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

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

OIKOS 115: 308320, 2006The evolution of density-dependent dispersal in a noisy spatialpopulation modelÁdám Kun and István ScheuringKun, Á. and Scheuring, I. 2006. The evolution of density-dependent dispersal in anoisy 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 spatially and temporallyheterogeneous habitats. However, the key question, what kind of dispersal strategy isoptimal in a particular situation, has remained unanswered. We studied the evolutionof density-dependent dispersal in a coupled map lattice model, where the populationdynamics are perturbed by external environmental noise. We used a very flexibledispersal function to enable evolution to select from practically all possible types ofmonotonous density-dependent dispersal functions. We treated the parameters of thedispersal function as continuously changing phenotypic traits. The evolutionary stabledispersal strategies were investigated by numerical simulations. We pointed 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 rateincreases in an accelerating manner above this threshold. Decreasing the cost ofdispersal increases the skewness of the population density distribution, while increasingthe environmental noise causes more pronounced bimodality in this distribution. Incase of positive temporal autocorrelation of the environmental noise, there is nodispersal below the threshold, and only low dispersal below it, on the other hand withnegative autocorrelation practically all individual disperses above the threshold. Wefound our results to be in good concordance with empirical observations.Á. Kun (, Dept of Plant Taxonomy and Ecology, EötvösLoránd Univ., Pázmány P. sétány 1/C, HU-1117 Budapest, Hungary. I. Scheuring,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 influences the dynamics and persistence of populations,the distribution and abundance of species, the levelof genetic diversity and community structure (reviewedby Dieckmann et al. 1999, Ferrière et al. 2000, Clobertet al. 2001). Dispersal is costly for the individual (Hanski1998), but it can give evolutionary benefit for severalreasons. Dispersal helps avoiding kin competition(Ronce et al. 1998, Gandon and Michalakis 2001,Lambin et al. 2001, Perrin and Goudet 2001) andprevents inbreeding (Hamilton and May 1977, Motro1991, Gandon and Michalakis 2001, O’Riain andBraude 2001, Perrin and Goudet 2001). Furthermore itcan be evolutionary favourable if the environment variesboth spatially 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, Parvinen et al. 2003).To make the model analytically tractable, theoreticalinvestigations of the evolution of dispersal rates generallyuse two important assumptions. First, it is frequentlyassumed that dispersal is unconditional (i.e. aconstant fraction of individuals disperse, regardless ofthe 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,Parvinen 1999, Mathias et al. 2001, Gyllenberg et al.2002, Kisdi 2002, Cadet et al. 2003, Parvinen et al.2003). Naturally, this unconditional dispersal is an oversimplification.In reality, most organisms will followmore sophisticated conditional dispersal rules. They 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, Scheuring2001, Buddle and Rypstra 2003). Second, as Travis andDytham (1998) pointed out, almost all theoreticalinvestigations of the evolution of dispersal employedspatially structured metapopulation models (Johst andBrandl 1997, Paradis 1998, Parvinen 1999, Mathias et al.2001, Gyllenberg et al. 2002, Kisdi 2002, Cadet et al.2003, Parvinen 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 spatially implicit representation, spatiallyexplicit models (Jánosi and Scheuring 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 thusdetermined by their position on the lattice. Travis andDytham (1998) also suggested that the evolution ofdispersal should be studied with spatially realisticmodels, which are both spatially explicit and incorporatespatial and/or temporal heterogeneity. While a numberof studies investigated the effect of environmentalvariation (McPeek and Holt 1992, Parvinen 1999,Mathias et al. 2001, Kisdi 2002, Poethke et al. 2003),these employed spatially implicit models. However,Poethke and Hovestadt (2002) found that their resultsdid not change if they employed a spatially implicitmodel instead of the spatially realistic one.The aim of this study was to investigate the evolutionof density-dependent dispersal with the help of aspatially realistic population model. To our knowledge,in