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Increasing accuracy of dispersal kernels in grid-based population models

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Ecological Modelling
222 (2011) 573–579
Contents lists available at ScienceDirect
EcologicalModelling
journal homepage: www.elsevier.com/locate/ecolmodel
Increasing accuracy of dispersal kernels in grid-based population models
D.H. Slone
∗
USGS Southeast Ecological Science Center, 2201 NW 40th Terrace, Gainesville, FL 32605, USA
a r t i c l e i n f o
Article history:
Received 19 January 2010Received in revised form20 November 2010Accepted 23 November 2010
Available online 14 December 2010
Keywords:
Map latticeSpatial modelInvasion modelNumerical simulationRedistribution
a b s t r a c t
Dispersalkernelsingrid-basedpopulationmodelsspecifytheproportion,distanceanddirectionofmove-mentswithinthemodellandscape.Spatialerrorsindispersalkernelscanhavelargecompoundingeffectsonmodelaccuracy.CircularGaussianandLaplaciandispersalkernelsatarangeofspatialresolutionswereinvestigated, and methods for minimizing errors caused by the discretizing process were explored. Ker-nelsofprogressivelysmallersizesrelativetothelandscapegridsizewerecalculatedusingcell-integrationand cell-center methods. These kernels were convolved repeatedly, and the ﬁnal distribution was com-pared with a reference analytical solution. For large Gaussian kernels (
>10 cells), the total kernel errorwas <10
−
11
compared to analytical results. Using an invasion model that tracked the time a populationtook to reach a deﬁned goal, the discrete model results were comparable to the analytical reference.WithGaussiankernelsthathad
≤
0.12usingthecellintegrationmethod,or
≤
0.22usingthecellcen-ter method, the kernel error was greater than 10%, which resulted in invasion times that were ordersof magnitude different than theoretical results. A goal-seeking routine was developed to adjust the ker-nelstominimizeoverallerror.Withthis,correctionsforsmallkernelswerefoundthatdecreasedoverallkernel error to <10
−
11
and invasion time error to <5%.
Published by Elsevier B.V.
1. Introduction
Spatially explicit population models are useful for forecastingspatial processes that cannot be solved with single-location ana-lyticalmodels.Theycombinetemporalreproductionandmortalityprocesses with spatial redistribution processes. The earliest spa-tial population models were analyzed with continuous time andspace equations (e.g. Kolmogorov et al., 1937), and this form is still
importantanduseful(e.g.Andowetal.,1990;Lutscheretal.,2007).
Continuoussystemsallowforexactsolutionstoresearchquestionssuchasdensityoforganismsandspeedofinvasionwavefrontsatagiven time and place. They simulate populations that have free, ornon-seasonal, reproduction throughout the time domain.For modeling organisms that have a distinct breeding season,integro-difference models that are discrete in time, but contin-uous in space, are often used (e.g. Neubert and Caswell, 2000;Lutscher and Lewis, 2004). Because of increased complexity inthe model system compared to all-continuous models, numericfast-Fouriertransformationsornumericalsolutionsarecommonlyfound in ecological applications of integro-difference dispersalmodels, rather than analytical solutions (e.g. Kot et al., 1996).Becauseofcomputationalcomplexity,theoreticalspatialprocessesare often developed ﬁrst in continuous space, and then demon-
∗
Tel.: +1 352 264 3551; fax: +1 352 374 8080.
E-mail address:
dslone@usgs.gov
strated with a grid-based discrete map lattice (e.g. Lutscher andLewis, 2004).
