The evolution of density-dependent dispersal in a noisy spatial ...

OIKOS 115: 308320, 2006**The** **evolution** **of** **density**-**dependent** **dispersal** **in** a **noisy** **spatial**population modelÁdám Kun and István Scheur**in**gKun, Á. and Scheur**in**g, I. 2006. **The** **evolution** **of** **density**-**dependent** **dispersal** **in** a**noisy** **spatial** population model. Oikos 115: 308320.It is well-known that **dispersal** is advantageous **in** many different ecological situations,e.g. to survive local catastrophes where populations live **in** **spatial**ly and temporallyheterogeneous habitats. However, the key question, what k**in**d **of** **dispersal** strategy isoptimal **in** a particular situation, has rema**in**ed unanswered. We studied the **evolution****of** **density**-**dependent** **dispersal** **in** a coupled map lattice model, where the populationdynamics are perturbed by external environmental noise. We used a very flexible**dispersal** function to enable **evolution** to select from practically all possible types **of**monotonous **density**-**dependent** **dispersal** functions. We treated the parameters **of** the**dispersal** function as cont**in**uously chang**in**g phenotypic traits. **The** **evolution**ary stable**dispersal** strategies were **in**vestigated by numerical simulations. We po**in**ted out thatirrespective **of** the cost **of** **dispersal** and the strength **of** environmental noise, thisstrategy leads to a very weak **dispersal** below a threshold **density**, and **dispersal** rate**in**creases **in** an accelerat**in**g manner above this threshold. Decreas**in**g the cost **of****dispersal** **in**creases the skewness **of** the population **density** distribution, while **in**creas**in**gthe environmental noise causes more pronounced bimodality **in** this distribution. Incase **of** positive temporal autocorrelation **of** the environmental noise, there is no**dispersal** below the threshold, and only low **dispersal** below it, on the other hand withnegative autocorrelation practically all **in**dividual disperses above the threshold. Wefound our results to be **in** good concordance with empirical observations.Á. Kun (kunadam@ludens.elte.hu), Dept **of** Plant Taxonomy and Ecology, EötvösLoránd Univ., Pázmány P. sétány 1/C, HU-1117 Budapest, Hungary. I. Scheur**in**g,Dept **of** Plant Taxonomy and Ecology, Hungarian Academy **of** Sciences and EötvösLoránd Univ., Pázmány P. sétány 1/C, HU-1117 Budapest, Hungary.Dispersal is one **of** the most important life-history traits it **in**fluences the dynamics and persistence **of** populations,the distribution and abundance **of** species, the level**of** genetic diversity and community structure (reviewedby Dieckmann et al. 1999, Ferrière et al. 2000, Clobertet al. 2001). Dispersal is costly for the **in**dividual (Hanski1998), but it can give **evolution**ary benefit for severalreasons. Dispersal helps avoid**in**g k**in** competition(Ronce et al. 1998, Gandon and Michalakis 2001,Lamb**in** et al. 2001, Perr**in** and Goudet 2001) andprevents **in**breed**in**g (Hamilton and May 1977, Motro1991, Gandon and Michalakis 2001, O’Ria**in** andBraude 2001, Perr**in** and Goudet 2001). Furthermore itcan be **evolution**ary favourable if the environment variesboth **spatial**ly and temporally (McPeek and Holt 1992);**dispersal** then helps populations to escape local catastrophes(Olivieri et al. 1995, Gyllenberg and Metz 2001,Metz and Gyllenberg 2001, Parv**in**en et al. 2003).To make the model analytically tractable, theoretical**in**vestigations **of** the **evolution** **of** **dispersal** rates generallyuse two important assumptions. First, it is frequentlyassumed that **dispersal** is unconditional (i.e. aconstant fraction **of** **in**dividuals disperse, regardless **of**the local population **density** and/or the environmentalAccepted 2 June 2006Subject Editor: Veijo KaitalaCopyright # OIKOS 2006ISSN 0030-1299308 OIKOS 115:2 (2006)

conditions, Paradis 1998, Travis and Dytham 1998,Parv**in**en 1999, Mathias et al. 2001, Gyllenberg et al.2002, Kisdi 2002, Cadet et al. 2003, Parv**in**en et al.2003). Naturally, this unconditional **dispersal** is an oversimplification.In reality, most organisms will followmore sophisticated conditional **dispersal** rules. **The**y maybe sensitive to such factors as local population size,habitat quality, age, social status, sex and behaviour(Johnson 1990, Motro 1991, McPeek and Holt 1992,Ronce et al. 1998, Ims and Hjermann 2001, Scheur**in**g2001, Buddle and Rypstra 2003). Second, as Travis andDytham (1998) po**in**ted out, almost all theoretical**in**vestigations **of** the **evolution** **of** **dispersal** employed**spatial**ly structured metapopulation models (Johst andBrandl 1997, Paradis 1998, Parv**in**en 1999, Mathias et al.2001, Gyllenberg et al. 2002, Kisdi 2002, Cadet et al.2003, Parv**in**en et al. 2003, Poethke et al. 2003), **in** whichlocal patches are connected to each other, and they arelocated at the same distance from all other patches. Incontrast to this **spatial**ly implicit representation, **spatial**lyexplicit models (Jánosi and Scheur**in**g 1997, Travis andDytham 1998, 1999, Travis et al. 1999, Travis 2001,Poethke and Hovestadt 2002) implement patches arrangedon a 2 D lattice: the distance **of** two patches is thusdeterm**in**ed by their position on the lattice. Travis andDytham (1998) also suggested that the **evolution** **of****dispersal** should be studied with **spatial**ly realisticmodels, which are both **spatial**ly explicit and **in**corporate**spatial** and/or temporal heterogeneity. While a number**of** studies **in**vestigated the effect **of** environmentalvariation (McPeek and Holt 1992, Parv**in**en 1999,Mathias et al. 2001, Kisdi 2002, Poethke et al. 2003),these employed **spatial**ly implicit models. However,Poethke and Hovestadt (2002) found that their resultsdid not change if they employed a **spatial**ly implicitmodel **in**stead **of** the **spatial**ly realistic one.