistical tool in criminology, where it uses the spatial locations of linked crimes (for example murder, rape or arson) to identify areas that are most likely to include the o!ender’s residence. "e technique has been successful in this field, and is now widely used by police forces and investigative agencies around the world. Here, we show that this novel technique can also be used to identify source populations of invasive species, using their current locations as input, as a prelude to targeted control measures. Our study has two main parts. In the first, we use computer simulations to compare GP to other simple measures of spatial central tendency (centre of minimum dis-tance, spatial mean, spatial median), as well as to a more sophisticated single parameter kernel density model. GP performs significantly better than any of these other approaches. In the second part of the study, we analyse historical data from the Biological Records Centre (BRC) for 53 invasive species in Great Britain, ranging from marine invertebrates to woody trees, and from a wide variety of habitats (including littoral habitats, woodland and man-made habitats). For 52 of these 53 data sets, GP outperforms spatial mean, spatial median and centre of minimum distance as a search strategy, particularly as the number of sources (or potential sources) increases. We analyse one of these data sets, for Heracleum mantegazzianum, in more detail, and show that GP also outperforms the kernel density model. Finally, we compare fitted parameter values between di!erent species, groups and habitat types, with a view to identifying general values that might be used for novel invasions where data are lacking. We suggest that geographic profiling could potentially form a useful component of inte-grated control strategies relating to a wide variety of invasive species.

Invasive species are now viewed as the second most important driver of world biodiversity loss behind habitat destruction and have been identified as a significant compo-nent of global change (Vitousek et al. 1996, Wilcove et al. 1998). "e cost of invasive species can run from millions to billions of dollars per occurrence (Mooney and Drake 1986, Pimentel et al. 2000). Invasive species have been shown to a!ect native species through predation and com-petition, by modifying ecosystem functions, by altering the abiotic environment and by spreading pathogens (Strayer et al. 2006, Ricciardi and Cohen 2007). For these reasons, prevention and control of invasive species has been identified as a priority for conservation organisations and government wildlife and agriculture ministries globally (Mooney and Drake 1986, Hulme 2006).

Geographic profiling (GP) is a spatial modelling tool originally developed in criminology to help prioritise large lists of suspects in cases of serial crime. Typically, such cases involve too many, rather than too few, suspects; for example, the investigation of the Yorkshire Ripper murders in the UK between 1975 and 1980 generated 268 000 names (Doney 1990). In criminology, GP uses spatial data concerning

the locations of connected crime sites to create a probability surface that is overlaid on the study area to produce a geoprofile. Geoprofiles do not provide an exact location for the criminal’s home, but rather allow the police to pri-oritise investigations by systematically checking suspects associated with locations in descending order of the height of these locations on the geoprofile, facilitating an opti-mal search process based on decreasing probability density (Rossmo 1993, 2000).

In essence, the model has two components. First, there is a distance decay function, such that the probability of a crime drops with increasing distance from the criminal’s anchor point (usually a home or workplace). Second, there is an area, the bu!er zone, surrounding the criminal’s home in which crimes are less likely (note that the bu!er zone radius can be set to zero). "e distance-decay function arises because travel – for either human criminals or inva-sive species – involves costs. "e bu!er zone, in crimino-logy, arises partly because criminals may avoid locations too near their home, because of increased risk of being iden-tified, and partly because plane geometry dictates that, if potential crime sites are randomly distributed, the number

Ecography 35: 704–715, 2012 doi: 10.1111/j.1600-0587.2011.07292.x

© 2012 "e Authors. Ecography © 2012 Nordic Society Oikos Subject Editor: Pedro Peres-Neto. Accepted 9 December 2011

705

of potential crime sites increases with the distance from the home. In biology, there is also ecological evidence for the existence of a bu!er zone, for example in trees and in bee species (Dramstad 1996, Saville et al. 1997, Singh et al. 2001).

Geographic profiling has been successfully applied in criminology, and it is now widely used by police forces around the world. Recently, it has been applied to biological data (Le Comber et al. 2006, 2011, Martin et al. 2009, Raine et al. 2009). Here, we extend this work to consider invasive species. In this analysis, sites colonised by the species in ques-tion are analogous to crime sites, while the source or sources of the invasion are analogous to the criminal’s home.

If geographic profiling is to provide an e!ective method for locating source populations of invasive species, it first of all needs to be demonstrated that it can do better than other methods of prioritizing searches. In addition, since it is likely that many or even most invasions will concern species where data are, at least initially, lacking, it would clearly be helpful if general model values could be used for di!erent groups of taxa (for example, what values of the bu!er zone radius are most appropriate for woody trees, or marine invertebrates?). In this study, we address both of these issues. Specifically, we ask 1) can geographic profil-ing be used to locate sources of invasive species? 2) If so, is GP more e#cient than other simple approaches such as spatial mean, spatial median or centre of minimum dis-tance, or a more complex single parameter kernel density model? 3) Is it possible to use general parameter values for di!erent species or groups or for species occupying di!erent types of habitats in cases where data may be lacking? 4) Do model variables alter over time, specifically in the earlier or later stages of invasions?