1.1. Discrete-space models
If a continuous-space model is analytically intractable or if afragmented, realistic map landscape is desired, the landscape of interest can be modeled directly in discrete space. Methods of dis-cretizing space include nodal models that simulate the measureddistances and directions among habitat nodes (spatially explicitpopulation model; Dunning et al., 1995), or simulate the “move-
ment cost” associated with movement between two nodes (Minorand Urban, 2007). These “graph models” are by deﬁnition uncon-cernedwiththespacebetweennodesofinterest.Spacecanalsobesubdivided into grids that simulate all of the landscape of interest.Irregulargrids,suchasunstructuredpolygonalmeshesorcurvilin-ear grids, allow for different sized cells to concentrate computingpowerandresolutioninthoselocationsthataremorecomplex.Thisapproachhasbeenusedtomodelhydrodynamics(e.g.Bockelmannet al., 2004; Crowder and Diplas, 2000), but apparently not for ani-
mal or plant models due to the complexity of calculating dispersalin cells that have various sizes and spatial arrangements.Subdividing the spatial domain into a map lattice of regularpolygons (generally squares) simpliﬁes dispersal modeling. Grid-basedkernelredistributionmodelsareusefulforsimulatingspatialdispersal processes such as invasion in complex, natural land-scapeswithvaryingfeaturesandmultipletypesofirregularhabitat
0304-3800/$ – see front matter.
Published by Elsevier B.V.
doi:10.1016/j.ecolmodel.2010.11.023
574
D.H. Slone / Ecological Modelling
222 (2011) 573–579
patches. Dispersal routines in grid-based population models spec-ifytheproportion,distanceanddirectionofmovementwithineachcellofthemodellandscape.Dispersalmethodsinregulargridmod-elsincludeglobalredistribution(e.g.KingandHastings,2003),and
nearest-neighborredistribution(e.g.cellularautomata;Ellneretal.,1998). A ﬂexible modeling paradigm that can be applied to bothgraphs and (with effort) grids is circuit theory (McRae et al., 2008).
Tosimulatelocaldispersalprocessesinagrid-basedspatialsys-tem, a continuous dispersal kernel, or probability density functionfor redistribution, can be split into component cells to generateandapplyadiscretekernelthroughspatialconvolution(Allenetal.,2001)orsimilarly,toapplyadisplacementmatrix(Sebert-Cuvillier
et al., 2008; Westerberg and Wennergren, 2003).
1.2. Problems with discrete spatial models
Standard terminology for cartographic standards, modiﬁed forecological use, will be used (Dungan et al., 2002). “Spatial extent”
is the overall size of the spatial domain, and “grain size” is the sizeof the cells in the landscape grid relative to the spatial extent. Asthe grain size decreases, the number of cells within a given spa-tial extent (the resolution) increases, and the accuracy of spatialprocesses, such as dispersal, increases.Kernelsmoothingandaccuracymeasuresfor“binned”data,andthe determination of what grain size is needed for a given levelof accuracy has been well documented ( Jones, 1989; González-Manteiga et al., 1996; Hall and Wand, 1996; Pielaat et al., 2006).
With classical numerical simulation of a complicated system thatcannot be solved analytically, the grain of the landscape can bedynamicallyadjustedtopreserveadeﬁned,lowerrorrate.Forthe-oreticalapplicationsindiscretesystemswithasmallspatialextent,a small grain size can be implemented for good accuracy. How-ever, if the landscape is large compared to the dispersal abilities of the organism, a ﬁne grain size for detailed dispersal kernels wouldlead to a very large number of cells across the spatial domain, thusrequiring large amounts of computer processor power, RAM anddata storage. Depending on the complexity of non-dispersal oper-ations, the amount of time needed to run grid-based models tendsto increase to the fourth power of the number of cells in any lineardimensionofthesimulation,sothetimetorunasimulationquicklyincreases as the spatial resolution becomes ﬁner.Forsomeapplications,theresolutionofthemodelisalreadysetdue to precedent models or available data resolution (e.g. satel-lite imagery), and the modeler must work within that framework.For example, there are several spatial population models that sup-portdecision-makinginthegreaterEvergladesrestorationprocess(CERP;http://www.evergladesplan.org/).Manyofthesemodelsaredrivenbyhydrologicalstatevariables,suchaswaterdepthorsalin-ity. Hydrological models that provide these variables are availablefor different regions and purposes in 2
×