**The** aim **of** this study was to **in**vestigate the **evolution****of** **density**-**dependent** **dispersal** with the help **of** a**spatial**ly realistic population model. To our knowledge,**in** the context **of** the **evolution** **of** **dispersal** rates the only**spatial**ly explicit model preced**in**g our study was themodel presented by Poethke and Hovestadt (2002). **The**topic is one **of** great theoretical **in**terest (Travisand French 2000, Armsworth and Roughgarden 2005,Bowler and Benton 2005) and ma**in**ly unexplored. Whilethere are ample examples **of** **density**-**dependent** **dispersal****in** nature (Ga**in**es and McClenaghan 1980, Hurd andEisenberg 1984, Denno et al. 1991, Bengtsson et al. 1994,Herzig 1995, Denno et al. 1996, Fonseca and Hart 1996,Gaona et al. 1998, Aars and Ims 2000, Albrectsenand Nachman 2001, French and Travis 2001, Rha**in**dset al. 2002, Lecomte et al. 2004, Moksnes 2004,Matthysen 2005), theory lags beh**in**d **in** elucidat**in**g themechanism beh**in**d it. Albeit there were other theoreticalstudies which **in**corporated **density**-**dependent** **dispersal**(Jánosi and Scheur**in**g 1997, Ruxton and Rohani 1999,Travis et al. 1999, Poethke and Hovestadt 2002,Amarasekare 2004a, 2004b), they used very rigid shapesfor the **dispersal** function. **The**se studies either employeda predef**in**ed function whose parameters were not subjectto **evolution**ary change (Ruxton and Rohani 1999,Amarasekare 2004a, 2004b), or, while allow**in**g someparameters **of** the **dispersal** function to change dur**in**gthe course **of** **evolution**, they restricted the shape **of**the **dispersal** function (Jánosi and Scheur**in**g 1997, Traviset al. 1999, Poethke and Hovestadt 2002).In this study we used a general **density**-**dependent****dispersal** function which can follow many qualitativelydifferent shapes, and thus allows selection to choose theoptimal shape for the **dispersal** function. **The** selected**dispersal** function allows us to **in**terpret empiricalobservations and to test the robustness **of** formertheoretical studies.**The** modelAssumptions **of** the modelDispersers have the same fecundity after **dispersal** asnon-dispersersThis assumption is based on some field observation:Johanessen and Andreasen (1998) found that the reproductiveoutput and mortality rates **of** immigrant andresident female root voles (Microtus oeconomus) werenot significantly different. In a review **of** the cost **of**migration **in** **in**sects, Rank**in** and Burchsted (1992) foundthat **in** some species dispersers actually had a higherreproductive output, but they also presented cases to theopposite (R**of**f 1977, 1984).Evolution was modelled as a succession **of** competitionevents between a resident population and an **in**itially raremutant populationThis assumption implies that mutation is **in**frequent, andthe population can settle to an ecological equilibriumbefore the next mutational event takes place. **The**seassumptions are generally applied **in** adaptive dynamics(Metz et al. 1992, 1996, Dieckmann 1997).**The** phenotypic trait was **in**herited asexuallyGenetic variation for migratory traits have been documentedand selection experiments were successful **in****in**duc**in**g changes **in** **dispersal** traits (R**of**f and Fairba**in**2001). **The** actual genetic background **of** a condition**dependent****dispersal** trait is unknown. Thus, **in**clusion**of** sexual **in**heritance would unduly complicate themodel and would not add to its realism. **The** assumption,that an **of**fspr**in**g has the same phenotype as its parentis employed **in** most strategic models (McPeek andHolt 1992, Jánosi and Scheur**in**g 1997, Paradis 1998,Travis and Dytham 1998, Dieckmann et al. 1999,OIKOS 115:2 (2006) 309

Parv**in**en 1999, Travis et al. 1999, Cadet et al. 2003,Parv**in**en et al. 2003, but see Poethke et al. 2003).Population dynamicsWe applied a coupled map lattice model to the study**of** the **evolution** **of** **density**-**dependent** **dispersal** rates.**The** square lattice conta**in**ed 100 rectangular patches.Follow**in**g the standard techniques the boundaries werewrapped-around (torus), to preclude edge effects.**The** equations govern**in**g the **in**ter patch populationdynamics were the same at each site. At each time stepthe population first experienced growth and then afraction **of** the local population dispersed accord**in**g toa **dispersal** function def**in**ed by the ‘‘pre-**dispersal**’’ localpopulation. Dispersal was assumed to be local, and**in**dividuals dispersed to the four closest neighbour**in**gpatches with equal probability. Dispersal events tookplace simultaneously for all populations. We consideredthe follow**in**g dynamics at site (x,y) and time t forthe resident (/N (res)tpopulation.(x; y)) and for the mutant (/N (mut)t(x; y))Growth**The** local dynamics were governed by a generalisedlogistic growth equation employed by Maynard-Smithand Slatk**in** (1973) as follows:N (res)t1(x; y)N(res)N (mut)t1t(x; y)(x; y)N(mut)1 [a(N (res)tt(x; y)j(s; x; y; t)l(x; y) N (mut) (x; y))] gtj(s; x; y; t)l(1)1 [a(N (res)t(x; y) N (mut)t(x; y))] gwhere l, a and g are the parameters **of** the populationdynamics, and a l1=g 1; where K is the carry**in**gKcapacity **of** a site. Depend**in**g on l and g, Eq. 1 may settleto a fixed po**in**t, follow a limit cycle or can be chaotic forother parameters. Chaotic local population dynamicsbehave qualitatively similar to **noisy** local populationdynamics, as both create many different local densities **in**the population. **The** only important difference is thatthe level **of** variance can be tuned precisely **in** thestochastic model, thus chaotic dynamics without loss**of** generality were not considered here. On the otherhand, both fixed po**in**t and limit cycle dynamics wereconsidered **in** the simulations. Furthermore, for themajority **of** the simulations, we employed parametercomb**in**ations that results **in** contest competition (g5/1),but scramble competition (g/1) was also studied./j(s; x; y; t) is a random perturbation on l withmagnitude s, at site (x,y) and time t. We **in**vestigatedthree scenario: (1) **spatial**ly heterogeneous and temporallyconstant environment; (2) **spatial**ly and temporallyheterogeneous environment, with uncorrelated environmentalnoise; and (3) **spatial**ly and temporally heterogeneousenvironment, with temporally correlatedenvironmental noise. In all cases, we assumed that the**spatial** correlation length **of** the environmental variationcorresponds to the size **of** the patches (**in** essence theextent **of** environmental variation def**in**es the patches).Consequently environmental noise was **spatial**ly uncorrelated.