Methods

General approach

Our general approach was to fit the GP model parameters using species locations at time t in such a way that they optimally predict species locations at time t–1, and then validate the model by using the same fitted parameters to test whether the locations at time t–1 predict the locations at time t–2. In this way, model fitting and model testing are independent. In this case, the time steps chosen were decades (e.g. 1960–1969, 1970–1979). For each of the spe-cies analysed, three decades were chosen (times t, t–1 and t–2). "e decades selected were not the same for each species, since decades where there were large data sets were preferen-tially chosen; early datasets, where data might be expected to be unreliable and susceptible to di!erences in sampling e!ort, were also avoided.

Geographic profiling model

A full description of the model can be found in Rossmo (2000). Here, we use a slight variant, introduced by Le Comber et al. (2006), which uses Euclidean rather than Manhattan distances (this approach was chosen as there is no reason that invasive species should move in the restricted pattern defined by urban (particularly North American) street layouts). "e geographic profiling func-tion generates a prioritised surface that describes the opti-mal search pattern for the sources of invasive species. For each point (i, j) within the target area, the score function (p) is calculated:

Sums probability across all ‘crimesites’ (invasion locations)

Turns first term off inside the bufferzone radius, and on outside

Uses parameter f to specify distancedecay moving outwards from the bufferzone radius. This reflects the fact that

dispersal probability declines withdistance

pij = k

Uses f and g to specify the increase in dispersalprobability moving away from the source,

reaching a maximum value at a distance equal tothe radius of the buffer zone. This reflects the

reduced probability of dispersal within thebuffer zone

Turns first term on inside thebuffer zone radius, and off outside

c

n=1 (xi – xn)2 + (yj – yn)2f

(1 – ) (Bg–f )+

(xi – xn)2 + (yj – yn)2g2B –

706

where

(xi – xn)2 + (yj – yn)2 B = 1

Sets phi to 1 if ‘crime site’ is outside the radius of thebuffer zone

and

(xi – xn)2 + (yj – yn)2 B = 0

Sets phi to 0 if ‘crime site’ is inside the radius of thebuffer zone

such that phi functions as a switch that is set to 0 for sites within the bu!er zone, and 1 for sites outside the bu!er zone. k is an empirically determined constant, B is the radius of the bu!er zone, C is the number of sightings of the invasive organism f and g are parameters that con-trol the shape of the decay function, (xi, yj) are the coordi-nates of point (i, j) and (xn, yn) are the coordinates of the nth site. "us, pij describes the likelihood that the anchor point occurs at point (i, j), given the crime site locations (Rossmo 2000). "e equation describes a two-part curve, which when plotted in three dimensions resembles the caldera of a volcano. When summed these ‘volcano’ shaped decay functions produce a surface that describes an e#cient search pattern for the location of species invasions. "e model was implemented using the R sta-tistical package ‘gp’ (Stevenson 2011, R development core team 2008).

Model fitting

To validate the model, we took advantage of the temporal resolution of data at the Biological Records Centre (Spatial data, below) to split species data into decades (e.g. 1960–1969, 1970–1979, 1980–1989). We then fitted the model by selecting the value of B that best predicted the locations of the species in question in (for example) 1970–1979, using as input the locations from 1980–1989. "is was done using a maximum likelihood approach with quasi-newton optimisation (Nocedal and Wright 1999) to fit B, and leaving f and g fixed at 1.2. "ese are the values typi-cally used in criminology, and previous studies have shown that the model is much more sensitive to changes in B than to changes in f and g (Le Comber et al. 2006, Raine et al. 2009). "is also helps to avoid the problem of overfitting. Overfitting (discussed by O’Leary (2009) in the context of geographic profiling) can result in a model that essen-tially reduces to a series of point estimates that perfectly predict the data used to fit the model, but that have little or no predictive power when applied to other data sets. A further advantage, in the context of this study, is that this allows the model to be constrained to a single parameter

allowing direct comparison with a single parameter kernel density model.

"e model’s degree of fit can be calculated using the hit score percentage (HS%), the proportion of the area covering the crimes (in this case, the locations of invasive species) in which the o!ender’s base (or the source of the invasive species) is located; in criminology, this is usually the area bounding the crimes, plus a ‘guard rail’ of 10% surrounding this. "e HS% is calculated by dividing the ranked score (pij) by the total search area and multiplying this result by 100. "e smaller the HS%, the more accu-rate the geoprofile; a hit score of 50% is what would be expected from a nonprioritized (i.e. random or uniform) search (Rossmo 2000). In our analysis, unlike criminology, there are multiple sources, so we calculated the mean hit score percentage across all locations. "is ‘learning hit score’ is reported in Table 1.

