The downward spiral: eco-evolutionary feedback loops lead to the emergence of ‘elastic’ ranges

In times of severe environmental changes and resulting shifts in the geographical distribution of animal and plant species it is crucial to unravel the mechanisms responsible for the dynamics of species’ ranges. Without such a mechanistic understanding, reliable projections of future species distributions are difficult to derive. Species’ ranges may be highly dynamic. One particularly interesting phenomenon is range contraction following a period of expansion, referred to as ‘elastic’ behaviour. It has been proposed that this phenomenon occurs in habitat gradients, which are characterized by a negative cline in selection for dispersal from the range core towards the margin, as one may find, for example, with increasing patch isolation. Using individual-based simulations and numerical analyses we show that Allee effects are an important determinant of range border elasticity. If only intra-specific processes are considered, Allee effects are even a necessary condition for ranges to exhibit elastic behavior. The eco-evolutionary interplay between dispersal evolution, Allee effects and habitat isolation leads to lower colonization probability and higher local extinction risk after range expansions, which result in an increasing amount of marginal sink patches and consequently, range contraction. We also demonstrate that the nature of the gradient is crucial for range elasticity. Gradients which do not select for lower dispersal at the margin than in the core (especially gradients in patch size, demographic stochasticity and extinction rate) do not lead to elastic range behavior. Thus, we predict that range contractions are likely to occur after periods of expansion for species living in gradients of increasing patch isolation, which suffer from Allee effects.

reaction-difusion models (Fisher 1937, Kolmogorov et al. 1937, Lubina and Levin 1988).Melbourne and Hastings (2009) demonstrated experimentally the intrinsic role of stochasticity during range expansions.hese results are underpinned by the work of Giometto et al. (2014), who showed that this stochasticity can indeed be predicted.However, range expansions rarely only include ecological dynamics.It is now clear that rapid evolutionary and resulting ecoevolutionary dynamics may play a major role (Perkins et al. 2013, Kubisch et al. 2014, Fronhofer and Altermatt 2015).For example, high emigration rates are selected for during the process of expansion due to spatial (Phillips et al. 2010, Shine et al. 2011, Fronhofer and Altermatt 2015, more dispersive individuals being at the front in combination with itness beneits through reduced competition) and kin selection (Kubisch et al. 2013) which leads to accelerating expansions.From a genetic point of view, range expansions usually lead to decreasing genetic diversity, either afecting the adaptation of species (Excoier et al. 2009) or their dispersal ability directly (Cobben et al. 2015).In addition, expanding populations are also likely to sufer from a mutation load, also called expansion load (Peischl et al. 2015).Finally, Allee efects, which can be the consequence of e.g.sexual reproduction or sociality Courchamp et al. (2010), can drastically inluence expansion dynamics, as they can lead to pulsed patterns of invasions (Johnson et al. 2006, Schurr et al. 2008).
Range contractions, however, are more diicult to investigate empirically and thus have been less intensively studied (Channell and Lomolino 2000).Yet, range contractions are highly relevant from a conservation and management point of view as they are usually assumed to be caused by extrinsic mechanisms, like climate change or human impacts (Li et al. 2015).
To further complicate the situation, besides simply expanding or contracting, species' ranges may also exhibit more complex dynamics such as 'elastic' behavior.Elasticity (as described by Kubisch et al. 2010) implies that a range expansion is immediately followed by a period of contraction due to evolutionary changes in dispersal.In his review of the work of MacArthur (1972), Holt (2003) irst described this phenomenon.He argued that after a period of increasing dispersal during range expansion there can be substantial selection against dispersal in marginal areas due to source-sink dynamics.If invasions occur along a gradient from source to sink populations, the latter would be sustained by initially high emigration rates which are typical for such expansions (Shine et al. 2011).Subsequent selection against dispersal due to an increased probability of arriving in sink patches characterized by low itness expectations will result in a contraction of the geographical range.
In a simulation study, Kubisch et al. (2010) could show that this phenomenon may indeed be likely to occur in nature, but that it crucially depends on the underlying gradient.he authors found that range border elasticity could only be observed in fragmentation gradients and, to a smaller extent, in fertility gradients.hey concluded that the mechanism explaining range elasticity is selection for lower emigration rates at range margins relative to core areas.In more recent work Henry et al. (2013) suggested that under climate change, elasticity should also be found in gradients of patch size, habitat availability, growth rate and local extinction risk.
Following the argumentation by MacArthur (1972) and Holt (2003), a crucial determinant of range border elasticity is the presence of actual sink patches at the initial wide range after expansion.Sink populations are populations with a negative growth rate and may be the result of altered abiotic conditions, which lead to maladaptation and reduced growth.Similarly, sink patches may be caused by new or altered biotic interactions, such as the occurrence of a predator at the range margin.Yet, previous studies (Kubisch et al. 2010, Henry et al. 2013) report the occurrence of range border elasticity even without changes in abiotic local conditions or the occurrence of novel biotic interactions.herefore, intra-speciic, biotic processes must be suicient to generate sink patches at the range margin.Under these conditions an important mechanism leading to the emergence of sinks are demographic Allee efects, which are deined as reduced growth rates at low population sizes or densities in comparison to populations at intermediate densities (Courchamp et al. 2010).
Here we argue that a negative cline in selection for dispersal from the range core to the margin is only one prerequisite for range elasticity caused by intraspeciic processes, and that the presence of Allee efects leading to sink populations at range margins is the second.he eco-evolutionary feedback loop created by these two forces leads to a spatio-temporally nonlinear cline in immigrant itness, which is caused by the emergence of sink patches and inally results in range contraction.