the context of the evolution of dispersal rates the onlyspatially explicit model preceding our study was themodel presented by Poethke and Hovestadt (2002). Thetopic is one of great theoretical interest (Travisand French 2000, Armsworth and Roughgarden 2005,Bowler and Benton 2005) and mainly unexplored. Whilethere are ample examples of density-dependent dispersalin nature (Gaines 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, Rhaindset al. 2002, Lecomte et al. 2004, Moksnes 2004,Matthysen 2005), theory lags behind in elucidating themechanism behind it. Albeit there were other theoreticalstudies which incorporated density-dependent dispersal(Jánosi and Scheuring 1997, Ruxton and Rohani 1999,Travis et al. 1999, Poethke and Hovestadt 2002,Amarasekare 2004a, 2004b), they used very rigid shapesfor the dispersal function. These studies either employeda predefined function whose parameters were not subjectto evolutionary change (Ruxton and Rohani 1999,Amarasekare 2004a, 2004b), or, while allowing someparameters of the dispersal function to change duringthe course of evolution, they restricted the shape ofthe dispersal function (Jánosi and Scheuring 1997, Traviset al. 1999, Poethke and Hovestadt 2002).In this study we used a general density-dependentdispersal function which can follow many qualitativelydifferent shapes, and thus allows selection to choose theoptimal shape for the dispersal function. The selecteddispersal function allows us to interpret 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 ofmigration in insects, Rankin and Burchsted (1992) foundthat in some species dispersers actually had a higherreproductive output, but they also presented cases to theopposite (Roff 1977, 1984).Evolution was modelled as a succession of competitionevents between a resident population and an initially raremutant populationThis assumption implies that mutation is infrequent, andthe population can settle to an ecological equilibriumbefore the next mutational event takes place. Theseassumptions are generally applied in adaptive dynamics(Metz et al. 1992, 1996, Dieckmann 1997).The phenotypic trait was inherited asexuallyGenetic variation for migratory traits have been documentedand selection experiments were successful ininducing changes in dispersal traits (Roff and Fairbain2001). The actual genetic background of a conditiondependentdispersal trait is unknown. Thus, inclusionof sexual inheritance would unduly complicate themodel and would not add to its realism. The assumption,that an offspring has the same phenotype as its parentis employed in most strategic models (McPeek andHolt 1992, Jánosi and Scheuring 1997, Paradis 1998,Travis and Dytham 1998, Dieckmann et al. 1999,OIKOS 115:2 (2006) 309

Parvinen 1999, Travis et al. 1999, Cadet et al. 2003,Parvinen et al. 2003, but see Poethke et al. 2003).Population dynamicsWe applied a coupled map lattice model to the studyof the evolution of density-dependent dispersal rates.The square lattice contained 100 rectangular patches.Following the standard techniques the boundaries werewrapped-around (torus), to preclude edge effects.The equations governing the inter patch populationdynamics were the same at each site. At each time stepthe population first experienced growth and then afraction of the local population dispersed according toa dispersal function defined by the ‘‘pre-dispersal’’ localpopulation. Dispersal was assumed to be local, andindividuals dispersed to the four closest neighbouringpatches with equal probability. Dispersal events tookplace simultaneously for all populations. We consideredthe following dynamics at site (x,y) and time t forthe resident (/N (res)tpopulation.(x; y)) and for the mutant (/N (mut)t(x; y))GrowthThe local dynamics were governed by a generalisedlogistic growth equation employed by Maynard-Smithand Slatkin (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 carryingKcapacity of a site. Depending on l and g, Eq. 1 may settleto a fixed point, 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 inthe population. The only important difference is thatthe level of variance can be tuned precisely in thestochastic model, thus chaotic dynamics without lossof generality were not considered here. On the otherhand, both fixed point and limit cycle dynamics wereconsidered in the simulations. Furthermore, for themajority of the simulations, we employed parametercombinations 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 investigatedthree scenario: (1) spatially heterogeneous and temporallyconstant environment; (2) spatially and temporallyheterogeneous environment, with uncorrelated environmentalnoise; and (3) spatially and temporally heterogeneousenvironment, with temporally correlatedenvironmental noise. In all cases, we assumed that thespatial correlation length of the environmental variationcorresponds to the size of the patches (in essence theextent of environmental variation defines the patches).Consequently environmental noise was spatially uncorrelated.