Model testing

To test the model’s performance, we used the fitted para-meters to test the model’s predictions on an earlier time step; in the example above, we would feed the locations in 1970–1979 back into the model, and calculate the hit score percentages of the locations in the previous time step (1960–1969) (the test hit score in Table 1), thus ensuring that the learning and test hit score percentages were independent.

Other measures of spatial central tendency

GP was compared to other simple measures of spatial ten-dency commonly used in both biology and criminology. "e two most basic approaches are to calculate the spatial mean (or centroid) and the spatial median. "e spatial mean is the point where the coordinates are the mean of the x and y coordinates. "e equation is shown below:

x n x y n yii

n

ii

n1 11 1

,

where xi and yi are the co-ordinates of the invasive species presence and n is the total number of locations where the species is present. "e spatial median is the median point of the x and y co-ordinates. Using the same notation as above it is calculated as follows:

x y y y12

x x 112 1n

2,n

2n2

n2

A slightly more complex approach is to use the centre of minimum distance (CMD). "e CMD is the location at which the sum of the distances to all other points is mini-mized and is described by the following equation:

W x y x y x yi

n

i i( , ) (( , ), , ))dist (1

W is the distance from each location of an invasive species (xi, yi) to the chosen point (x–, y–). W is then minimised using an iterative algorithm. "e function of W is a convex hull

707

Tabl

e 1.

Ani

mal

and

pla

nt s

peci

es u

sed

in th

is a

naly

sis,

with

Eng

lish

Nat

ure

cate

gory

list

ings

and

EU

NIS

hab

itat d

ata.

EU

NIS

hab

itats

are

as

follo

ws:

A1

litto

ral r

ock

and

othe

r har

d su

bstra

te; C

3 lit

tora

l zo

ne o

f inl

and

surfa

ce w

ater

way

s; G

1 br

oadl

eave

d de

cidu

ous

woo

dlan

d; G

4 m

ixed

dec

iduo

us a

nd c

onife

rous

woo

dlan

d; I1

ara

ble

land

and

mar

ket g

arde

ns; I

2 cu

ltiva

ted

area

s of

gar

dens

and

par

ks;

J2 lo

w d

ensi

ty b

uild

ings

; J4

trans

port

netw

orks

and

oth

er c

onst

ruct

ed h

ard-

surfa

ced

area

s; N

A n

ot a

pplic

able

. ‘Ti

me

step

’ sho

ws

the

data

use

d to

fit t

he m

odel

, prio

r to

test

ing.

B is

the

fitte

d va

lue

of th

e bu

ffer

zone

rad

ius.

The

lear

ning

hit

scor

e is

sho

wn,

alo

ng w

ith th

e te

st a

nd, f

or c

ompa

rison

, hit

scor

es fo

r th

e ce

ntre

of m

inim

um d

ista

nce

(CM

D),

spat

ial m

edia

n (S

SM) a

nd s

patia

l mea

n (S

MM

). Fo

r ea

ch s

peci

es a

naly

sed,

the

mos

t effe

ctiv

e of

thes

e fo

ur s

earc

h st

rate

gies

is s

how

n as

a s

hade

d ce

ll; in

52

of 5

3 (9

8%) c

ases

, GP

out-p

erfo

rmed

all

thre

e ot

her s

trate

gies

. Dat

a fro

m E

nglis

h N

atur

e au

dit

of n

on-n

ativ

e sp

ecie

s (H

ill e

t al.

2005

).

Spec

ies

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on n

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ish

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ure

listin

g ca

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ry

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tego

ry

M

inor

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egor

y

D

omin

ant

habi

tat

Ti

me

step

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lear

ning

hi

t sco

re

G

P te

st

hit

scor

e

C

MD

te

st h

it sc

ore

SS

M

test

hit

scor

e

SM

M

test

hit

scor

e

Ach

eta

dom

estic

usH

ouse

cric

ket

55an

imal

sin

sect

sN

A19

71–1

980;

19

81–1

990

0.38

7 00

E-69

0.25

0.52

50.

581

0.57

9

Agl

ossa

pin

guin

alis

Larg

e ta

bby

53an

imal

sin

sect

sI2

1981

–199

0;

1991

–200

00.

425

29E-

060.

230.

471

0.65

40.

557

Chr

ysal

ina

amer

ican

aRo

sem

ary

beet

le52

anim

als

inse

cts

NA

1991

–200

0;

2001

–201

00.

353

72E-

240.

140.

423

0.83

00.

431

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rano

palp

us ra

mos

usH

arve

stm

an44

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als

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rtebr

ate

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inse

ctI2

1981

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0;

1991

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00.

443

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940.

120.

335

0.46

90.

373

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ella

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n-in

sect

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20

01–2

010

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6 60

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0.22

0.48

50.

563

0.51

4

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naria

agr

estis

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o sp

ider

44an

imal

sin

verte

brat

e no

n-in

sect

I219

91–2

000;

20

01–2

010

0.39

1 35

E-71

0.26

0.51

90.