(1) In a temporally constant environment j(s; x; y)was **in****dependent**ly set for each site to j(s; x; y)(8s1) where 8 is a random number chosenfrom a standard normal distribution. j(s; x; y) wasnot allowed to be less than 0, so random values lessthan 0 were set to 0. j(s; x; y) was set before thefirst time step, and was left unchanged afterward,thus creat**in**g a **spatial**ly heterogeneous, but temporallyconstant environment.(2) In an environment with uncorrelated environmentalnoise, j(s; x; y; t) was **in****dependent**ly set for eachsite at each time step t to j(s; x; y; t)(8s1);where 8 and constra**in**ts on values **of** j is as above.(3) We have created correlated environmental noisewith the method presented **in** the study **of** Ripaand Lundberg (1996), see also Petchey, et al.1997, He**in**o 1998, Cudd**in**gton and Yodzis1999). Namely, j(s; x; y; t1)(j(s; x; y; t)1)/pk8sffiffiffiffiffiffiffiffiffiffiffiffi1k 2 1; where k governs the degree**of** autocerrelation, 8 is a random number chosenfrom a standard normal distribution, and s ispffiffiffiffiffiffiffiffiffiffiffiffiscaled by 1k 2 so the magnitude **of** environmentalfluctuation is constant for any value **of** k.Because we employed a 100 time steps **in**itial phase(see below) the asymptotic level **of** variance can beused without unduly affect**in**g the outcome (He**in**oet al. 2000).DispersalN (mut)t1(x; y)N(mut)tN (res)t1(x; y)N(res)t(x; y)N (mut)t(x; y)D (mut) (x; y) 1 X4(1s) D (mut) (x neigh:;i ; y neigh:;i )4i1N (mut)t(x neigh:;i ; y neigh:;i ) (2:1)(x; y)N (res)t(x; y)D (res) (x; y) 1 X4(1s) D (res) (x neigh:;i ; y neigh:;i )4i1N (res)t(x neigh:;i ; y neigh:;i ) (2:2)where s is mortality dur**in**g **dispersal** (05/s5/1); andD(x,y) is the **dispersal** function for the resident(/D (res) (x; y)) or the mutant (/D (mut) (x; y)) population.Dur**in**g **dispersal** N t (x; y)D(x; y) **in**dividuals disperse310 OIKOS 115:2 (2006)

from the site at (x,y). Each **of** the neighbour**in**g sitescan be the dest**in**ation **of** the emigration with equalprobabilities. Similarly, -consider**in**g four neighbour**in**gpatches one fourth **of** the dispers**in**g **in**dividuals fromeach neighbour**in**g patch head toward the site at (x,y)(third term **of** the equations). s portion **of** the dispers**in**g**in**dividuals die dur**in**g **dispersal**, and only (1/s) portionarrive to their dest**in**ations.Density **dependent** **dispersal**We chose a general **density**-**dependent** **dispersal** functionD(N(x; y)) (Eq. 3.), that measures the fraction **of****in**dividuals dispers**in**g from a patch at co-ord**in**ates(x,y) as function **of** population **density**.D 0D(N(x; y))(3)1 Exp[(N(x; y) b) × a]where, N denotes the total population size (/NN (res) N (mut) ) **in** a patch at co-ord**in**ates (x,y); D 0 is the maximal**dispersal** rate (/D 0 (0; 1)); b is the **in**flection po**in**t **of** thefunction (/b (0:01; 30)); and a governs the sharpness **of**the **in**crease at the **in**flections po**in**t (/a (10 4 ; 10));however a and D 0 together determ**in**e the slope **of** thefunction at the **in**flection po**in**t, which is aD 04 : Similar**density**-**dependent** **dispersal** function was used **in** thestudy on population dynamics by Ylikarjula et al.(2000). Please note that the restriction on the values **of**a, b are dictated either by biological reality or by thetechnical constra**in**ts **of** the simulation. Resident andmutant each have three parameters (D 0 , a and b) todeterm**in**e the shape **of** the **dispersal** function.With the right set **of** parameters the **dispersal** functioncan have the shape **of** a saturat**in**g curve, exponentialgrowth, sigmoid curve or be (nearly) l**in**ear (Fig. 1) **in** therange **of** densities that are realized by the population.Also the function can be both positively and negatively**density**-**dependent**, depend**in**g on the sign **of** a (Fig. 1c).S**in**ce reproduction success is the greatest when population**density** tends to zero **in** Eq. 1 we assumed that**dispersal** **in**creases with **density** (a/0). Negative **density**dependenceis possible if low population **density** causessocial dysfunctions or makes f**in**d**in**g a mat**in**g partnerdifficult. However, the Allee effect (Stephens and Sutherland1999) is not **in**cluded **in** our model.Evolutionary dynamicsWe followed the **evolution**ary change **of** the threeparameters **of** the **dispersal** function (D 0 , b, a). Eachround **of** **evolution** consisted **of** (a) an **in**itial phase,dur**in**g which the resident population could reachequilibrium; (b) the **in**troduction **of** a rare mutant; and(c) the competition between the resident and the mutant.**The** w**in**ner **of** the competition was resident **in** the nextround, and a new mutant was chosen (d). In either case**evolution** cont**in**ued with a new **evolution**ary step, untilthe values **of** the evolv**in**g traits settled to a non**in**vadableoptimal value.This ‘‘end-po**in**t’’ is a global **evolution**ary stable**dispersal** function (ESDF), as mutant can have arbitrarytrait values from the trait space.(a) Initial phaseInitially all patches conta**in**ed K resident **in**dividuals.**The** resident was allowed to grow and disperse alone for100 time steps. In the first **evolution**ary step the traits **of**the resident (D 0 (res) , b (res) , a (res) ) were determ**in**ed randomly.