Download - Can landscape-scale characteristics be used to predict plant invasions along rivers?

Can landscape-scale characteristics be usedto predict plant invasions along rivers?G. S. Campbell1*, P. G. Blackwell2 and F. I. Woodward1 1Department of Animal and Plant

Sciences, University of Sheffield, Sheffield, UK, and 2Department of Probability and Statistics,

University of Sheffield, Sheffield, UK

Abstract

Aim To determine whether the invasions of hydrochorus plants, that is those which canmake use of rivers to transport their propagules, can be predicted using informationderived at the landscape scale. This is desirable to avoid the need for the difficult tomeasure parameters required by detailed invasion models.

Methods A model for plant propagule dispersal was developed that simulated bothlocal dispersal (autochory) and aided dispersal along river corridors (hydrochory). Thisprovided the simulated invasion behaviour that was to be predicted by the simpleanalytical method. This latter was based on readily available river network character-istics. The analytical summary was then tested for its ability to predict the results of aseries of simulation experiments.

Results Predicted dispersal rates derived from the analytical summary method werestrongly correlated (R2 of 0.8941) to the mean seed displacement simulated by the plantdispersal model.

Main conclusion The simple analytical summary of the river networks provides a goodprobabilistic description of the simulated invasion process. This means that readilyavailable information might be able to be used to predict real invasions by alien plantspecies. This method should now be tested against observed invasions by alien plants.

Keywords

Hydrochory, invasion, simulation, landscape.

INTRODUCTION

This paper describes the development of a model for theinvasion of the UK by alien plant species, specifically thosethat are associated with riparian habitats. There are manywell-documented examples of invasion by alien plant speciesin the UK (see for example Ellis, 1993; Beerling & Palmer,1994; Dawson, 1994; Perrins et al., 1993). In the ecologicalcontext an invasion is the dispersal of a non-endemic speciesand its subsequent spread into a new environment (Cousens& Mortimer, 1995). The term dispersal is used here to meanthe movement of propagules away from the parent (Speller-berg & Sawyer, 2000) and migration is used for the advanceof a species from one area to another (Allaby, 1992). An alienis a species, which has arrived because of human activity(Clement & Foster, 1994) and in the British context may be

defined as a species which was not present in the area nowforming the British Isles, at the end of the last Ice age (Ellis,1993). Modelling such invasions is made difficult by thediscrepancy between the scale at which the mechanisms ofinvasion occur, i.e. at the scale of individual plants and that atwhich its progress is recorded nationally, generally at aresolution of 10 km. This paper reports an approach to themodelling of plant invasions that attempts to reconcile thehighly random nature of the processes involved with theavailable spatial and autecological data.

Much work has been aimed at developing predictivemodels of the spatial response of plants to natural andanthropogenic environmental disturbances (see Higgins &Richardson, 1996 for a review). Such disturbances includeclimate change, edaphic alteration and, perhaps mostdramatically, the introduction of plants to novel environ-ments. The reaction–diffusion approach adopted in theclassic work by Skellam (1951) assumed that the rate ofspecies spread was a constant given that the asymptotic rate

*Correspondence: Department of Plant and Soil Sciences, University of

Aberdeen, Aberdeen AB24 3UU, UK. E-mail: [email protected]

Journal of Biogeography, 29, 535–543

� 2002 Blackwell Science Ltd

of spread has been reached. The central assumption in thismodel that dispersal is effectively a random process exhib-iting Brownian motion, does not allow for a non-uniformlandscape or for more than one mode of dispersal. In fact, itwas the underestimation of the post-glacial migration byoak trees in England, using this model which lead Skellam(1951) to the conclusion that aided dispersal in the form ofornithochory (dispersal by birds) must have played a part inthe migration (Higgins & Richardson, 1996). Similarpalaeoecological evidence for hazel (Corylus avellana L.)(Huntley, 1993) and alder (Alnus glutinosa L.) (Chambers& Elliot, 1989) is indicative of a post-glacial migration rategreater than that, which would be expected from autochorusdispersal alone.