2 mile squares (SFWMM;SFWMD, 2005), 500
×
500m (ATLSS; DeAngelis et al., 1998), or
400
×
400m (EDEN; Liu et al., 2009). Land managers generallyexpect ecological model output to be in the same grid system asthehydrologicalmodelsforconsistencyandeaseofinterpretation.These pre-deﬁned grid sizes can lead to very coarse-grained spa-tialprocessesandsmalldispersalkernels.Climatemodelsoftenarecalculated with grid sizes of several kilometers. Downscaling to aﬁner scale presents substantial challenges and effort (Araújo et al.,2005),sotoolstousecoarse-scalemodelsdirectlywouldbeuseful.When dispersal of organisms is introduced to a discrete spa-tial model, the square shape of the landscape cells introduceserrors in distance and direction as compared to the analyti-cal dispersal process. Simple discrete dispersal methods such asnearest-neighbor, where propagules are redistributed only to thenearest contiguous cells, limit dispersal patterns, and are inap-propriate for wide-ranging organisms. Using a discrete form of thecontinuousintegro-differenceredistributionkernelmayreducespatial errors compared to a simple nearest-neighbor distributionprocess, but each cell in the kernel can only contain one constantdensity,whileacontinuousdispersalkernelcanchangevalueoverthe same space. For some applications, a ﬁne temporal scale maybe desired. As the time step decreases, dispersal kernels becomesmaller and more coarse-grained. As the grain size is increasedand the number of cells in a dispersal kernel shrinks, the informa-tion contained within the kernel also shrinks, and the kernel tailsbecome less well deﬁned. Very coarse kernels can essentially bereduced to a nearest-neighbor situation. These coarse kernels canbe expected to contain large spatial errors (Hall and Wand, 1996;Fig. 1).Errors that are generated by the discretization of spatial pro-cesses have always been tacitly acknowledged by researchersperforming traditional numerical solutions to continuous spatialprocesses, so they use very ﬁne grain sizes or error-controllingnumerical methods (such as Runge-Kutta) in their simulations.Recently,uncontrollederrorthatappearsindiscretemodelsystemswhere the grain size is pre-selected and large is receiving atten-tion in the literature (Chesson and Lee, 2005; Holland et al., 2007;
O’SullivanandPerry,2009).Signiﬁcantdispersalerrorswerefoundin a model described by Slone et al. (2003), caused by small size
dispersal kernels on a large-grained landscape. For that model, theauthors corrected the speciﬁc kernels used on an ad-hoc basis, butquestions remained about the general error rates of small kernels.Measuring and correcting these errors will be the central focus of this paper.
2. Methods
2.1. Deﬁning and correcting errors
Accurate dispersal kernels are necessary for spatial models tobe reliable tools for answering management questions. As grainsizedecreasestowardszero,resultsfromadiscretesimulationwillasymptoticallyapproachthatofacontinuous-spacesimulation(i.e.– have zero error). Two questions that arise are (1) at what grainsize does the error between the discrete and continuous systembecome negligible for answering research questions, and (2) atcoarserresolutions,cantheerrorbecorrectedsothatmoreefﬁcientcoarse-grained simulated landscapes can be used?The research has the following three objectives:1) quantifyerrorindiscreteGaussianandLaplacedispersalkernels,andtheinvasionspeedofthesekernelswhenappliedtoaspatialmodel;2) explore methods to correct kernel error, thus allowing theoutput from coarse-grained discrete spatial systems to matchtheoretical or ﬁeld-measured dispersal rates; and3) determineaminimumgrainsizewherenocorrectionisrequired(<5% error in invasion speed).Though population redistribution functions are often non-normal (Kot et al., 1996), the bivariate Gaussian distribution was
exploredﬁrstbecauseitexhibits“closure”(ChessonandLee,2005):that is, as organisms disperse from a single cell through time witha Gaussian dispersal kernel, their overall distribution will remainGaussian, with known parameters. This property enables a sim-ple but powerful test: as a Gaussian dispersal kernel becomes verysmall, do the propagules still disperse in the expected pattern andretain the expected Gaussian distribution? Other kernel shapes donotlendthemselvessoreadilytothistypeofanalysis.TheGaussiankernel – assuming non-directional circular dispersal (
x
=
y
=0;
x
=
y
=
)–hasanadditionalsimplifyingpropertythatithasonlyone parameter (
; see Table 1 for notation).