The model
We use an individual-based model of a spatially structured population of an asexually reproducing species with discrete generations.his approach has been used in several previous studies (Poethke et al. 2011, Fronhofer et al. 2013, Kubisch et al. 2014).

Landscape
We implement linear unidirectional environmental gradients.his means that along the x-axis of the landscape, one speciic habitat characteristic changes from favorable to unfavorable conditions with respect to the survival of the species (see below for details).he simulated landscape consists of x  y  200  50 habitat patches, arranged on a rectangular grid.Larger landscapes (speciically in x-direction) would lead to larger elasticity efects.Yet, we were computationally limited in the total number of populations.

Individuals
Every patch may contain a population of the species, assuming a carrying capacity K x,y (see below).Local populations consist of individuals, which are determined by their speciic location x,y and one heritable trait deining their probability to emigrate.

Population dynamics
Local population dynamics follow the discrete logistic growth model developed by Beverton and Holt (1957).his model is extended by the implementation of a direct Allee efect, the strength of which depends on population density instead of size (see also Kubisch et al. 2011).We draw the individuals' average ofspring number for every patch and generation () ,, Λ xyt from a log-normal distribution with mean l x,y and standard deviation s. he latter represents the degree of environmental stochasticity.Afterwards every individual in a patch gives birth to a number of ofspring drawn from a Poisson distribution with mean () ,, Λ xyt .
Density-dependent competition then acts on ofspring survival probability s, which is given by with K x,y being carrying capacity of a patch, N x,y,t denoting the population size of a focal patch and a deining the strength of the Allee efect.Note that propagule survival is thus actually regulated by the number of parental individuals, the competition between which is thus phenomenologically implemented.We assume a sigmoid increase in survival probability with the number of inhabitants in a patch (Eq.1c).Generally, increasing a leads to a decreased probability of survival.For example, individuals in a population of density N x,y,t /K x,y  a will have a decrease in their survival of 50%.
A newborn inherits the dispersal allele from its parent.During this process the allele may mutate with probability m  10  4 .In case of a mutation, a Gaussian distributed random number with mean 0 and standard deviation 0.2 is added to the allele's value.As has been shown by Kubisch et al. (2010), range limit elasticity is robust against changes in mutation rate, as long as increasing dispersal during range expansion is provided.
We assume a moderately low level of population turnover as is characteristic for metapopulations (Fronhofer et al. 2012).Here, this turnover is driven extrinsically due to local patch extinctions.herefore, following reproduction, every population may go extinct by chance with probability e  0.05.Changing this extinction rate did in previous analyses not result in qualitative changes of the presented results.