Another approach to modelling plant invasions, which hasproved partially successful, is spatio-phenomenologicalmodelling, that is the use of historical data to determinemathematical relationships between elapsed time and areainvaded (see for example Perrins et al., 1993; Pysek &Prach, 1993). However for some species, such modelsproduce estimates of the mean rate of invasion thatunderestimates the observed invasion rate by an order ofmagnitude (Andow et al., 1990). An example of such aspecies is Himalayan Balsam (I. Glandulifera), which has anobserved maximum invasion rate 38 km year)1, whereas thedehiscent (explosive seed pod) dispersal alone should give aninvasion rate of 2–3 m year)1 (Williamson, 1996). As withthe post-glacial migration of tree species this discrepancy inpredicted and observed invasion rates is attributed to thephenomenon of aided dispersal (Higgins & Richardson,1996) whereby organisms make use of secondary dispersalvectors to augment autochorus mechanisms. A furtherweakness of spatio-phenomenological models is that theydo not consider the heterogeneous nature of the landscapewhen estimating dispersal rates.

A contrasting and in many ways complementary approachis that taken in spatio-mechanistic models, which useempirically derived growth and dispersal functions todeterministically model the behaviour of plant populations.Unlike spatio-phenomenological models, spatio-mechanisticmodels such as MIGRATE (Collingham et al., 1997) canallow for biological differences between species. The spa-tially explicit, spatio-mechanistic model allows for landscapeheterogeneity and can incorporate functions for modellingmultiple dispersal vectors, including occasional long-dis-tance dispersal of propagules. These models perform well atcertain landscape scales, providing a platform for sensitivityanalysis and invasion rate prediction (see for example Williset al., 1997). However, such models deal with the fate ofindividual propagules in a probabilistic, albeit biologicallyrealistic manner, consequently the resulting populationinvasion predictions, whilst being entirely feasible, repre-sents just one of the many possible scenarios. The resultsfrom such models need careful interpretation as the effect ofone or more rare and long-distance dispersal events on theoverall pattern of propagule dispersal can be substantial(Perrins et al., 1993).

An alternative modelling approach is not to attempt tomodel the fine scale mechanisms of the invasion process butrather to use analytical methods to model the process at alandscape and population scale. This approach parallels thatused by Yves et al. (1998), to investigate the ability of simpleanalytical summaries of landscape characteristics (spatialarrangements of habitats) to predict population dynamics.The predictions of a simple analytical description of habitatsuitability were regressed against the results from a demo-graphic simulation to test the predictive ability of the former.The strong relationship between the values predicted by theanalytical method and those simulated by the demographicmodel (R2 ¼ 0.88, where R2 is the proportion of variation inthe observed values explained by the model) suggest thatsuch analytical approaches could be used to study landscape-scale population dynamics in the absence of detailedecological and geographical data. This type of analyticalmodel was developed to predict the invasion of hydrochorusalien plants in the UK and is described here.

For species that are able to use rivers as agents of aideddispersal it was hypothesized that the contribution toinvasion rate made by hydrochorus dispersal is such thatfor certain plants, predictions based on this alone willexplain the observed pattern of invasion (Campbell, 2001).To assess the potential of such an analytical summarymethod the following approach was adopted.

1. Construct a model of plant propagule dispersal, whichsimulates local, plant-mediated (autochorus) and long-range aided dispersal by rivers (hydrochory).

2. Carry out a series of simulation experiments to developrelationships between a relevant landscape characteristic(river network distribution) and simulated rates ofdispersal.

3. Assess the capacity of the analytical summary method topredict the dispersal rate of an independent series ofsimulation experiments.

4. If step 3 is successful, apply the analytical summarymodel to historical plant invasions in the UK and assessits predictive ability in real world situations.