600

0.50

8

Cor

ophi

um s

exto

nae

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mar

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mar

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and

es

tuar

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inve

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ates

A1

1981

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0;

1991

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00.

361

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270.

240.

466

0.58

80.

578

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ssos

trea

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oys

ter o

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pane

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mar

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estu

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verte

brat

esA

119

81–1

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19

91–2

000

0.6

4 88

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0.16

0.39

40.

843

0.47

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ata

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mon

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1981

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193

96E-

630.

180.

467

0.48

00.

470

Elm

iniu

s m

odes

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acle

14m

arin

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arin

e an

d es

tuar

ine

inve

rtebr

ates

A1

1971

–198

0;

1981

–199

00.

557

20E-

410.

140.

373

0.44

80.

391

Petr

icol

a ph

olad

iform

isFa

lse

ange

l win

g

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eric

an P

iddo

ck13

mar

ine

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ine

and

estu

arin

e in

verte

brat

esA

119

81–1

990;

19

91–2

000

0.56

1 70

E-14

0.21

0.44

60.

562

0.52

3

Stye

la c

lava

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ked

or le

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ry s

ea

squi

rt18

mar

ine

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ine

and

es

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ates

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1981

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0;

1991

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00.

131

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220.

210.

449

0.53

70.

481

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natifi

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0.18

0.45

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545

0.52

2

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tano

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way

map

le77

plan

tsva

scul

ar p

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sG

119

51–1

960;

19

61–1

970

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1 35

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0.25

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573

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3

(Con

tinue

d)

708

Tabl

e 1.

(C

ontin

ued)

.

Spec

ies

C

omm

on n

ame

Engl

ish

Nat

ure

listin

g ca

tego

ry

M

ajor

ca

tego

ry

M

inor

cat

egor

y

D

omin

ant

habi

tat

Ti

me

step

B

GP

lear

ning

hi

t sco

re

G

P te

st

hit

scor

e

C

MD

te

st h

it sc

ore

SS

M

test

hit

scor

e

SM

M

test

hit

scor

e

Cry

ptom

eria

japo

nica

Japa

nese

red

ceda

r76

plan

tsva

scul

ar p

lant

sG

419

51–1

960;

19

61–1

970

0.29

4 71

E-21

0.11

0.74

30.

920

0.85

3

Erin

us a

lpin

usFa

iry fo

xglo

ve77

plan

tsva

scul

ar p

lant

sJ2

1971

–198

0;

1981

–199

00.

648

68E-

130.

140.

419

0.48

30.

398

Fallo

pia

japo

nica

Japa

nese

kno

twee

d77

plan

tsva

scul

ar p

lant

sC

319

31–1

940;

19

41–1

950

0.38

5 29

E-09

0.19

0.41

80.

576

0.49

1

Fuch

sia

mag

ella

nica

Fuch

sia

77pl

ants

vasc

ular

pla

nts

I119

31–1

940;

19

41–1

950

0.42

9 98

E-45

0.14

0.35

40.

755

0.41

1

Fum

aria

reut

eri

Mar

tin’s

ram

ping

- fu

mito

ry77

plan

tsva

scul

ar p

lant

sI1

1981

–199

0;

1991

–200

00.

541

34E-

580.

120.

330

0.46

50.

458

Gal

anth

us n

ival

is77

plan

tsva

scul

ar p

lant

sG

119

31–1

940;

19

41–1

950

0.26

4 14

E-17

0.19

0.44

10.

549

0.48

3

Ger

aniu

m d

isse

ctum

Cut

-leav

ed c

rane

’s-

bill

77pl

ants

vasc

ular

pla

nts

I119

31–1

940;

19

41–1

950

0.71

7 68

E-25

0.19

0.40

10.

510

0.50

3

Her

acle

um

man

tega

zzia

num

Gia

nt h

ogw

eed

77pl

ants

vasc

ular

pla

nts

J419

31–1

940;

19

41–1

950

0.77

00.

130.

401

0.42

70.

411

Jugl

ans

regi

aC

omm

on w

alnu

t, Pe

rsia

n w

alnu

t or

Engl

ish

wal

nut

77pl

ants

vasc

ular

pla

nts

G1

1971

–198

0;

1981

–199

00.

236

65E-

710.

130.

401

0.52

00.

423

Kick

xia

spur

iaRo

undl

eaf c

ance

rwor

t77

plan

tsva

scul

ar p

lant

sI1

1951

–196

0;

1961

–197

00.

37

83E-

420.

110.

385

0.48

40.

377

Lact

uca

serr

iola

Pric

kly

lettu

ce77

plan

tsva

scul

ar p

lant

sJ4

1931

–194

0;

1941

–195

00.

353

45E-

110.

160.

123

0.21

10.

234

Larix

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ants

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ular

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710

and as such has a unique minimum. We used the algorithm developed by Weiszfeld (1936):

where the initial values of x and y may be taken as any reason-able value (the centroid for example). "e method has been shown to always converge on the CMD, yet the length of time taken for this to occur is an unknown function depen-dant on the size and distribution of the data (Kuhn and Kuenne 1962). We ran the method for k 100 to ensure its convergence.