(1) In a temporally constant environment j(s; x; y)was independently 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 creating a spatially heterogeneous, but temporallyconstant environment.(2) In an environment with uncorrelated environmentalnoise, j(s; x; y; t) was independently set for eachsite at each time step t to j(s; x; y; t)(8s1);where 8 and constraints 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, Heino 1998, Cuddington and Yodzis1999). Namely, j(s; x; y; t1)(j(s; x; y; t)1)/pk8sffiffiffiffiffiffiffiffiffiffiffiffi1k 2 1; where k governs the degreeof 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 initial phase(see below) the asymptotic level of variance can beused without unduly affecting the outcome (Heinoet 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 during 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.During dispersal N t (x; y)D(x; y) individuals disperse310 OIKOS 115:2 (2006)

from the site at (x,y). Each of the neighbouring sitescan be the destination of the emigration with equalprobabilities. Similarly, -considering four neighbouringpatches one fourth of the dispersing individuals fromeach neighbouring patch head toward the site at (x,y)(third term of the equations). s portion of the dispersingindividuals die during dispersal, and only (1/s) portionarrive to their destinations.Density dependent dispersalWe chose a general density-dependent dispersal functionD(N(x; y)) (Eq. 3.), that measures the fraction ofindividuals dispersing from a patch at co-ordinates(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-ordinates (x,y); D 0 is the maximaldispersal rate (/D 0 (0; 1)); b is the inflection point of thefunction (/b (0:01; 30)); and a governs the sharpness ofthe increase at the inflections point (/a (10 4 ; 10));however a and D 0 together determine the slope of thefunction at the inflection point, which is aD 04 : Similardensity-dependent dispersal function was used in thestudy on population dynamics by Ylikarjula et al.(2000). Please note that the restriction on the values ofa, b are dictated either by biological reality or by thetechnical constraints of the simulation. Resident andmutant each have three parameters (D 0 , a and b) todetermine the shape of the dispersal function.With the right set of parameters the dispersal functioncan have the shape of a saturating curve, exponentialgrowth, sigmoid curve or be (nearly) linear (Fig. 1) in therange of densities that are realized by the population.Also the function can be both positively and negativelydensity-dependent, depending on the sign of a (Fig. 1c).Since reproduction success is the greatest when populationdensity tends to zero in Eq. 1 we assumed thatdispersal increases with density (a/0). Negative densitydependenceis possible if low population density causessocial dysfunctions or makes finding a mating partnerdifficult. However, the Allee effect (Stephens and Sutherland1999) is not included in our model.Evolutionary dynamicsWe followed the evolutionary change of the threeparameters of the dispersal function (D 0 , b, a). Eachround of evolution consisted of (a) an initial phase,during which the resident population could reachequilibrium; (b) the introduction of a rare mutant; and(c) the competition between the resident and the mutant.The winner of the competition was resident in the nextround, and a new mutant was chosen (d). In either caseevolution continued with a new evolutionary step, untilthe values of the evolving traits settled to a noninvadableoptimal value.This ‘‘end-point’’ is a global evolutionary stabledispersal function (ESDF), as mutant can have arbitrarytrait values from the trait space.(a) Initial phaseInitially all patches contained K resident individuals.The resident was allowed to grow and disperse alone for100 time steps. In the first evolutionary step the traits ofthe resident (D 0 (res) , b (res) , a (res) ) were determined randomly.(b) Introduction of a rare mutantMutant individuals were introduced into one of thepatches in a number equal to 10% of the residentsubpopulation in that patch (see part (d) below for themethod of creating the trait values of the new mutant).The number of mutants introduced is equivalent to 1%of the whole population. Introducing exactly 1 mutantresults in the quantitatively same result, but simulationprogresses more slowly.