Dispersal
Surviving ofspring may emigrate from their natal patch.We implemented dispersal in two diferent modes.

Nearest-neighbor dispersal
In this standard scenario, which is used throughout the main text, the probability to disperse for any given ofspring is given by its dispersal allele (e).If an individual disperses, it may die with a certain probability m, which includes all potential costs that may be associated with dispersal, like predation risk or energetic costs (Bonte et al. 2012). he dispersal mortality is calculated as the arithmetic mean between the patch-speciic dispersal mortalities m x,y of the natal and the target patch.he target patch is randomly drawn from the eight surrounding patches.To avoid edge efects we wrap the world in y-direction, thus forming a tube along the x-dimension of the world.Individuals leaving the world along the x-dimension are relected.

Dispersal kernel
To test the validity of our results against alternative implementations of dispersal we performed additional simulations with dispersal distances evolving instead of propensities.
In these cases the dispersal alleles coded for the mean of a negative exponential distance distribution (kernel; see also Henry et al. 2013).Given that the dispersal mortality m in our original approach means a per-step mortality (as the step length for nearest neighbor dispersal is one) we have based the implementation of mortality in the kernel scenario on the same rationale and assumed that the probability of dying during the transition phase, m d , is given by: with d denoting the traveled dispersal distance (for more details see Fronhofer et al. 2015).m is calculated as described above as the mean of m x,y of the natal and the target patch.

Simulation experiments and analysis
In order not to bias the results by using artiicial initial values for the dispersal traits we implemented a burn-in period allowing for the adaptation of dispersal strategies to local conditions in the range core.herefore we added additional 10 rows to the landscape in front of the position x  1, all patches there being deined by conditions found at position x  1 and illed these patches with K 1,y individuals.We then let the simulation run for 2500 generations, assuming torus conditions of the burn-in region (i.e.individuals leaving this region in xand y-direction were wrapped around).In the case of dispersal propensities evolving, the alleles were initially drawn from a uniform distribution between 0 and 1.
In the alternative scenario with dispersal distances evolving we initialized the individuals with mean dispersal distance values from a uniform distribution between 0 and 10. he fact that dispersal distances did not closely approach the maximum values indicates that our burn-in region was suficiently large.
After the burn-in phase (i.e. in generation 2501), we allowed the populations to spread further than the irst 10 rows for 5000 generations, allowing for range expansion and assuring the formation of a stable range limit.Although we focus on a gradient in dispersal mortality (i.e.habitat fragmentation), we tested a range of other possible gradients.A summary of gradient dimensions and references to according igures are given in Table 1.
Respective parameters, which were not changing across space in a given simulation were set to standard values (K x,y  100, l x,y  2, m x,y  0.2, s x,y  0.2, e x,y  0.05), To account for the fact that fragmentation gradients, as they  the margin is fully populated (i.e.all patches are at N x,y,t  K x ).hus the number of colonizers is given by N c,x  K x  e  (1  m x ), with e denoting emigration rate.Our approach is conservative in the sense that populations directly behind the margin may not yet have reached carrying capacity.We therefore overestimate the number of colonizers, which leads to a systematic underestimation of the impact of the phenomenon of interest, the Allee efect.
2) As described above for the evolutionary simulations we assume that the mean fecundities of the colonizers Λ x follow a lognormal distribution with mean l x and standard deviation s x .he parameter s accounts for environmental stochasticity.
3) Demographic stochasticity is taken into account by assuming that reproduction can be described as a Poisson process with Λ x as mean.Subsequently, density regulation is applied according to Eq. 1 (with N c,x as the population size).
4) Finally, local extinctions are represented as a binomial process with mean (1  e) acting on the ofspring numbers.
his algorithm allows us to calculate a distributionW of potential per capita ofspring numbers for each x-location in all gradients, assuming that the margin of a saturated range would lie directly at its front.We approximated this distribution by sampling 1 000 000 times.To average the mean colonizer itness we used an approximated geometric mean calculation.he arithmetic mean would be a poor estimate of itness, as it is comparably insensitive to the distribution's variance.he true geometric mean is, however, too sensitive against zero ofspring numbers, which are drawn with a high probability based on above method.A Jean series approximation of the geometric mean, however, provides a sensible estimate, as it is sensitive towards variation but not zero, when such ofspring numbers are included.According to Jean and Henry (1983) we calculate the mean colonizer itness w c thus as: withW denoting the arithmetic mean of the distribution.In summary, w c thus is the average number of ofspring an immigrant would get, if it would colonize a habitat patch at the very margin of the species' distribution.his metric depends strongly on the dispersal rate, local environmental conditions and the strength of the Allee efect.We performed the analysis for all gradients and two values of Allee efect strength (a  0 and a  0.05).We varied emigration rate e in 10 equidistant steps from 0.05 to 0.5.Comment source iles for above analysis and the simulation program can be found in Supplementary material Appendix 4.