"ese methods were then compared to GP by searching outwards from the central point in concentric bands. Using this search strategy, a figure comparable to the hit score could be calculated based on a directed search originating out from these measures of spatial central tendency.

Kernel density method

In addition to the simple methods described above, we also compared GP to a kernel density method of the type that underlies gravity models. In this study, we adapted the kernel density method for estimating home range size described by Worton (1989) which underlies the backcast-ing application in MacIsaac et al. (2004). "is kernel den-sity approach uses a fixed kernel based on the summation of unimodal bivariate normal probability density functions. Whilst similar in some ways to the geographic profiling model, this approach uses a single parameter normal distribu-tion in place of the geographic profiling function. "e single parameter bivariate normal kernel is given by the following equation (note that the notation used in this equation, but not the form, have been modified to make the comparison with the GP model more explicit):

Pnh

x x y yhij

i n

i

ni n1 1

2 22 21

. ( ) ( )exp

where pij is the probability of each point being a source. (xi, yi) and (xn, yn) are the same as in the geographic profiling model. h is the only free parameter in the model and can be used to smooth out or concentrate the surface; it is referred to as the smoothing parameter. "is parameter may be fitted in much the same way as the geographic profiling model’s parameter B as discussed above.

Simulations

Simulated data were also created to test and compare the GP model alongside real species invasions. "e simulations were created using the statistical modelling environment R (R development core team), and consisted of a 100 by 100 grid in which sources were uniformly distributed in the central 36 36 region (the constraint was necessary to avoid edge e!ects). From each source a normal distribution of spread

( , )(( , ),( , ))((

( ) ( )( ) ( )

x yx x y x y

xk k i i i

k kin

1 1 1

1/dist/dist ii i

k kin

i i ik k

in

y x yy x y x y

, ),( , )),

(( , ),( , ))( ) ( )

( ) ( )

1

1 /dist111 /dist(( , ),( , ))( ) ( )x y x yi i

k kin

points was simulated with a standard deviation of 5. "e simulations tested every possible combination of 1, 2 and

5 sources and 10, 20 and 30 spread points. Each variation was replicated 1000 times and the hit score for each model calculated to test how well they identified the source loca-tions. Where there were multiple sources, the mean hit score of all of the individual sources was used.

Two sets of simulations were conducted. In the first, GP was tested against the spatial mean, spatial median and centre of minimum distance. In this set of simulations (as in most real-world examples), the dispersal parameters are unknown, and GP uses half the mean nearest-neighbour distance between points to estimate B. In the second set of simula-tions, GP was compared to the kernel density method. "e kernel density method uses as an input h, which corresponds to the standard deviation of the normal distribution used to simulate dispersal, and in this set of simulations both h and B were set to the true standard deviation.

Spatial data

Fifty three British invasive species were chosen that had extensive invasion histories as recorded in the R662 audit of non-native species of England, produced by Natural England (Hill et al. 2005). "e R662 audit was a desk-based survey that collated all of the standard British resources and checked these against up-to-date studies. Data on the locations and date of presence (taken as the centroid within 10 km2 grid) were obtained from the Biological Records Centre (BRC) database downloaded from NBN gateway ( www.nbn.org.uk ). "e BRC database contains datasets from hundreds of contributors across the U.K., ranging from government bodies, NGOs and scientific surveys. "ere is significant variation in the quality of the data collected, but all species chosen had multiple records collected from di!erent organi-sations. A full list of the species chosen, years analysed and results obtained for each species can be found in Table 1.

Taxonomic differences

Di!erent species have very di!erent reproductive habits and dispersal traits, and this is likely to be reflected in dif-ferences in the radius of the bu!er zone (parameter B) in our model. To test for di!erences in B between species belonging to di!erent taxa, we examined the distribution of B values in major and minor taxonomic functional groups as defined by English Nature’s Audit of non-native species (Hill et al. 2005).

Habitat differences

Di!erent species have preferred habitat types and the abundance of suitable habitats, and their distribution, may a!ect the rates at which species are able to spread. In

711

sources number of spread points: F1,17992 0.1, p 0.32; model type number of sources number of spread points: F1,17992 0.4, p 0.55). Tukey post-hoc tests showed that GP performed significantly better than all other methods except when the number of sources was 1, when the kernel density model performed as well as GP (Table 2b).

Model performance

Across each of the 53 species examined, GP’s mean hit score percentage was lower than the 50% that characterises a random search (mean SD: 18% 18.4%; log trans-formation: t 37.338, DF 52, p 2.2e-16). GP also out-performed the spatial mean, spatial median and centre of minimum distance in 52 out of 53 datasets. GP had a mean hit score percentage of 18.4% across all datasets (SD 6%) compared to 58.1% (SD 14%) for the spatial mean, 48.7% (SD 9%) for the spatial median and 43.4% (SD 9%) for the CMD. ANOVA reveals significant di!erences between these (F3,50 100.01, p 2.2e-16) (Table 1).