(c) CompetitionThe simulation of the population dynamics was continueduntil 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 byrepeating 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 winning mutant wasgenerated by reducing D 0 by 0.1, then the D 0 of the nextmutant was again decreased by 0.1 compared to thewinning mutant. This rule was implemented for practicalreasons, as it speeds up the course of evolution, but hasno effect on the final outcome. (Please note that allowingmore than one parameter to mutate does not change theresult, albeit it takes more evolutionary steps to reachthe ESDF. The slowing-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 parametersof the dispersal function was changed in the mutant.OIKOS 115:2 (2006) 311

Environmental fluctuation ( σ ) (%)Percentage (%)Percentage (%)Percentage (%)0.30 PopulationD 0 = 0.26Dispersal0.25 α = 0.09β = 1.360. 0.5 1.0 1.5 0.5 1.0 1.5 0.5 1.0 1.5 0 = 0.33α = 0.07β = 1.24D 0 = 0.41α = 0.06β = 1.21D 0 = 0.50α = 0.05β = 1.28γ Contest competitionScramble competition0. (%)Percentage (%)Percentage (%)Percentage (%)0.300. 0.5 1.0 1.5 2.00.300. 0.5 1.0 1.5 2.00.300. 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 = 0.5 1.0 1.5 2.0 2.5 0.5 1.0 1.5 2.0Population densityFig. 4. The effect of increasing environmental fluctuation.Evolutionary stable dispersal function (ESDF) (solid line) andthe distribution of population sizes (grey bars) at different levelsof environmental fluctuation. Environmental fluctuation (s)increases 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 increase in dispersal propensity abovethe carrying capacity (Fig. 6). As good years are unlikelyto be followed by another good year, individuals arebetter off leaving favourable sites. On the other hand, athigh temporal autocorrelation the propensity to dispersefrom a good site is low, and only a small fraction ofthe population disperses to offset the effect of crowding(Fig. 6). Both high and low population densities arerelatively more frequent compared to the uncorrelatedcase.We also tested the effect of dispersal distance bycomparing local dispersal to the frequently used globalFig. 5. The effect of the type of local competition. ESDF (solidline) and the distribution of population sizes (grey bars) atdifferent g (increases 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 individuals candisperse to every patch with the same probability). Wefound that our results are robust with regard to dispersaldistance as employing global dispersal did not changethe quantitative predictions of our model. Poethke andHovestadt (2002) have found similar result in theirmodels.DiscussionThe evolution of dispersal strategies has attractedconsiderable theoretical interest for years. As we emphasizedin the Introduction, most of these works studiedspatially 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 (%) 0.5 1.0 1.5 2.00.25D 0 = 0.26α = 0.110.20 β = 1.310. 0.5 1.0 1.5 2.00.300. 0.5 1.0 1.5 2.00.25D 0 = 0.20α = 0.090.20 β = 1.330. 0.5 1.0 1.5 0 = 0.37α = 0.12β = 1.32D 0 = 0.26α = 0.09β = 1.36D 0 = 0.04α = 1.30β = 0.5 1.0 1.5 2.0Population densityFig. 6. Correlated environmental noise. ESDF (solid line)and the distribution of population sizes (grey bars) at differentautocorrelation of environmental fluctuation. Temporalautocorrelation (k) increases 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 independent. Eventhose models that considered density-dependent dispersal,used either a linear dispersal function (Travis et al.1999), a highly nonlinear threshold-like function (Jánosiand Scheuring 1997), or based on the theory of idealfree distribution considered a fixed saturating functionof density (Poethke and Hovestadt 2002) all of whichconstrain the possible outcome of evolution.We studied the evolution of density-dependent dispersalin a spatially realistic coupled map lattice model.The employed dispersal function has three parameters,all three of which could be varied in the course ofevolution. These varying phenotypic traits potentiallyallow a wide range of possible types of density-dependencefor the dispersal (Fig. 1). With the help of ourmodel, earlier findings can be tested for their robustness,and deeper insights into the corresponding ecologicaland evolutionary processes can be established. In thefollowing we compare our results with the theoreticalfindings of earlier studies and the results of experimentalworks.Extinction rateThe threat of