The first three stages are designed to develop a relationshipbetween river network characteristics and plant dispersalrates and to assess the ability of the analytical method toexplain the behaviour of a simulated invasion using onlyvery limited input data. The important point about this inputdata is that unlike the multiple parameters required fordetailed mechanistic models, it is readily available at thenational scale for the UK. The fourth step, as describedabove, is designed to see if this model can predict realinvasions using only this minimal input data. This paperreports on the first three stages of this process, the modeldevelopment and testing against simulated invasions.

LANDSCAPE GENERATION PROGRAM

The experimental landscapes were generated using aprogram written in C and were represented by a grid of

� 2002 Blackwell Science Ltd, Journal of Biogeography, 29, 535–543

536 G. S. Campbell et al.

cells arranged in a two-dimensional array forming thedispersal surface. The grid dimensions were set by a user-defined parameter at the start of the simulation. Each cellwas assigned a code, which dictated the direction a simu-lated propagule that moves into that cell takes in thesubsequent iteration. For the purposes of description com-pass directions will be used to describe the movements acrossthe two-dimensional landscapes, with north being towardsthe top of the dispersal area. A directional code of zerorepresents dry land in the simulation model with no pre-determined dispersal direction, i.e. with an equal probabilityof movement in any of the eight allowable directions. Thedispersal surface was initialized with an array of zero values;this represented a landscape containing no rivers to aidpropagule transport. This homogeneous dispersal surfacewas then overlain with a simulated river network in whicheach cell has a downstream movement direction code,corresponding to the directionalized network function ofthe ARC/INFO Geographical Information System (ESRI,1998). This code determines the direction of the nextpropagule move, for example, a code of two would meana downward diagonal move to the southeast and a code of16 would mean a horizontal movement to the west (Fig. 1).

THE EXPERIMENTAL LANDSCAPES

To determine the relationship between dispersal efficiencyand drainage characteristics, a series of random networkswere generated with drainage densities (area of river cellsas a percentage of the total area) ranging from 0 to 25%.These were consistent with observed drainage characteris-tics of the UK river network at a 0.5-km resolution. Arange of networks was generated with 0, 1, 2, 4, 8, 16, 32,50, 64, 100, 128, 200 and 256 stream heads (the upstreamend of the river) to provide a range of drainage densities,which produced realistic-looking river networks (seeFig. 2). The number of stream heads was based on ageometric progression with the addition of three values(50, 100 and 200) to avoid extended breaks in thecontinuum of generated densities. The second characteristicof the simulated river networks, which was experimentallyvaried, was the directional bias as determined by thenetwork generation program. The experiments describedhere were carried out on 325 simulated river networks thatranged from those with very meandering rivers (series 5) tothose with a strong westeast bias (series 1) via intermediatedegrees of directional bias (series 2, 3 and 4). Each seriescontained five replicates each of a range of thirteennetwork densities with between 0 and 256 stream heads.This gave a total of 325 simulated landscapes but forreasons of space, only nine examples of these networks areshown (Fig. 2).

To assess the predictive ability of the model, two furtherseries of networks were generated. Series X, having riverswith a directional bias intermediate between those of series 3and 4; and series Y with completely random river networks(Fig. 3). Neither of these network series (X and Y) was usedin the regression analysis described below but was rather

used as an independent data set to test the predictive abilityof the relationships derived for series 1–5. For the experi-ments described here, all simulated dispersal landscapeswere 200 by 200 cells in extent.

SIMULATION MODEL

A simulation program was written in the C language inwhich plant propagules (seeds) disperse across the experi-mental landscapes in a random manner, with all land-basedmoves being one cell in any of the eight directions (Fig. 1).Each move made by a ‘seed’ represents one generation, i.e.from seed to adult, which then reproduces, before dying. Thepropagule produced by the previous generation then makessubsequent moves, until the requisite number of generationshas been simulated. The model simulates the dispersal of alarge number of individuals each producing a singleoffspring before themselves dying, i.e. they are ‘annual’plants. The offspring, which are produced during oneiteration, are able to disperse during the subsequent iter-ation, and so a single line of descent is followed in each case.One of the fundamental features of dispersal is the random

Figure 1 An example of a section of a simulated landscape withsimulated river flow direction indicated by the codes correspondingto those in the key. For clarity, the zero codes of terrestrial cells arenot shown.