D.H. Slone / Ecological Modelling
222 (2011) 573–579
575
Fig. 1.
Dispersal kernels of different standard deviations: smaller kernels are more spatially inaccurate, and also converge to a common shape (only the center portions of the larger distributions are displayed).
For kernels that do not exhibit closure, a practical dispersal testcan be conducted to ascertain whether speed of invasion remainsconsistent over a ﬁxed distance as the grain size of the simulationchanges. The circular Laplace (exponential decay) kernel and alsothe Gaussian kernel were tested in this way.
2.2. Dispersal kernel functions
Dispersal in continuous space integro-difference models isaccomplished through use of a spatial convolution model of theform:
N
t
(
i, j
)
=
x,y
k
(
x, y
)
f
(
N
t
−
1
(
i
−
x, j
−
y
))
dxdy,
(1)(Kot et al., 1996; Lutscher and Lewis, 2004), where
k
(
x
,
y
)represents the dispersal kernel for the species in question, and
f
(
N
t
−
1
(
i
,
j
))issomefunctionthattransfersthepopulationfromone
Table 1
Notation.Variable Deﬁnition
k
Distribution kernel function
x
,
y
Spatial coordinates of distribution kernel
N
Population density at each location
i
,
j
Spatial coordinates of model landscape
t
Time step of model
f
Population reproduction or mortality function
Mean of the Gaussian kernel
Standard deviation (spread) of the Gaussian kernel
′
Corrected standard deviation of the Gaussian kernel
b
Scale (spread) of the Laplace distribution
b
′
Corrected scale of Laplace distribution
n
Number of convolutions applied* Spatial convolution operator
time step to the next. The dispersal kernel
k
can be any prob-ability density function (pdf). For grid-based discrete ecologicalmodels (Allen et al., 2001), the continuous convolution process
above becomes the analogous discrete convolution:
N
t
(
i, j
)
=
x
y
k
(
x, y
)
f
(
N
t
−
1
(
i
−
x, j
−
y
))
.
(2)This equation may be solved by multiplying the 2-d fast Fouriertransforms of
k
and
f
together and then inverting in a time loop tovisualize the spatial dynamics, and when
k
is large relative to thespatial domain, this may be computationally efﬁcient. In practice,however,mostecologistsaresimulatingverylargespatialdomainsrelative to the dispersal of their organisms, implying that
k
is verysmall relative to the spatial extent of
f
. In this case, the direct con-volution is usually faster.Two methods were compared for calculating discrete kernels.For the ﬁrst method, the value of the appropriate continuous dis-tribution was calculated at the center of each cell of the array, andused as the value for that cell (center method):
k
[
x,y
]
=
12
2
exp
−
x
2
+
y
2
2
2
,
(3)where
is measured in grid cell units.Forthesecondmethod,whichwasmorecomputationallyinten-sive but more indicative of the actual density of organisms movingtoeachcell,thepartofthecontinuousprobabilitydistributionthatfell within each cell was integrated. The resulting value was usedas the value of that cell (integrated method):
k
[
x,y
]
=
x
+
0
.
5
x
−
0
.
5
y
+
0
.
5
y
−
0
.
5
12
2
exp
−
x
2
+
y
2
2
2
.
(4)
576
D.H. Slone / Ecological Modelling
222 (2011) 573–579
Most testing was performed using these two variations of theGaussian kernel. Additional invasion speed tests were performedusingthecenter-calculatedcircularLaplacedistribution.Italsohasonly one parameter (
b
), which is analogous to the
parameter of the Gaussian distribution:
k
[
x,y
]
=
1
b
·
exp
−
1
b
x
2
+
y
2
.