(b) Introduction **of** a rare mutantMutant **in**dividuals were **in**troduced **in**to one **of** thepatches **in** a number equal to 10% **of** the residentsubpopulation **in** that patch (see part (d) below for themethod **of** creat**in**g the trait values **of** the new mutant).**The** number **of** mutants **in**troduced is equivalent to 1%**of** the whole population. Introduc**in**g exactly 1 mutantresults **in** the quantitatively same result, but simulationprogresses more slowly.(c) Competition**The** simulation **of** the population dynamics was cont**in**ueduntil either the size **of** the mutant populationbecame negligible (less than 10 15 ), the size **of** theresident population fell below 1, or the 50 000th timestep was reached, whichever occurred first.(d) Outcome **of** competitionIf the mutant won the competition, i.e. the populationsize **of** the resident fell below 1, then the mutant becamethe new resident. **The** new mutant was then generated byrepeat**in**g the mutation that generated the successfulmutant **in** the previous round: the trait affected by thesuccessful mutation was modified further by the samedegree and **in** the same direction as **in** the previousmutational step. For example, if the w**in**n**in**g mutant wasgenerated by reduc**in**g D 0 by 0.1, then the D 0 **of** the nextmutant was aga**in** decreased by 0.1 compared to thew**in**n**in**g mutant. This rule was implemented for practicalreasons, as it speeds up the course **of** **evolution**, but hasno effect on the f**in**al outcome. (Please note that allow**in**gmore than one parameter to mutate does not change theresult, albeit it takes more **evolution**ary steps to reachthe ESDF. **The** slow**in**g-down **of** **evolution** is due to thefact, that **in** this case the above rule **of** ‘‘make amutational step **in** the previously successful direction’’cannot be implemented.)On the other hand, if the resident won the competition,then the new mutant was chosen randomly.Compared to the resident, only one **of** the parameters**of** the **dispersal** function was changed **in** the mutant.OIKOS 115:2 (2006) 311

Environmental fluctuation ( σ )0.60.81.01.2Percentage (%)Percentage (%)Percentage (%)Percentage (%)0.30 PopulationD 0 = 0.26Dispersal0.25 α = 0.09β = 1.360.200.150.100.050.000.0 0.5 1.0 1.5 2.00.50.40.30.20.10.00.0 0.5 1.0 1.5 2.00.50.40.30.20.10.00.0 0.5 1.0 1.5 2.00.50.40.30.20.1D 0 = 0.33α = 0.07β = 1.24D 0 = 0.41α = 0.06β = 1.21D 0 = 0.50α = 0.05β = 1.28γ Contest competitionScramble competition0.81.01.11.5Percentage (%)Percentage (%)Percentage (%)Percentage (%)0.300.250.200.150.100.050.000.0 0.5 1.0 1.5 2.00.300.250.200.150.100.050.300.250.200.150.100.050.000.0 0.5 1.0 1.5 2.00.300.250.200.150.100.05D 0 = 0.24α = 0.12β = 0.98D 0 = 0.26α = 0.09β = 1.36PopulationDispersal0.000.0 0.5 1.0 1.5 2.0D 0 = 0.25α = 0.08β = 1.55D 0 = 0.26α = 0.06β = 2.290.000.0 0.5 1.0 1.5 2.0 2.5 3.00.00.0 0.5 1.0 1.5 2.0Population **density**Fig. 4. **The** effect **of** **in**creas**in**g environmental fluctuation.Evolutionary stable **dispersal** function (ESDF) (solid l**in**e) andthe distribution **of** population sizes (grey bars) at different levels**of** environmental fluctuation. Environmental fluctuation (s)**in**creases from the top toward the bottom panel, values **of** s areshown next to the panels. Dispersal mortality is s/0.4 **in** allcases. (l/2, K/1 and g/1).Negative autocorrelation **in** environmental noise selectsfor sharper **in**crease **in** **dispersal** propensity abovethe carry**in**g capacity (Fig. 6). As good years are unlikelyto be followed by another good year, **in**dividuals arebetter **of**f leav**in**g favourable sites. On the other hand, athigh temporal autocorrelation the propensity to dispersefrom a good site is low, and only a small fraction **of**the population disperses to **of**fset the effect **of** crowd**in**g(Fig. 6). Both high and low population densities arerelatively more frequent compared to the uncorrelatedcase.We also tested the effect **of** **dispersal** distance bycompar**in**g local **dispersal** to the frequently used globalFig. 5. **The** effect **of** the type **of** local competition. ESDF (solidl**in**e) and the distribution **of** population sizes (grey bars) atdifferent g (**in**creases from the top toward the bottom panel,values **of** g are shown next to the panels). (l/2, K/1, s/0.4and s/0.6).**dispersal** rule (i.e. **in** the case where **in**dividuals candisperse to every patch with the same probability). Wefound that our results are robust with regard to **dispersal**distance as employ**in**g global **dispersal** did not changethe quantitative predictions **of** our model. Poethke andHovestadt (2002) have found similar result **in** theirmodels.Discussion**The** **evolution** **of** **dispersal** strategies has attractedconsiderable theoretical **in**terest for years. As we emphasized**in** the Introduction, most **of** these works studied**spatial**ly structured models, where the **spatial** structurewas generally present **in** an implicit manner and it wasOIKOS 115:2 (2006) 315

Temporal autocorrelation ( κ )–0.9–0.20+0.2+0.9Percentage (%)Percentage (%)Percentage (%)Percentage (%)Percentage (%)0.250.200.150.100.05population**dispersal**0.000.0 0.5 1.0 1.5 2.00.25D 0 = 0.26α = 0.110.20 β = 1.310.150.100.050.000.0 0.5 1.0 1.5 2.00.300.250.200.150.100.050.000.0 0.5 1.0 1.5 2.00.25D 0 = 0.20α = 0.090.20 β = 1.330.150.100.050.000.0 0.5 1.0 1.5 2.00.250.200.150.100.05D 0 = 0.37α = 0.12β = 1.32D 0 = 0.26α = 0.09β = 1.36D 0 = 0.04α = 1.30β = 1.130.000.0 0.5 1.0 1.5 2.0Population **density**Fig. 6. Correlated environmental noise. ESDF (solid l**in**e)and the distribution **of** population sizes (grey bars) at differentautocorrelation **of** environmental fluctuation. Temporalautocorrelation (k) **in**creases from negatively correlated (atthe top) toward positively correlated (at bottom), values **of** kare shown next to the panels. (l/2, K/1, g/1, s/0.4 ands/0.6).assumed that **dispersal** is **density** **in****dependent**. Eventhose models that considered **density**-**dependent** **dispersal**,used either a l**in**ear **dispersal** function (Travis et al.1999), a highly nonl**in**ear threshold-like function (Jánosiand Scheur**in**g 1997), or based on the theory **of** idealfree distribution considered a fixed saturat**in**g function**of** **density** (Poethke and Hovestadt 2002) all **of** whichconstra**in** the possible outcome **of** **evolution**.We studied the **evolution** **of** **density**-**dependent** **dispersal****in** a **spatial**ly realistic coupled map lattice model.