We also selected one dataset, Heracleum mantegazzianum, for which there are good data, for more detailed analysis, examining the distribution of hit scores for individual sites, as well as considering the average across all sites (Fig. 2). For 35 of 51 1941–1950 sources the hit score percentage was below 10%. "e mean hit score percentage for all the sources was 13% and never exceeded 24% of the searchable area. We also ran a kernel density model on the Heracleum mantegazzianum data set, to compare its performance to that of GP. For 18 of 1941–1950 sources the hit score percentage was below 10%. "e mean hit score was 31% and reached a maximum of 65% of the search area. "is result was sig-nificantly worse than that of GP (Wilcoxon rank sum test: W 382, n 51, p 8.052e-10).

Taxonomic differences

"ere were no significant di!erences in fitted B values across the English Nature major groups animals, marine, plants (mean SD: animals: 0.43 0.08; marine 0.56 0.48; plants 0.42 0.18; log transformation: F4,49 0.60, p 0.44), although we noted that the variance for the cat-egory ‘animals’ was lower than in the other two categories. However, when we looked within the category plants to compare categories 76 (woody stemmed conifers) and 77 (deciduous flowering plants), we did find significantly di!er-ent values of B (median values: category 76: 0.17; category 77: 0.16; Wilcoxon rank sum test: W 238, n 11, 30, p 0.005) (Fig. 3).

Habitat differences

When we fitted the model to di!erent EUNIS habitat listings, we found significant di!erences in fitted values of B (log transformation: ANOVA F4,46 3.26, p 0.02), although this was driven largely by di!erences between cat-egories G (woodland or forest) and J (industrial, constructed or artificial habitats) (Fig. 4).

its current formulation, GP makes no attempt to include habitat information, but it is still possible to check whether there is an association between the values of B produced by species living in di!erent habitats. We examined "e European Nature Information System (EUNIS) habitat types for all 53 of the species analysed to test for patterns in B values associated with habitat di!erences. "e EUNIS habitat classification is a pan-European system, which was developed between 1996 and 2001 by the European Environment Agency (EEA) in collaboration with experts from throughout Europe, and covers all types of natural and artificial habitats, both aquatic and terrestrial (Davies et al. 2004).

Temporal differences

Invasions can span decades or even centuries. In this area the biologist has an advantage over the criminologist, since data can be divided into di!erent temporal units, allow-ing examination of changes in model parameters as inva-sions progress. Invasions can change over time, and the notion of an invasion delay, followed by an explosion of expansion, is well established (Crooks and Soulé 1999). We selected three species with extensive invasion histories and repeated our analysis over multiple decades. In this way we could determine if the fit of our model alters over the course of an invasion history and could also compare species to determine if they had consistently di!erent B values across time.

Results

Simulations

"e mean hit scores for each combination of conditions used in the simulation are presented in Table 2 and Fig. 1. "e results were analysed using a full factorial ANOVA with model type, number of sources and number of spread points as factors, and all interactions included. "e first test compared GP to spatial mean, spatial median and centre of minimum distance. "e results showed significant di!erences between model type, but not between other factors or interactions (model type: F3,35984 75386.2, p 2e-16; number of sources: F1,35984 3.6, p 0.056; number of spread points: F1,35984 1.4, p 0.23; model type number of sources: F3,35984 0.5, p 0.66; model type number of spread points: F3,35984 0.1, p 0.94; number of sources number of spread points: F1,35984 0.0, p 0.89; model type number of sources number of spread points: F3,35984 2.1, p 0.1). Tukey post-hoc tests showed that GP performed significantly better than all other methods in each case (Table 2a). "e second test compared GP to the kernel density method. Again, the results showed significant di!erences between the mod-els (model type: F1,17992 81216.9, p 2e-16; number of sources: F1,17992 0.1, p 0.75; number of spread points: F1,17992 2.3, p 0.13; model type number of sources: F1,17992 0.2, p 0.67; model type number of spread points: F1,17992 2.7, p 0.19; number of

712

Figure 1. Boxplot of simulation results, with GP compared to (a) centre of minimum distance (CMD), spatial mean (SMM) and spatial median (SSM), and (b) a kernel density model. Mean hit score % across all 9000 runs of each method (1000 replicates three levels of source points three levels of spread points) are shown on the y axis. GP performs well across all tests and never took 7% of the search area to find all of the sources. All other methods show much greater range in success. "e kernel density method performs well when number of sources equals 1 but then rapidly begins to decline in e#ciency as the number of sources increase.

not significantly di!erent from 0 in all cases). However, the method detected significant di!erences in fitted values of B between individual species (F2,20 8.5248, p 0.0005474) (Fig. 5).