(5)The spatial extent of each kernel was set to 12
+1 cells for theGaussian kernels and 12
b
+1 cells for the Laplace kernels, with aminimum of 5
×
5 cells. For example, a Gaussian kernel with
of 10 was 121
×
121 cells, while a Gaussian kernel with
of 0.1 was5
×
5cells,because12(0.1)+1=2.2.Thisrulegeneratedkernelsthathaddensitiesoflessthan10
−
16
attheirperiphery.Allkernelswerenormalized by dividing each cell of the kernel by its sum:
k
∗
[
x,y
]
=
k
[
x,y
]
x
y
k,
(6)thusassuringthateachkernelsummedto1andmaintainedacon-stant population abundance as it was applied.
2.3. Quantifying discrete dispersal kernel error
A Gaussian kernel exhibits “closure” (Chesson and Lee, 2005),
such that a kernel
k
(
,
x
,
y
) that is convolved
n
times (repeateddispersalevents)istheoreticallyequivalenttoasingleconvolutionof a large Gaussian dispersal kernel of size
k
(
√
n, x, y
):
k
(
, x, y
)
1
∗
k
(
, x, y
)
2
∗···∗
k
(
, x, y
)
n
∗
N
(
x, y
)
=
k
(
√
n, x, y
)
∗
N
(
x, y
)
,
(7)where * is the convolution operator. To measure the shape error of dispersal kernels, 2-D discreteconvolution was applied repeatedlytoasimulatedpopulationofsize10,000placedinthecentercellofalarge discrete arena. The resulting distribution was then comparedto the theoretical result.Askernelsizeapproachesinﬁnity(i.e.,grainsizeshrinkstowardszero), the error induced by the “pixelation” of the discrete land-scape also approaches zero. So, we might expect a large discretedispersal kernel to show high ﬁdelity to its theoretical parame-ters, while a smaller one would show less ﬁdelity (Fig. 1). Throughpreliminary testing, a Gaussian kernel with a
of 24 was foundto contain negligible spatial error. To test the effect of size onspatial error, Gaussian kernels (
k
small
) with 0.01
≤
≤
4, were con-volved (24/
)
2
times, so the resulting distribution (
k
conv
) could becompared to a Gaussian kernel of
=24 (
k
large
). Larger Gaussiankernels with 4
≤
≤
24 were convolved 36 times and then com-pared to a
k
large
kernel with
large
=6
small
. This methodology wasrepeated for each of the two kernel generation methods (centerand integrated), with
k
large
and
k
small
being generated with thesame method. There was no simulation of immigration, emigra-tion, reproduction or mortality, so the total population size in allsimulations remained constant.To measure error in the
k
conv
kernels, the pairwise sum-of-squares error (SSE) was calculated for the entire spatial domain:SSE
=
x
y
k
conv
(
x, y
)
−
k
large
(
x, y
)
2
.
(8)
2.4. Correcting dispersal kernel error
For each dispersal kernel of size
,
there was an associated
n
,or the number of convolutions used to generate the uncorrected
k
large
.Anadaptivegradientdescentalgorithmwasusedtotestnewvalues of
for each value of
n
, to minimize the SSE between
k
conv
and
k
large
. This procedure ultimately resulted in a corrected valuefor each
, called
′
.
Fig.2.
SpatialerrorwasmeasuredafterrepeatedconvolutionofGaussiandispersalkernels of a single population placed in the center cell, with smaller kernels show-ing dramatic errors in population distribution. Integrated=kernel calculated usingintegrated method; center=kernel calculated using center method.
Finally, a simple invasion scenario was developed to test thespeed of invasion of the uncorrected and corrected values of
.