**The** employed **dispersal** function has three parameters,all three **of** which could be varied **in** the course **of****evolution**. **The**se vary**in**g phenotypic traits potentiallyallow a wide range **of** possible types **of** **density**-dependencefor the **dispersal** (Fig. 1). With the help **of** ourmodel, earlier f**in**d**in**gs can be tested for their robustness,and deeper **in**sights **in**to the correspond**in**g ecologicaland **evolution**ary processes can be established. In thefollow**in**g we compare our results with the theoreticalf**in**d**in**gs **of** earlier studies and the results **of** experimentalworks.Ext**in**ction rate**The** threat **of** ext**in**ction **of** a local subpopulation is one**of** the driv**in**g factors promot**in**g the **evolution** **of****dispersal** (Gandon and Michalakis 2001). In **spatial**lystructured models it has been shown that disregard**in**gk**in**-competition and **in**breed**in**g depression, and tak**in**g**in**to consideration that **dispersal** is costly, local ext**in**ctionis necessary for non-zero **dispersal** rate to be**evolution**ary favourable (Parv**in**en 1999). Most theoretical**in**vestigations predict that **dispersal** rate is positivelycorrelated with the probability **of** local ext**in**ction(Paradis 1998, Gandon and Michalakis 2001, Poethkeet al. 2003). But Parv**in**en and co-workers (2003)demonstrated that without demographic stochasticitythe adapted **dispersal** rate exhibits a maximum for**in**termediate rates **of** disturbance. On the other hand,Poethke and co-workers (2003) concluded that ext**in**ctioncaused solely by demographic fluctuation has ambiguouseffect on **dispersal** rates. Accord**in**g to the results **of** theirmodel the propensity **of** **in**dividuals to disperse maycorrelate positively, negatively or ambiguously with localext**in**ction rates. In our model, ext**in**ction was solelycaused by environmental fluctuation. In concordancewith most earlier studies we concluded that **in**creas**in**gext**in**ction rates **in**crease the **evolution**arily stable level**of** the mean **dispersal** rates. **The** available experimentalevidence also suggests that **dispersal** rate mostlycorrelates positively with the local ext**in**ction rate(Friedenberg 2003). Friedenberg (2003) exposed twostra**in**s **of** C. elegans, where one has a higher propensity**in** disperse to a patchy environment with the possibility**of** local ext**in**ction. After six generations the stra**in** withhigher **dispersal** rate was clearly dom**in**ant **in** thepopulation, suggest**in**g that frequent ext**in**ction selectsfor higher **dispersal** rate. Also **in** the review **of** Dennoet al. (1991), the percentage **of** macroptery **in** planthopperswas negatively correlated with habitat persistence,**in**dicat**in**g that ext**in**ction rate was **in**deed positivelycorrelated with **dispersal** propensity.316 OIKOS 115:2 (2006)

Cost **of** **dispersal**Dispersal is generally considered to be costly for the**in**dividuals (Hanski 1998, Gandon and Michalakis 2001,Ims and Hjermann 2001), mostly because **of** the**in**creased mortality dur**in**g **dispersal**, albeit other factorsmight also have some role **in** this cost (e.g. the metaboliccost **of** **dispersal** ands its effect on fecundity, R**of**f 1977,1984, Rank**in** and Burchsted 1992). While it is notoriouslydifficult to measure **dispersal** mortality, Hanskiet al. (Hanski et al. 2000, Petit et al. 2001) were able toderive the **dispersal** mortality **of** two butterfly species byanalyz**in**g markreleaserecapture data with the virtualmigration model (Hanski et al. 2000). Furthermore it iswell known that animals cross**in**g roads experience**in**creased mortality (Ashley and Rob**in**son 1996, Bonnetet al. 1999, Carr and Fahrig 2001, Aresco 2003). It hasbeen shown that **in** the idealistic case when **dispersal** hasno cost, **in**dividuals would disperse unconditionally(Parv**in**en 1999), while the realistic case, when **dispersal**is costly, is expected to select for lower rates **of** **dispersal**.A number **of** theoretical studies have concluded that**dispersal** rate is a decreas**in**g function **of** **dispersal** cost(Travis et al. 1999, Poethke and Hovestadt 2002, Cadetet al. 2003, Parv**in**en et al. 2003, Poethke et al. 2003). Inthe special case **of** **density**-**dependent** **dispersal** when**in**dividuals only disperse when the local population**density** reaches a certa**in** threshold value, this threshold**density** **in**creases with **in**creas**in**g cost **of** **dispersal**(Poethke and Hovestadt 2002). When a l**in**ear **dispersal**function is considered, the **in**crease **in** the cost **of****dispersal** lowers the **in**tercept **of** the l**in**e, but has nosignificant effect on the slope (Travis et al. 1999). **The****in**tercept **of** the l**in**ear **dispersal** function ma**in**ly **in**fluencesthe **dispersal** rates at very low densities. We found**in** our more general model that from low densities todensities close to the carry**in**g capacity **of** the patch the**dispersal** function was nearly l**in**ear (Fig. 3). **The****in**tercept **in** our case was also negatively correlatedwith **dispersal** mortality, and the **in**flection po**in**t (b)**of** the evolved **dispersal** function (which is analogous tothe threshold **density** **of** the model **of** Poethke andHovestadt (2002)) was positively