Discussion

Our study shows that geographic profiling can correctly pre-dict the sources of invasive species, using as input their cur-rent locations. Crucially, it can also do so in the early stages of invasions with data that is possible to acquire quickly, and when control e!orts are most likely to be e!ective. Geographic profiling outperformed other widely used spa-tial statistics such as the centre of minimum distance, spatial mean and spatial median in locating invasion sources. In addition, a simple kernel density approach was found to be less e!ective than GP, with GP’s advantage increasing as the number of sources of invasion increased. Our study also sug-gests that it may be possible to use general, taxon- or habitat-specific values for the model parameters in cases where data on individual species are lacking, since di!erent fitted values of B were obtained in flowering plants and conifers.

"ere are marked similarities between the problems faced by police investigators and invasion managers. Both can face situations with large volumes of data and di#-culty in determining which data are informative and which uninformative. In both cases, resources and time are lim-ited (Williamson and Fitter 1996a). Methods for optimis-ing searches for source locations of invasive organisms could therefore provide a valuable tool in the rapid assessment and control in the critical early stage of invasion history (Leung et al. 2002), just as they have done in criminology (Rossmo 2000).

Temporal differences

For the three species for which we analysed multiple time periods, fitted values of B did not di!er within species across di!erent time steps (linear regressions of slopes were

Table 2. Results of computer simulations comparing GP to (a) centre of minimum distance (CMD), spatial mean (SMM) and spatial median (SSM), and (b) a kernel density model. The results shown are the mean and SD hit score percentage. Asterisks mark cases in which GP sig-nificantly outperformed the other methods. GP performs well across the entire study, never exceeding a mean search efficiency of 4.44% of the simulated area. GP’s advantage over other methods increases as the number of sources increases. Other methods fail to replicate this and become increasingly inaccurate as the number of sources increases.

Mean hit score % (SD)

Number of sources Number of spread points CMD SSM SMM GP (fitted B)

(a) GP versus SSM, SMM and CMD1 10 9.5 (0.06) 14.6 (0.08) 19.2 (0.08) 3.9 (0.05)1 20 9.4 (0.06) 14.4 (0.06) 19.3 (0.08) 4.3 (0.05)1 30 9.2 (0.06) 14.2 (0.07) 19.0 (0.09) 3.3 (0.05)2 10 19.4 (4.13) 24.4 (4.44) 29.2 (4.42) 3.4 (0.03)2 20 22.6 (4.26) 27.7 (4.15) 32.5 (4.35) 3.6 (0.03)2 30 27.1 (4.13) 32.1 (4.15) 36.9 (4.41) 3.5 (0.04)5 10 28.0 (4.26) 38.1 (4.49) 42.9 (4.55) 2.7 (0.02)5 20 35.9 (4.10) 32.6 (4.35) 47.4 (4.4) 3.1 (0.02)5 30 40.3 (4.20) 50.9 (4.20) 52.4 (4.52) 2.3 (0.02)

(b) GP versus kernel density model Kernel (fixed h) GP (fixed B)1 10 4.9 (0.06) 4.4 (0.05)1 20 4.9 (0.05) 4.3 (0.05)1 30 4.7 (0.07) 4.2 (0.05)2 10 5.2 (3.06) 3.5 (0.05)2 20 5.5 (3.07) 3.3 (0.05)2 30 6.8 (3.10) 3.3 (0.05)5 10 8.7 (3.06) 2.7 (0.05)5 20 7.7 (3.07) 2.7 (0.05)5 30 13.5 (3.12) 2.6 (0.05)

713

Many invasive organisms show distinctive time lag between initial invasion, establishment and subsequent expansion (Crooks and Soulé 1999). Often repeated inva-sions will occur by the same pathway and only after a period of time will one of these invasions begin to expand beyond the initial source of invasion (Drake and Lodge 2004). GP operates at this critical early stage of the invasion pro-cess, locating the source locations of rapidly expanding populations allowing e!ective allocation of limited resources available for control measures (Puth and Post 2005, Keller et al. 2008).

Following the work of numerous scientific workers such as weed scientists, resource managers, conservation biolo-gists, restoration biologists, field ecologists and economists,

Figure 3. Boxplot showing fitted values of B for English Nature category listings 76 (conifers and gingko) and 77 (flowering plants).

Figure 2. Search strategies for Heracleum mantegazzianum, based on (a) geographic profiling; (b) centre of minimum distance, and (c) kernel density model. Black circles show the locations in 1941–1950, and white circles the locations in 1931–1940. Contours are shown in bands of 5%. Geoprofiling successfully locates 35 of 51 source populations after searching just 10% of the target area, with a mean hit score for all sources of 13%. Other approaches do not perform as well. "e centre of minimum distance (CMD), spatial mean (S mean) and spatial median (S median) are also shown in blue, black and red respectively.