Landscape-scale characteristics 537

nature of the trajectory of each propagule in the absence ofexternal vectors such as wind, gravity or as in this caseflowing water. What is required to simulate such randomdispersal behaviour is a model that captures the essential

properties of this sequence of independent steps (i.e. themovement of propagules away from the parent plant). Onesuitable model for this purpose is the random walk, a wellstudied mathematical process (for example Renshaw, 1993).

Figure 2 Examples of random drainage net-works generated with different numbers ofstream heads and direction code probabili-ties.

Figure 3 Examples of the independent setsof simulated river networks X and Y.



This model describes a series of single steps the direction ofeach being independent of the proceeding one. This wouldallow the modelling of a plant producing a single propagulethat moves away from the parent plant before becomingfixed and in turn producing a single propagule. In reality,few sessile species do produce a single offspring, but rather anumber each of which is free to move independently in arandom direction before producing more than one offspringthemselves. This form of dispersal behaviour is referred to abranching random walk (Fig. 4).

A well-developed mathematical relationship exists linkingthese two processes (see for example Biggins, 1996) and thisallows the less computationally demanding simple randomwalk to be simulated and the results subsequently convertedinto a branching random walk by specifying the distributionof the number of offspring. This distribution can besummarised by a Malthusian parameter which allowsspecies-dependent factors such as fecundity and survivorshipto be incorporated into the dispersal rate calculations. Thiscalculation can be carried out independently of the simula-tions and so a generic set of simulation results can be usedto model a number of biologically different species, whichhave a similar dispersal mechanism. This conversion to apopulation wave front and application to observed inva-sions is not reported here.

If a simulated propagule encounters a river, there is achance that it may become entrained in the river and carrieddownstream for a distance before being stranded. Once thepropagule is stranded, it resumes its random progress acrossthe simulated landscape until the simulation is complete. Theaided downstream (hydrochorus) dispersal events take placein the course of one generation and as a consequence suchvectors are more efficient at dispersal than the land-based(autochorus) dispersal events. The likelihood of entrainmentand the distribution of downstream stranding are experi-mental parameters; thus, the relative efficiency of the twodispersal vectors may be experimentally manipulated. Otherparameters which may be varied include the number of seedsto be simulated and the number and spatial distribution ofthe initial foci of dispersal. As it moves across the simulatedlandscape, each seed generates a movement history file forlater analysis.

NETWORK ANALYSES

The analytical summary method developed was the netdirectional bias (NDB), which in the case of the experimentsdescribed here was bias in an easterly direction. This wascalculated for each experimental landscape by summingthose river cells with an easterly component and subtractingfrom this those cells with a westerly component. Forexample, for a network with an equal number of river cells‘flowing’ in each direction, the net directional bias would bezero. This would mean that a dispersing propagule wouldhave an equal probability of being carried in any direction,so that the overall effect of the river network was zero, i.e.simulated dispersal of a large number of propagules acrosssuch a landscape would have a mean dispersal distance ofzero. By contrast, a landscape with a preponderance of cellsin one direction would have a relatively large NDB in thatdirection and would be expected to exhibit greater simula-ted dispersal distances in that direction. As the relationshipbetween the value of the NDB and the simulated dispersaldistances was not directionally dependent, a one-dimen-sional analysis was applied to the two-dimensional simula-tions. For clarity, only the east–west bias and dispersaldistances were considered and for the purpose of thisexperiment, the north and south river, cells were not used inthe calculation. The NDB was expressed as a density, i.e. asa proportion of the total number of cells in the area inquestion.

SIMULATION ANALYSIS

The movement histories for each seed as it moved across thesimulated landscape were analysed using a suite of C andSplus programs. The net displacement of each seed wascalculated from the start and end points of the seed’smovement during the simulation. The mean and variance ofthe Euclidean (straight line) displacements of the entireexperimental population of 10,000 seeds was calculated foreach simulated landscape and set of experimental parame-ters. These values were then calculated for each simulation,and the results plotted against the corresponding set oflandscape summary characteristics (NDB). The graphs soproduced were then used to derive regression equationsrelating NDB to the mean and variance of the propaguledisplacements.