correlated with **dispersal**mortality (Fig. 2). So, by us**in**g a general evolv**in**g**dispersal** function we were able to comb**in**e the results **of**both previous models that considered **density**-**dependent****dispersal**.Dispersal functionDensity-**dependent** **dispersal** has been experimentallydemonstrated **in** a number **of** studies (e.g. Microtusoeconomus: Aars and Ims 2000; Paroxyna plantag**in**is:Albrectsen and Nachman 2001; Onychiurus armatus:Bengtsson et al. 1994; Prokelisia marg**in**ata andP. dolus: Denno et al. 1996; Simulium vittatum: Fonsecaand Hart 1996; Anisopteromalus calandrae: French andTravis 2001; Lynx pard**in**us: Gaona et al. 1998; Trirhabdavirgata: Herzig 1995; Tendora s**in**ensis: Hurd andEisenberg 1984; Lacerta viviparia: Lecomte et al. 2004;Carc**in**us maenas: Moksnes 2004; Metisa plana: Rha**in**dset al. 2002; reviewed by Ga**in**es and McClenaghan 1980,Denno et al. 1991, Matthysen 2005). Moreover, an **in**creas**in**gnumber **of** studies are available where **density****dependent****dispersal** is demonstrated by measur**in**g**dispersal** rates at more than two densities, thus thecharacter **of** **density**-dependence can be estimated(Denno et al. 1991, 1996, Fonseca and Hart 1996,Albrectsen and Nachman 2001, Rha**in**ds et al. 2002,Moksnes 2004). **The** common nature **of** these studies isthat the observed **dispersal** functions were always nonl**in**ear.Dispersal is very moderate or miss**in**g at low**density** and starts to **in**crease abruptly at a thresholdlevel. For example, Fonseca and Hart (1996) studiedthe **density** dependence **of** the **dispersal** **of** black fly(Simulium vittatum) neonates. At low densities(02 larvae mm 2 ) **dispersal** was hardly detectable(Fig. 2 **in** Fonseca and Hart 1996), at moderate densities(46 larvae mm 2 ) the **dispersal** rate began to **in**crease,and **in**creased with an **in**creas**in**g rate at higher densities(8/ larvae mm 2 ). Albrectsen and Nachman (2001)obta**in**ed similar results **in** their **in**vestigation **of** the**dispersal** **of** the females **of** the tetripid fly, Paroxynaplantag**in**is (the **dispersal** **of** the male flies was not**density**-**dependent**). **The**y measured the **dispersal** rateat densities **of** 10, 40 and 100 flies per patch. **The**equilibrium **density** **of** the female flies was 12.009/3.86**in**dividual per patch. **The** **dispersal** rate **in**creasedexponentially above the equilibrium **density** (see Fig. 3**in** Albrectsen and Nachman 2001). **The** **density**-**dispersal****of** juvenile shore crabs (Carc**in**us maenas) also exhibit amarked **in**crease at higher densities (Fig. 3 **in** Moksnes2004). At densities **of** 2 and 6 **in**dividuals per musselpatch around 20% **of** the crabs emigrated, whereasproportional emigration **in**creased to 45% at a **density****of** 18 crab/mussel patch. Furthermore, **in** certa**in**planthopper species (Homoptera: Delaphidacidae) thefemales exhibit **density**-**dependent** **dispersal**, where thepercentage **of** macroptery (and thus **dispersal** rate) eitherdepends exponentially on **density** (for example forSogatella furcifera, Fig. 3C **in** Denno et al. 1991), or**in** other planthoppers (Nilaparvata lugens, Javsellapellucida, Laodelphax stritellus) the **dispersal** functionexhibits a threshold, above which **dispersal** rates **in**creaseat a higher rate (Fig. 3C **in** Denno et al. 1991).Our results are **in** a remarkably good agreement withthese experimental f**in**d**in**gs. Thus it is **of** paramountimportance to use such a flexible **dispersal** function **in**the study **of** the **evolution** **of** **dispersal** rates. **The** exist**in**gmodels that employed **density**-**dependent** **dispersal** functionswere able to capture some **of** the features **of** theevolved **dispersal** function. Travis and co-workers (1999)OIKOS 115:2 (2006) 317

demonstrated **in** their model, which allowed for l**in**ear**density**-**dependent** **dispersal**, that **in**dividuals tend not todisperse if local population **density** is below the equilibrium,and always disperse if the local population**density** is above twice the equilibrium **density**. Because**dispersal** is costly, it is worthwhile to disperse only iflocal conditions are very bad (**in** this case because **of**crowd**in**g). Poethke and Hovestadt (2002) put this theory**in**to their model, by assum**in**g a **dispersal** function whichhad zero **dispersal** rate below a threshold **density**, andthen it was a saturat**in**g function **of** **density**. In somecases such saturat**in**g **dispersal** functions were alsodemonstrated experimentally (for females **of** Prokelisiamarg**in**ata and both sexes **of** Prokenisia dolus (Fig. 2 **in**Denno et al. 1991) and for Anisoptermalus calandrae(French and Travis 2001)). Furthermore, Rha**in**ds et al.(2002) found similar trends **in** the balloon**in**g rate **of**bagworm larvae. But **in** these experiments the steep**in**crease **in** **dispersal** rate occurred at very low densities,well below the carry**in**g capacity **of** the habitat, andthus probably demonstrates a different phenomenon.Accord**in**gly, assum**in**g such a **dispersal** function a prioriunduly restricts the **evolution**ary outcome. Poethke andHovestadt (2002) and Metz and Gyllenberg (2001)derived the above mentioned saturat**in**g **dispersal** functionfrom the theory **of** ideal free distribution (IFD).IFD theory assumes that **in**dividuals’ **dispersal** strategyleads to equal fitness for all local habitats. However thelimited **in**formation about the environment frustrates theemergence **of** IFD. Ranta et al. (1999) showed that withlimited knowledge **of** the environment populations couldnot achieve ideal free distribution. Furthermore, goodpatches will be underpopulated (population size will bebelow the actually carry**in**g capacity), and bad patcheswill be overpopulated (Kennedy and Gray 1993, Rantaet al. 1999). We experienced the same **in** our model where**in**dividuals