extinction of a local subpopulation is oneof the driving factors promoting the evolution ofdispersal (Gandon and Michalakis 2001). In spatiallystructured models it has been shown that disregardingkin-competition and inbreeding depression, and takinginto consideration that dispersal is costly, local extinctionis necessary for non-zero dispersal rate to beevolutionary favourable (Parvinen 1999). Most theoreticalinvestigations predict that dispersal rate is positivelycorrelated with the probability of local extinction(Paradis 1998, Gandon and Michalakis 2001, Poethkeet al. 2003). But Parvinen and co-workers (2003)demonstrated that without demographic stochasticitythe adapted dispersal rate exhibits a maximum forintermediate rates of disturbance. On the other hand,Poethke and co-workers (2003) concluded that extinctioncaused solely by demographic fluctuation has ambiguouseffect on dispersal rates. According to the results of theirmodel the propensity of individuals to disperse maycorrelate positively, negatively or ambiguously with localextinction rates. In our model, extinction was solelycaused by environmental fluctuation. In concordancewith most earlier studies we concluded that increasingextinction rates increase the evolutionarily stable levelof the mean dispersal rates. The available experimentalevidence also suggests that dispersal rate mostlycorrelates positively with the local extinction rate(Friedenberg 2003). Friedenberg (2003) exposed twostrains of C. elegans, where one has a higher propensityin disperse to a patchy environment with the possibilityof local extinction. After six generations the strain withhigher dispersal rate was clearly dominant in thepopulation, suggesting that frequent extinction selectsfor higher dispersal rate. Also in the review of Dennoet al. (1991), the percentage of macroptery in planthopperswas negatively correlated with habitat persistence,indicating that extinction rate was indeed positivelycorrelated with dispersal propensity.316 OIKOS 115:2 (2006)

Cost of dispersalDispersal is generally considered to be costly for theindividuals (Hanski 1998, Gandon and Michalakis 2001,Ims and Hjermann 2001), mostly because of theincreased mortality during dispersal, albeit other factorsmight also have some role in this cost (e.g. the metaboliccost of dispersal ands its effect on fecundity, Roff 1977,1984, Rankin 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 byanalyzing markreleaserecapture data with the virtualmigration model (Hanski et al. 2000). Furthermore it iswell known that animals crossing roads experienceincreased mortality (Ashley and Robinson 1996, Bonnetet al. 1999, Carr and Fahrig 2001, Aresco 2003). It hasbeen shown that in the idealistic case when dispersal hasno cost, individuals would disperse unconditionally(Parvinen 1999), while the realistic case, when dispersalis costly, is expected to select for lower rates of dispersal.A number of theoretical studies have concluded thatdispersal rate is a decreasing function of dispersal cost(Travis et al. 1999, Poethke and Hovestadt 2002, Cadetet al. 2003, Parvinen et al. 2003, Poethke et al. 2003). Inthe special case of density-dependent dispersal whenindividuals only disperse when the local populationdensity reaches a certain threshold value, this thresholddensity increases with increasing cost of dispersal(Poethke and Hovestadt 2002). When a linear dispersalfunction is considered, the increase in the cost ofdispersal lowers the intercept of the line, but has nosignificant effect on the slope (Travis et al. 1999). Theintercept of the linear dispersal function mainly influencesthe dispersal rates at very low densities. We foundin our more general model that from low densities todensities close to the carrying capacity of the patch thedispersal function was nearly linear (Fig. 3). Theintercept in our case was also negatively correlatedwith dispersal mortality, and the inflection point (b)of the evolved dispersal function (which is analogous tothe threshold density of the model of Poethke andHovestadt (2002)) was positively correlated with dispersalmortality (Fig. 2). So, by using a general evolvingdispersal function we were able to combine the results ofboth previous models that considered density-dependentdispersal.Dispersal functionDensity-dependent dispersal has been experimentallydemonstrated in a number of studies (e.g. Microtusoeconomus: Aars and Ims 2000; Paroxyna plantaginis:Albrectsen and Nachman 2001; Onychiurus armatus:Bengtsson et al. 1994; Prokelisia marginata andP. dolus: Denno et al. 1996; Simulium vittatum: Fonsecaand Hart 