a clear understanding has emerged of the stages that make up invasions and the relevant modelling and available man-agement options to prevent and manage invasive species (Williamson and Fitter 1996b, Sakai et al. 2001). Sakai et al. (2001) present an example invasion framework (modified from Lodge (1993)) that highlights the general steps in the invasion process and their relationship to management steps that can be taken. "e movement, establishment and subse-quent spread of invasive species are best characterized by a series of discrete steps, each of which poses di!erent problems to both manager and modeller (Vitousek 1997). Some of these stages are more relevant to prevention; others are more relevant for issues of control and restoration. "ere has been increasing understanding that feedback may occur between many of these steps (Sakai et al. 2001). Within each of the phases, di!erent types of predictive and analytical models can be used to gather information and make assessments of the risks presented by invasive species. "e results reported in this study suggest that geographic profiling could form part of an integrated strategy and allow more precise target-ing of source populations and thus more e!ective allocation of resources. "is will be most useful in the establishment and lag phase, but may still be useful in populations that have started to expand rapidly, since the model may be used to di!erentiate between existing populations that are acting as sources and those that are not, again improving the e#-ciency of intervention.

GP models present a new approach based on two simple spatial concepts, distance decay and the bu!er zone. Distance decay has obvious relevance to the spread of invasive species, since movement involves costs, either in terms of energy expenditure or exposure to risk of predation, both of which limit dispersal distance. "e bu!er zone applies even with-out active avoidance of nearby sites since, if suitable habitats are randomly distributed, the number of sites increases geo-metrically with distance from the source. In fact, there may

714

Figure 4. Boxplot showing (a) fitted values of B for EUNIS habitat categories A (marine habitats), C (inland surface waters), G (woodland, forest and other wooded land), I (regularly or recently cultivated agricultural, horticultural and domestic habitats) and J (constructed, industrial and other artificial habitats); (b) Tukey 95% post-hoc comparisons for each of these categories, showing that only categories J and G are significantly di!erent.

Second, the mathematics of GP is under continual development and di!erent functions have been explored (Canter et al. 2000, Levine 2009, O’Leary 2009). O’Leary (2009) proposes a movement into a Bayesian framework, and future work will compare di!erent modelling approaches in terms of model fit, ease of use and practical application, as well as continuing to assess its use in biological systems.

We suggest that invasive species managers and conser-vation biologists should strongly consider the use of GP,

be cases where bu!er zones arise from aspects of a species’ biology. In some species, o!spring avoid occupying the same area as their parents due to the competition for similar niche space that will inevitable result (sometimes only in the adult phase); for example, allelopathy in British trees (Singh et al. 2001). "ere is also evidence for bu!er zones in bee species (Dramstad 1996, Saville et al. 1997).

GP models operate in a similar fashion to gravity and kernel density methods, but have the distinct advantage of performing well with multiple sources of invasion, as well as being possible to run quickly and easily on a small number of data points; crucially, GP models run using only presence/absence data, without the need to estimate parameters for mechanisms such as outflows and inflows between sources and destinations, risk ranks etc. Rossmo (2000) has shown using Monte Carlo simulations that GP is capable of pro-ducing reliable profiles with as few as five data points.

"e results presented here demonstrate the feasibility of the method, but there is also significant development potential. In two key areas there is immediate improvement that can be made in the model. First, GP can be easily placed within existing modelling frameworks in invasion biology. GP models can work alongside population growth models and be informed by trait-based risk analysis (Leung et al. 2002). One exciting area for future research would be to integrate GP with niche-based modeling (Peterson 2003, "uiller et al. 2005). GP models are based on data that describe where species are now and how they have histori-cally spread, a spatial relationship that does not include any information on preferable habitat types. Because niche-based models do not include any spatial data but are based upon habitat preferences (Leung et al. 2004, "uiller et al. 2005), these two modeling types could be combined to produce a risk map that incorporates where species are, how they spread and how likely they are to settle in certain areas.

Figure 5. Time series plots for Heracleum mantegazzianum, Larix decidua and Acheta domesticus. In none of these three cases did fitted values of B di!er within species across di!erent decades (linear regressions were non-significant in all cases). However, the method detected significant di!erences in fitted values of B between individual species (F2,20 8.5248, p 0.0005474).

715

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especially when it is likely that there are multiple source populations of the species in question. Although GP and kernel models perform equally well when there is only a single source, our computer simulations show that, as the number of sources increases, simple kernel density models rapidly become less e!ective (for example, with five sources and 30 data points (a more than reasonable biological case), kernel models searched 13% of the area before finding all the sources, while GP searched on average 2.6% of the area before finding all sources). Managers are cautioned that using kernel models in the case of multiple sources could lead to searching/targeting significantly more of the target area than is necessary, involving a corresponding increase in e!ort. In real-world examples, where resources are likely to be limiting, geographic profiling o!ers an increase in search e#ciency over other methods – such as spatial mean, spa-tial median, centre of minimum distance and simple kernel density models – that is likely to lead to improved targeting of interventions, and more e#cient use of scarce resources.

Acknowledgements – We thank Annabel Dennis for help in assembling data for analysis.

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