only sense the local **density**, but not **in**formedabout the actual carry**in**g capacities **of** the sites theydisperse to.While we have considered a wide range **of** possibleenvironmental and life-history traits, some types **of**environment was left out **of** the current **in**vestigation.Most notably **spatial**ly correlated and temporally and**spatial**ly correlated environment rema**in**s to be studied **in**the future.To summarize, we corroborate that **in** constantenvironment it is generally unfavourable to disperse. Ina **noisy** environment, we found that the key feature **of** theevolved **dispersal** function are that (1) below a certa**in**threshold the propensity **of** **in**dividuals to disperse is verylow, (2) above this threshold the **dispersal** rate grows **in**an accelerat**in**g manner (exponentially **in** our model)with further **in**crease **in** **density**. This result proved to berobust to changes **in** the type **of** population dynamics(fixed po**in**t vs limit-cycle); local competition (contest vsscramble) and temporal autocorrelation **of** the environmentalnoise.Acknowledgements We thank Veijo Kaitala, ThomasHovestadt and Viktor Müller for their helpful comments onan earlier version **of** the manuscript. **The** project was subsidizedby the Hungarian National Research Fund (OTKA T037726and T049692). ÁK is a postdoctoral fellow **of** OTKA(D048406).ReferencesAars, J. and Ims, R. A. 2000. Population dynamic and geneticconsequences **of** **spatial** **density**-**dependent** **dispersal** **in**patchy populations. Am. Nat. 155: 252265.Albrectsen, B. and Nachman, G. 2001. Female-biased **density****dependent****dispersal** **of** a tetripid fly **in** a fragmented habitatand its implication for population regulation. Oikos 94:263272.Amarasekare, P. 2004a. **The** role **of** **density**-**dependent** **dispersal****in** source-s**in**k dynamics. J. **The**or. Biol. 226: 159168.Amarasekare, P. 2004b. Spatial variation and **density**-**dependent****dispersal** **in** competitive coexistence. Proc. R. Soc. Lond.B 271: 14971506.Aresco, M. J. 2003. Highway mortality **of** turtles and otherherpet**of**auna at Lake Jackson, Florida, USA and theefficacy **of** a temporary fence/culvert system to reduce roadkills. In: Irw**in**, C. L., Garrett, P. and McDermott, K. P.(eds), Proc. Int. Conf. on Ecology and Transportation, pp.433449.Armsworth, P. R. and Roughgarden, J. E. 2005. **The** impact **of**directed versus random movement on population dynamicsand biodiversity patterns. Am. Nat. 165: 449465.Ashley, E. P. and Rob**in**son, J. T. 1996. Road mortality **of**amphibians, reptiles and other wildlife on the Long Po**in**tcauseway, Lake Erie, Ontario. Can. Field-Nat. 110: 403412.Bengtsson, G., Hedlund, K. and Rundgren, S. 1994. Food- and**density**-**dependent** **dispersal**: evidence from a soil collembolan. J. Anim. Ecol. 63: 513520.Bonnet, X., Naulleau, G. and Sh**in**e, R. 1999. **The** dangers **of**leav**in**g home: **dispersal** and mortality **in** snakes. Biol.Conserv. 89: 3950.Bowler, D. E. and Benton, T. G. 2005. Causes and consequences**of** animal **dispersal** strategies: relat**in**g **in**dividual behaviourto **spatial** dynamics. Biol. Rev. 80: 205225.Buddle, C. M. and Rypstra, A. L. 2003. Factors **in**itiat**in**gemigration **of** two wolf spider species (Areneae: Lycosidae)**in** an agroecosystem. Environ. Entomol. 32: 895.Cadet, C., Ferriere, R., Metz, J. A. J. et al. 2003. **The** **evolution****of** **dispersal** under demographic stochasticity. Am. Nat.162: 427441.Carr, L. W. and Fahrig, L. 2001. Effect **of** road traffic on twoamphibian species **of** different vagility. Conserv. Biol. 15:10711078.Clobert, J., Danch**in**, E., Dhondt, A. A. et al. 2001. Dispersal. Oxford Univ. Press.Cudd**in**gton, K. M. and Yodzis, P. 1999. Black noise andpopulation persistence. Proc. R. Soc. Lond. B 266: 969973.Denno, R. F., Roderick, G. K., Olmstead, K. L. et al. 1991.Density-**dependent** migration **in** planthoppers (Homoptera:Delphaciadea): the role **of** habitat persistence. Am. Nat.138: 15131541.Denno, R. F., Roderick, G. K., Peterson, M. A. et al. 1996.Habitat persistence underl**in**es **in**traspecific variation **in** the**dispersal** strategies **of** planthoppers. Ecol. Monogr. 66:389408.Dieckmann, U. 1997. Can adaptive dynamics **in**vade. Trends.Ecol. Evol. 12: 128131.318 OIKOS 115:2 (2006)

Dieckmann, U., O’Hara, B. and Weisser, W. 1999. **The****evolution**ary ecology **of** **dispersal**. Trends Ecol. Evol. 14:8890.Ferrière, R., Belth**of**f, J. R., Olivieri, I. et al. 2000. Evolv**in**g**dispersal**: where to go next. Trends Ecol. Evol. 15: 5 7.Fonseca, D. M. and Hart, D. D. 1996. Density-**dependent****dispersal** **of** black fly neonates is mediated by flow. Oikos75: 4958.French, D. R. and Travis, J. M. J. 2001. Density-**dependent****dispersal** **in** host-parasitoid assemblages. Oikos 95: 125135.Friedenberg, N. A. 2003. Experimental **evolution** **of** **dispersal** **in**spatiotemporally variable microcosms. Ecol. Lett. 6: 953959.Ga**in**es, M. S. and McClenaghan, L. R.., Jr. 1980. Dispersal **in**small mammals. Annu. Rev. Ecol. Syst. 11: 163196.Gandon, S. and Michalakis, Y. 2001. Multiple causes **of** the**evolution** **of** **dispersal**. In: Clobert, J., Danch**in**, E.,Dhondt, A. A. et al. (eds), Dispersal. Oxford Univ. Press.Gaona, P., Ferreras, P. and Delibes, M. 1998. Dynamics andviability **of** a metapopulation **of** the endangered Iberian lynx(Lynx pard**in**us). Ecol. Monogr. 