Foreachvalueof
,asquaregridmapwascreatedwithasizethatwasthe larger of either 100
+1, or 41 cells. In the center cell of thismap, a population of 10,000 individuals was added, and then theuncorrected Gaussian kernel was convolved with the map until acell at the edge of the map had a population>1. This number of convolutions was recorded. Next, the respective corrected kernelswereconvolvedwiththesamescenario,andthenumberofconvo-lutionsneededforthepopulationtoreachtheedgeofthemapwasrecorded.Toassesswhetheradispersalkernelcouldbecorrectedwithoutresortingtoanalysisofthekernelsizeandshape,anadaptivegradi-entdescentalgorithmwasusedtodirectlyoptimize
bymatchingthenumberoftimestepsthediscreteinvasionmodeltooktoreachtheedgetothetheoreticalnumberoftimestepsthatanequivalentcontinuous model took. After several values of
were corrected, apolynomial regression was ﬁtted to ln(
′
/
), and then this regres-sion was tested using the full range of
values in the invasionmodel, again by comparing the number of time steps the discretemodel took relative to the continuous model.Finally,asimilarinvasionscenariowasrunusingtheLaplacedis-tribution. Again using a gradient descent goal-seeking algorithm,the invasion model was processed for several levels of
b
until opti-mum corrected values (
b
′
) were found that matched the expectedcontinuous-space results. A polynomial regression was ﬁtted toln(
b
′
/
b
), and this equation was validated by choosing a wide rangeof values of
b
′
to compare with the continuous-space results.AnalysiswasperformedinMatlab,2007a(TheMathworks,Inc.,Natick,MA,USA)withdouble-precisionﬂoatingpointcalculations.
3. Results and discussion
3.1. Quantifying discrete dispersal kernel error
Dispersal error increased rapidly in both center and integratedmethods for
>1 (Fig. 2). Whereas the center method produceda somewhat better ﬁt than the integrated method for medium-sized kernels (
from 0.45 to 7.5), it produced much larger errorsfor small kernels below
=0.45. For simulation models with lowresolution, uncorrected kernels generated with the center methodwouldgenerallyshowslowerinvasionspeedthanthecorrespond-ing analytical result. This pattern has been seen in the literaturewhere the analytical and numerical results were plotted together(e.g. Fig. 5 in Kot et al., 1996; Figs. 1 and 2 in Méndez et al., 2002).
Numericalsimulationswithsmallkernelsgeneratedwiththeinte-grated method might show slower or faster invasion wavefronts,
D.H. Slone / Ecological Modelling
222 (2011) 573–579
577
Fig. 3.
Results of invasion test with Gaussian dispersal kernels generated by theintegrated method (black lines) and by the center method (gray lines). CorrectedkernelswerederivedfrompolynomialEqs.(9)and(10)thatwereﬁttedtothecurvesplotted in this ﬁgure. Small residual errors in the corrected kernels were the resultof error in the polynomial equation to the optimized kernels – optimizing each sizeof kernel directly would lead to an accurate invasion model for every size.
depending on the resolution of the kernel and the precise ratio of the dispersal kernel to the map cell size.
3.2. Correcting dispersal kernel error
The invasion test conﬁrmed that the error increased rapidly insmall kernels. Accurate results were obtained without correctionfrom the integrated method down to
=2 (Fig. 3), and with thecentermethoddownto
=0.8(Fig.3).Reasonableaccuracy(within10% error) was retained down to
=0.3 for the integrated method
Fig. 4.
The corrected kernels were generally larger than the uncorrected kernels –that is, if uncorrected, dispersal would be underestimated. Dispersal kernels withstandard deviations that were smaller than 1 required more correction, with thecenter method requiring the most correction. Integrated=kernel calculated usingintegrated method; center=kernel calculated using center method.
and
=0.5forthecentermethod,butbelowthosevalues,accuracydegraded rapidly with decreasing standard deviation values.For each method and each
, a corrected kernel size (
′
) wasfound that brought the total SSE to very small levels (<1e-11,Fig. 2), and also produced accurate invasion speed results (Fig. 3).
Asthesizeof
decreased,themagnitudeofcorrectionnecessarytominimizethetotalSSEincreased(Fig.4).Thecentermethodgener-ally required a larger correction, especially for very small kernels.Regressionsfromtheinvasionmodeloptimizationwerecalculated
Fig. 5.
An arbitrarily-chosen small Gaussian kernel using the Center calculation method and
of 0.2134. The srcinal uncorrected kernel is shown on the left, and thecorrected kernel is on the right (only the center portions of the larger distributions are displayed). The correction reduced the population density error by approximately 8orders of magnitude. Other sizes show similar results. Note that the scales may differ in each graph.

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