EXPERIMENTAL PARAMETERS

These experiments were designed to test the predictiveability of the chosen landscape analysis method rather thanmodel the progress of any particular plant species. Theexact choice of experimental parameters was not crucial,however, it was important that the values chosen allowedthe analytical summary methods to be assessed in realisticsituations. The rationale for the choice of each parametervalue used in the non-scaled simulation experiments isgiven here. The river entrainment probability used in theFigure 4 A diagrammatic representation of a random walk (i) and a

branching random walk (ii).



simulations described here was set at a value, which wasprobably much higher than would be realistic, i.e. 90%.The more realistic alternative to this approach would havebeen to set the value much lower and to simulate thedispersal of very many more propagules. This latterapproach was avoided, as it would be computer intensiveand would not effect the outcome. The downstreamdispersal coefficient was set at a range of values from 90to 98 meaning that once a propagule was entrained in ariver, there was a 90–98% chance that the propagulewould not become stranded on the river bank. There is anequal probability of stranding at each subsequent down-stream cell, giving rise to a downstream distance distribu-tion with a negative exponential form. Such a leptokurticdownstream distribution is similar in form to that observedin the field (Johansson & Nilsson, 1993). The relativelyhigh number of 1000 foci was chosen in order that theeffect of a starting point for simulated dispersal wasminimized. This was important, as the randomly chosenstarting points were held constant for all experimentswhilst the experimental landscapes were varied. It wasassumed that propagules would not start in the river andthere was a function in the simulation program, whichprecluded this, the potential effects of river juxtapositionwould be reduced if the number of starting points wererelatively large. The number of propagules per focus, incombination with the number of foci used, determined thetotal number of propagules used in each simulation. Arelatively large number of simulated propagules were usedto give statistical validity to the use of mean and standarddeviation of the propagule dispersal distance. Each prop-agule ‘line’ was allowed to progress the selected number ofsimulated generations before the dispersal distance wasmeasured. The number of generations was chosen to allowthe propagules to disperse away from the foci but notallow a significant number to ‘escape’ from the dispersallandscape. The figure of thirty-five simulated generationswas selected after tests with a range of simulatedlandscapes.

SIMULATION EXPERIMENTS

The experimental parameters and experimental landscapesdescribed above were used to carry out a range ofexperiments. The fates of 100 propagules were simulatedas they dispersed from 100 random foci, the coordinates ofwhich were generated by a separate program. The samestarting locations were used for each experiment in turn. Inorder to minimize potential edge effects these foci werelocated in a 40 · 40 zone placed centrally within the200 · 200 experimental grid. The simulations were run andthe resulting propagule dispersal behaviour analysed usingthe programs described above to produce the mean andvariance of the dispersal distances of the simulated propa-gules. The NDBs of the landscapes used in each simulationexperiment were also calculated as previously described andthe data obtained used to generate the results discussedbelow.

EXPERIMENTAL SIMULATION RESULTS

The NDB for the 10,000 seeds in each of the 325experiments was plotted against the mean displacement(the horizontal distance between the initial and final locationof each seed) in the easterly direction. Figure 5 shows anexample of such a graph, which represents the results from asimulation with a particular set of downstream transportand entrainment coefficients. In order to establish therelationship between landscape character, expressed asNDB and the mean seed displacement, linear regressionswere performed over the range of experimental parametervalues. The regression models fitted each use 325 data pointsand estimate a single parameter, giving 324 degrees offreedom. The results of these regression analyses for simu-lated rivers with an entrainment coefficient of 90 aresummarized in Table 1. The multiple R2 value indicatesthe strength of the relationship between mean displacementand NDB as measured by the linear regression. Thedisplacement NDB ratio is in effect the slope of the line ofbest fit produced by the linear regression. The variance of thesimulated seed displacements after twenty-five generationswas treated in the same manner as the mean displacementand an example of the results from a particular set ofdownstream transport and entrainment coefficients are

Figure 5 Mean displacement of 10,000 seeds after twenty-fivegenerations plotted against net directional bias (NDB) of thesimulated river network. The net directional bias and the meandisplacement are both expressed in numbers of cells.