Results
Strong elasticity is only detected for a gradient in dispersal mortality, i.e. patch fragmentation, in the presence of an Allee efect (Fig. 1).We found a very weak elastic behavior in a gradient of decreasing per capita growth rate (l 0 ; Supplementary material Appendix 1, Fig. A2) -a result consistent with the indings of Kubisch et al. (2010).All other gradients (K, s, e) do not lead to range border elasticity occur in nature, usually afect not only the isolation of habitat patches, but also imply decreasing patch sizes, we have additionally tested a 'mixed' gradient, in which along the x-axis habitat capacity K was reduced and dispersal mortality m increased, using the same parameters as given above.hese results can be found in the Supplementary material, Appendix 1, Fig. A5.We repeated the simulations for the ive standard gradients (i.e.m, K, l, s, e) for a scenario with dispersal distance instead of propensity evolution.he results are provided in the Supplementary material, Appendix 2, Fig. A6-A10).We tested 11 values for the strength of the Allee efect (a) in equidistant steps from 0 to 0.1.For all scenarios we performed 50 replicate simulations.For all runs we assessed the absolute range border position R, deined as the x-position of the foremost populated patch.We analyzed the marginal emigration rate as the fraction of individuals emigrating from their natal patch in the ive columns of patches behind the range border.
To quantify the presence and degree of range elasticity, we analyzed range border position as a function of time.We itted a function to the resulting progression of relative range size r, which is calculated as the absolute range size position along the landscape's x-dimension R divided by the maximum extent of that dimension (x max  200). he function we used (Eq.3) is lexible enough to quantify elasticity and its parameters can be directly interpreted in biological terms: with v e denoting the speed of range expansion,  r the equilibrium range size, v c the speed of range contraction and ∆ t the time to reach equilibrium.Using non-linear least squares regressions (R language for statistical computing ver.3.1.0function nls(), R Core Team) we it the curve to the respective simulation output (i.e.relative range border position over time).he relative amplitude of the elasticity efect was calculated using the resulting function as: Figure 3E shows a typical example of this calculation.