68: 349370.Gyllenberg, M. and Metz, J. A. J. 2001. On fitness **in** structuredmetapopulations. J. Math. Biol. 43: 545560.Gyllenberg, M., Parv**in**en, K. and Dieckmann, U. 2002.Evolutionary suicide and **evolution** **of** **dispersal** **in** structuredmetapopulations. J. Math. Biol. 45: 79105.Hamilton, W. D. and May, R. M. 1977. Dispersal **in** stablehabitats. Nature 269: 578581.Hanski, I. 1998. Metapopulation dynamics. Nature 396: 4149.Hanski, I., Alho, J. and Moilanen, A. 2000. Estimat**in**g theparameters **of** survival and migration **of** **in**dividuals **in**metapopulations. Ecology 81: 239251.He**in**o, M. 1998. Noise colour, synchrony and ext**in**ctions **in****spatial**ly structured populations. Oikos 83: 368375.He**in**o, M., Ripa, J. and Kaitala, V. 2000. Ext**in**ction risk undercoloured environmental noise. Ecography 23: 177184.Herzig, A. L. 1995. Effects **of** population **density** on longdistance**dispersal** **in** the goldenrod beetle Trirhabda virgata. Ecology 76: 20442054.Hurd, L. E. and Eisenberg, R. M. 1984. Experimental **density**manipulation **of** the predator Tendora s**in**ensis (Orthoptera:Mantidae) **in** an old-field community. I. Mortality, developmentand **dispersal** **of** juvenile mantids. J. Anim. Ecol. 53:269281.Ims, R. A. and Hjermann, D. Ø. 2001. Condition-**dependent****dispersal**. In: Clobert, J., Danch**in**, E., Dhondt, A. A.et al. (eds), Dispersal. Oxford Univ. Press, pp. 203216.Jánosi, I. M. and Scheur**in**g, I. 1997. On the **evolution** **of** **density****dependent** **dispersal** **in** an **spatial**ly structured populationmodel. J. **The**or. Biol. 187: 397408.Johanessen, E. and Andreasen, H. P. 1998. Survival andreproduction **of** resident and immigrant female root voles(Microtus oeconomus ). Can. J. Zool. 76: 763766.Johnson, M. L. 1990. Evolution **of** **dispersal**: theoretical modelsand empirical tests us**in**g birds and mammals. Annu. Rev.Ecol. Syst. 21: 449480.Johst, K. and Brandl, R. 1997. Evolution **of** **dispersal**: theimportance **of** the temporal order **of** reproduction and**dispersal**. Proc. R. Soc. Lond. B 264: 2330.Kennedy, M. and Gray, R. D. 1993. Can ecological theorypredict the distribution **of** forag**in**g animals? A criticalanalysis **of** experiments on the ideal free distribution. Oikos 71: 163166.Kisdi, É. 2002. Dispersal: risk spread**in**g versus local adaptation. Am. Nat. 159: 579596.Lamb**in**, X., Aars, J. and Piertney, S. B. 2001. Dispersal,**in**traspecific competition, k**in** competition and k**in** facilitation:a review **of** the empirical evidence. In: Clobert, J.,Danch**in**, E., Dhondt, A. A. et al. (eds), Dispersal. OxfordUniv. Press, pp. 123142.Lecomte, J., Boudjemadi, K., Sarraz**in**, F. et al. 2004. Connectivityand homogenisation **of** population sizes: anexperimental approach **in** Lacerta viviparia. J. Anim.Ecol. 73: 179189.Mathias, A., Kisdi, É. and Olivieri, I. 2001. Divergent **evolution****of** **dispersal** **in** a heterogeneous landscape. Evolution 55:246259.Matthysen, E. 2005. Density-**dependent** **dispersal** **in** birds andmammals. Ecography 28: 403416.Maynard-Smith, J. and Slatk**in**, M. 1973. Maynard Smith, J.and Slatk**in**, M. 1973. **The** stability **of** predator preysystems. Ecology 54: 384391.McPeek, M. A. and Holt, R. D. 1992. **The** **evolution** **of** **dispersal****in** **spatial**ly and temporally vary**in**g environments. Am.Nat. 140: 10101027.Metz, J. A. J. and Gyllenberg, M. 2001. How should we def**in**efitness **in** structured metapopulation models? Includ**in**g anapplication to the calculation **of** **evolution**ary stable **dispersal**strategies. Proc. R. Soc. Lond. B 268: 499508.Metz, J. A. J., Nisbet, R. and Geritz, S. A. H. 1992. How shouldwe def**in**e ’’fitness’’ for general ecological scenarios? Trends Ecol. Evol. 7: 198202.Metz, J. A. J., Geritz, S. A. H., Meszéna, G. et al. 1996.Adaptive dynamics: a geometrical study **of** the consequence**of** nearly faithful reproduction. In: van Strien, S. J. andVerduyn Lunel, S. M. (eds), Stochastic and **spatial** structures**of** dynamical systems. North Holland, pp. 183231.Moksnes, P. O. 2004. Interference competition for space **in**nursery habitats: **density**-**dependent** effects on growth and**dispersal** **in** juvenile shore crabs Carc**in**us maenas. Mar.Ecol. Progr. Ser. 281: 181191.Motro, U. 1991. Avoid**in**g **in**breed**in**g and sibl**in**g competition:the **evolution** **of** sexual dimorphism for **dispersal**. Am.Nat. 137: 108115.Olivieri, I., Michalakis, Y. and Gouyon, P. H. 1995. Metapopulationgenetics and the **evolution** **of** **dispersal**. Am.Nat. 146: 202228.O’Ria**in**, M. J. and Braude, S. 2001. Inbreed**in**g versus outbreed**in**g**in** captive and wild populations **of** naked mole-rats. In: Clobert, J., Danch**in**, E., Dhondt, A. A. et al. (eds),Dispersal. Oxford Univ. Press.Paradis, E. 1998. Interaction between **spatial** and temporalscales **in** the **evolution** **of** **dispersal** rate. Evol. Ecol. 12:235244.Parv**in**en, K. 1999. Evolution **of** migration **in** a metapopulation. Bull. Math. Biol. 61: 531550.Parv**in**en, K., Dieckmann, U., Gyllenberg, M. et al. 2003.Evolution **of** disersal **in** metapopulation with local **density**dependence and demographic stochasticity. J. Evol. Biol.16: 143153.Perr**in**, N. and Goudet, J. 2001. Inbreed**in**g, k**in**ship, and the**evolution** **of** natal **dispersal**. In: Clobert, J., Danch**in**, E.,Dhondt, A. A. et al. (eds), Dispersal. Oxford Univ. Press, pp.143154.Petchey, O. L., Gonzalez, A. and Wilson, H. B. 1997. E¡ects onpopulation persistence: the **in**teraction between environmentalnoise colour, **in**traspecific competition and space. Proc.R. Soc. Lond. B 264: 18411847.Petit, S., Moilanen, A., Hanski, I. et al. 2001. Metapopulationdynamics **of** the bog fritillary butterfly: movements betweenhabitat patches. Oikos 92: 491500.Poethke, H. J. and Hovestadt, T. 2002. Evolution **of** **density**andpatch-size-**dependent** **dispersal** rates. Proc. R. Soc.Lond. B 269: 637645.Poethke, H. J., Hovestadt, T. and Mitesser, O. 2003. Localext**in**ction and the **evolution** **of** **dispersal** rates: causes andcorrelations. Am. Nat. 161: 631640.Rank**in**, M. A. and Burchsted, J. C. A. 1992. **The** cost **of**migration **in** **in**sects. Annu. Rev. Entomol. 37: 533559.Ranta, E., Lundberg, P. and Kaitala, V. 1999. Resourcematch**in**g with limited knowledge. Oikos 86: 383.Rha**in**ds, M., Gries, G., Ho, C. T. et al. 2002. Dispersal bybagworm larvae, Matisa plana: effects**of** opulation **density**,OIKOS 115:2 (2006) 319