Table 1 Linear regressions of mean displacement against netdirectional bias (NDB) for simulated rivers with an entrainmentcoefficient of 90

Downstream transportefficiency (%)

MultipleR2 value

DisplacementNDB ratio

90 0.9254 0.157092 0.9251 0.226094 0.9186 0.334596 0.8899 0.486798 0.8120 0.6540



shown in Fig. 6. Linear regressions were performed over therange of experimental parameter values and the results aresummarized in Table 2. The results show a strong linearrelationship between dispersal efficiency (as indicated bymean displacement) and both density and NDB of thesimulated river networks. This was true for all downstreamdispersal efficiencies. The linearplacement bias ratio washigher with higher downstream dispersal efficiencies, as canbe seen by the steeper lines of best fit (Fig. 7). These resultsshow a strong linear relationship between dispersal effi-ciency (mean displacement) and simulated river density forall downstream dispersal efficiencies.

The results shown in Fig. 7 are for a range of downstreamdispersal efficiencies between 90 and 98% with the NDB andthe mean displacement both expressed in numbers of cells.The linear regression results showed that there was a strongrelationship (R2 values of around 0.9) between the simulatedmean seed displacement against NDB of simulated rivernetworks. However, would the relationships established asdescribed above hold true for simulated dispersal networksother than those used to generate those relationships? Inorder to answer this question an arbitrary set of inputparameters were chosen, in this case an entrainment prob-ability of 90 and a downstream transport efficiency of 96and the appropriate regression equation selected from the

results from the previous experiment. This regression equa-tion was subsequently used to predict the mean and varianceof the displacement of seeds dispersing across the independ-ent set of 130 landscapes formed by series X and Y (Table 1)using only the calculated NDB of the landscapes as inputparameters. These formed the predicted mean dispersaldisplacements.

Simulations were then carried out using the same inputparameters and the same 130 dispersal landscapes the resultsof these simulations constituted the ‘observed’ mean disper-sal displacements. The predicted results were plotted againstthose ‘observed’ from the simulations and regression analysisperformed. Figure 8 shows the regression of ‘observed’ meandisplacement against predicted mean displacement, thisregression had a multiple R2 value of 0.8941 (with 129degrees of freedom). A similar plot for the regression ofobserved displacement variance against predicted displace-ment variance is shown in Fig. 9 (multiple R2 value 0.8867and with 129 degrees of freedom). Neither of these

Table 2 Linear regressions of displacement variance against netdirectional bias (NDB) for simulated rivers with an entrainmentcoefficient of 90

Downstream transportefficiency (%)

MultipleR2 value

DisplacementNDB ratio

90 0.9453 0.006492 0.9452 0.00894 0.9428 0.010296 0.9346 0.013498 0.9118 0.0179

Figure 6 Displacement variance of 10,000 seeds after twenty-fivegenerations plotted against net directional bias (NDB) of thesimulated river network. The net directional bias is expressed innumbers of cells, while the variance of displacement is given in termsof cells2.

Figure 7 Lines of best fit for the linear regression of meandisplacement of 10,000 seeds after twenty-five generations againstnet directional bias (NDB) of the simulated river network.

Figure 8 The regression of observed mean seed displacement fromsimulations plotted against predicted mean displacement aftertwenty-five generations. Both the predicted and observed meandisplacements are expressed in numbers of cells.



relationships was perfectly linear and it may have beenpossible to improve the fit of the models and correspondingR2 values. However, even a simple linear model producedsufficiently good results (high R2 values) to accept that thesummary method (NDB) was able to predict simulationbehaviour.