1996; Anisopteromalus calandrae: French andTravis 2001; Lynx pardinus: Gaona et al. 1998; Trirhabdavirgata: Herzig 1995; Tendora sinensis: Hurd andEisenberg 1984; Lacerta viviparia: Lecomte et al. 2004;Carcinus maenas: Moksnes 2004; Metisa plana: Rhaindset al. 2002; reviewed by Gaines and McClenaghan 1980,Denno et al. 1991, Matthysen 2005). Moreover, an increasingnumber of studies are available where densitydependentdispersal is demonstrated by measuringdispersal 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, Rhainds et al. 2002,Moksnes 2004). The common nature of these studies isthat the observed dispersal functions were always nonlinear.Dispersal is very moderate or missing at lowdensity and starts to increase 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 increase,and increased with an increasing rate at higher densities(8/ larvae mm 2 ). Albrectsen and Nachman (2001)obtained similar results in their investigation of thedispersal of the females of the tetripid fly, Paroxynaplantaginis (the dispersal of the male flies was notdensity-dependent). They measured the dispersal rateat densities of 10, 40 and 100 flies per patch. Theequilibrium density of the female flies was 12.009/3.86individual per patch. The dispersal rate increasedexponentially above the equilibrium density (see Fig. 3in Albrectsen and Nachman 2001). The density-dispersalof juvenile shore crabs (Carcinus maenas) also exhibit amarked increase at higher densities (Fig. 3 in Moksnes2004). At densities of 2 and 6 individuals per musselpatch around 20% of the crabs emigrated, whereasproportional emigration increased to 45% at a densityof 18 crab/mussel patch. Furthermore, in certainplanthopper 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), orin other planthoppers (Nilaparvata lugens, Javsellapellucida, Laodelphax stritellus) the dispersal functionexhibits a threshold, above which dispersal rates increaseat a higher rate (Fig. 3C in Denno et al. 1991).Our results are in a remarkably good agreement withthese experimental findings. Thus it is of paramountimportance to use such a flexible dispersal function inthe study of the evolution of dispersal rates. The existingmodels 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 lineardensity-dependent dispersal, that individuals tend not todisperse if local population density is below the equilibrium,and always disperse if the local populationdensity is above twice the equilibrium density. Becausedispersal is costly, it is worthwhile to disperse only iflocal conditions are very bad (in this case because ofcrowding). Poethke and Hovestadt (2002) put this theoryinto their model, by assuming a dispersal function whichhad zero dispersal rate below a threshold density, andthen it was a saturating function of density. In somecases such saturating dispersal functions were alsodemonstrated experimentally (for females of Prokelisiamarginata and both sexes of Prokenisia dolus (Fig. 2 inDenno et al. 1991) and for Anisoptermalus calandrae(French and Travis 2001)). Furthermore, Rhainds et al.(2002) found similar trends in the ballooning rate ofbagworm larvae. But in these experiments the steepincrease in dispersal rate occurred at very low densities,well below the carrying capacity of the habitat, andthus probably demonstrates a different phenomenon.Accordingly, assuming such a dispersal function a prioriunduly restricts the evolutionary outcome. Poethke andHovestadt (2002) and Metz and Gyllenberg (2001)derived the above mentioned saturating dispersal functionfrom the theory of ideal free distribution (IFD).IFD theory assumes that individuals’ dispersal strategyleads to equal fitness for all local habitats. However thelimited information 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 carrying capacity), and bad patcheswill be overpopulated (Kennedy and Gray 1993, Rantaet al. 1999). We experienced the same in our model whereindividuals only sense the local density, but not informedabout the actual carrying capacities of the sites theydisperse to.While we have considered a wide range of possibleenvironmental and life-history traits, some types ofenvironment was left out of the current investigation.Most notably spatially correlated and temporally andspatially correlated environment remains to be studied inthe 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 certainthreshold the propensity of individuals to disperse is verylow, (2) above this threshold the dispersal rate grows inan accelerating manner (exponentially in our model)with further increase in density. 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