DISCUSSION

The simulations and results described above are based on amodified random walk and not the movement of a wavefront, which would require a branching random walk, whereplants could have multiple offspring at each generation, eachof which would be able to move independently of each other.In addition, the simulations described here do not allow forsuch species-dependent factors as mortality rate, fecundityrate and age of sexual maturity. In order to maintain theflexibility of this approach, our simulation model generatesmovement patterns for one seed at a time, following onedescendant in each generation. It is effectively generating amovement pattern for one line of descent at a time. Suchmovement patterns can then, as we show in the presentpaper, be described concisely in terms of a few keyparameters.

To make predictions at the population level, and forcomparison with observed data, it is necessary to transformthese individual-level descriptions into an idea of how thepopulation as a whole will spread. An essential idea in ourapproach is that this step can, under reasonable modellingassumptions, be achieved algebraically. The movementprocess in our simulations (or in fact almost any line-of-descent model) can be thought of mathematically as arandom walk (see for example Renshaw, 1993); the positionof a seed at a particular time-step is given by its location atthe previous time-step plus some random change (albeit onethat depends on the local properties of the network). Themultiple lines of descent existing in the population as awhole can be represented using a branching process (Biggins,

1995). This latter is a mathematical model for populationsize in which each individual has a random number ofoffspring (possibly zero), from some given distribution,independently of others in the population. Such models forma very rich class, but much of their behaviour can bedescribed by a single growth rate or Malthusian parameterderived from published empirical studies of the species ofinterest. These two mathematical ideas can be combined tocreate a model known as a branching random walk (Biggins,1996), simultaneously representing spatial movement andpopulation dynamics. Furthermore, mathematical resultsexist for the rate of spread of a branching random walk(Biggins, 1995, 1996). Thus given the individual movementprocess (as in this paper) and given a description ofpopulation dynamics, at its simplest, a value for theMalthusian parameter, deriving the population rate ofspread is relatively simple. This approach has the advantagethat the separate roles of the movement and reproductionprocesses are made clear; while the two interact crucially inderiving the final answer, we can model the movementprocess without making assumptions about the reproductionprocess and vice versa. The results would seem to indicatethat even a simple summarization of network propertiesmight be used to provide estimates of dispersal rates acrossheterogeneous landscapes.

CONCLUSIONS

This paper describes a simple analytical summary of simu-lated landscapes based on the relative number and directionof river cells flowing in each of the four directions. Thissummary (NDB) is able to predict aspects (mean and varianceof the dispersal rate) of the dispersal behaviour of a largegroup of simulated individual plant propagules dispersingacross a heterogeneous landscape with a reasonable degree ofprecision. The main implication of this result is that arelatively simple analytical summary such as NDB is capableof predicting the dispersal behaviour exhibited by a morecomplex simulation model. If this proves applicable to realworld situations it would overcome the problems of param-eterizing spatio-mechanistic models and avoid probabilisticsimulation at the point process level. In turn this might allowpredictions of the course of future invasions by hydrochorusalien plants based on readily available input data.

ACKNOWLEDGMENTS

The authors would like to express thanks to YvonneCollingham for the 0.5 km resolution directionalized mapof part of the UK drainage system.

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BIOSKETCHES

Gary Campbell is a Research Fellow in the Department of Plant and Soil Sciences at the University of Aberdeen. Early researchincluded nutrient measurements in oak woodlands followed by several years teaching Environmental Impact Assessment andEcology. A PhD at the University of Sheffield led to a move into the fields of landscape-scale modelling and geographicinformation systems. Dr Campbell’s modelling interests include plant invasions and emissions of greenhouse gases at the regionalscale.

F. Ian Woodward is Professor of Plant Ecology, Department of Animal and Plant Sciences, University of Sheffield. Researchinterests include climate and plant distribution and the relationship between CO2 concentration and plant processes. He editsseveral journals including New Phytologist, and Global Ecology and Biogeography.

Dr P. G. Blackwell is a lecturer, Department of Probability and Statistics, University of Sheffield. Research interests includestochastic processes, mathematical ecology, evolutionary game theory and Bayesian inference.



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