Numerical analysis of the mean fitness of colonizers
To mechanistically investigate the eco-evolutionary feedback loop underlying the observed range dynamics, we performed additional numerical analyses.We quantiied the mean itness expectations (reproductive success) of potential colonizers at the range margin as a function of their dispersal strategy and of the landscape gradient.For a better comparability, we calculate the colonizers' itness expectations over one generation for every potential location of the range margin in close analogy to the individual-based simulation model described above.1) We calculate the number of colonizers at the margin based on the conservative assumption that the range behind elastic ranges for a mixed gradient-scenario, including declining patch size and increasing patch isolation (Supplementary material Appendix 1, Fig. A5).
he evolving emigration rate at the range margin is negatively afected by the Allee efect strength (Fig. 2).Consequently, with increasing Allee efects, range expansion (Supplementary material Appendix 1, Fig. A1, A3, A4).Importantly, elasticity only occurs in the presence of an Allee efect and the degree of elasticity (amplitude) increases with increasing Allee efect strength (Fig. 1, 3).For values of a exceeding 0.07, the entire spatially structured population goes extinct.It is also important to note that we did ind  he numerical analyses of mean colonizer itness show characteristically diferent patterns between the various environmental gradients (Fig. 4).For the dispersal mortality gradient without Allee efects an increase in colonizer itness deeper in the gradient can be observed (Fig. 4A), which sets this gradient apart from all the others.Increasing emigration (darker lines) results in overall lowered itness expectations, especially for range core areas (Fig. 4A).Allee efects strongly interact with this pattern (Fig. 4B): for high dispersal rates the relationship between colonizer itness and spatial location changes from monotonically increasing to unimodal with a rapid decrease of itness in regions with harsher conditions (i.e. higher dispersal mortality; Fig. 4B). he lower the dispersal rate is, the sooner this decrease sets in.
is slower (Fig. 3A) and the resulting equilibrium range size is smaller (Fig. 3B). he enhanced elasticity for increasing Allee efects (which is apparent in Fig. 1) is characterized by 1) an increase in the velocity of contraction (followed by a slight decrease for a very strong Allee efect; Fig. 3C) and 2) an increase in the amplitude, i.e. the diference between maximum and equilibrium range size (Fig. 3D).As we had hypothesized, a considerable Allee efect must be present for elasticity to emerge (A  0 for a  0.02, Fig. 3D).
he results of simulations allowing for the evolution of dispersal distance instead of propensities show the same qualitative behavior.Elasticity can only be found in a gradient of dispersal mortality (per-step mortality) and for strong Allee efects (Supplementary material Appendix 2, Fig. A6-A10).l 0 , Fig. 1, Supplementary material Appendix 1, Fig. A2).For all other gradients, we either found patterns of decreasing itness deeper in the gradient (Fig. 4C-F, I, J) or no spatial relationship in the case of the gradient in carrying capacity (Fig. 4G, H).However, for all these gradients we found the same impact of the Allee efect: whereas in its absence decreased dispersal rates (lighter colors in Fig. 4) result in consistently higher itness, an Allee efect inverts this pattern.Importantly, no non-monotonic or interaction efects can be found in contrast to the results for dispersal mortality (Fig. 4A and B; and fertility to a smaller extent, Fig. 4C and D).

Discussion
Our results show that range border elasticity can only be observed in speciic abiotic habitat gradients and in the presence of Allee efects if one focuses on intra-speciic processes.Speciically, we show that an eco-evolutionary feedback loop, caused by the interplay between Allee efects, landscape structure and dispersal evolution, is necessary for ranges to show elastic behavior.As a range expansion proceeds into a fragmentation gradient (m), the selective pressures at the range margin change: initially, spatial selection (more dispersive individuals being at the front in combination with itness beneits through reduced competition; Phillips et al. 2010) leads to the evolution of increased dispersal.Yet, once the range has expanded more deeply into the gradient, dispersal is reduced by natural selection because of high habitat isolation (Fig. 2).As a result of this decrease, more and more patches turn into sinks because the reduction of the numbers of immigrants reduces population sizes.hese small populations decrease even further due to the presence of the Allee efect.his means that from the populations' perspective the itness gradient gets steeper over time, ultimately resulting in a range contraction (Fig. 4B). he strength of this efect is further increased due to selection against dispersal as a result of the marginal source/sink-dynamics -a downward spiral of eco-evolutionary feedbacks leading to the observed strong range contraction.

Eco-evolutionary feedbacks lead to range border elasticity
he itness-relevant consequences of these non-linearly interacting efects can be seen directly in Fig. 4B.High dispersal rates due to spatial selection (dark lines) lead to a fast range expansion, as colonizer itness is initially increasing or at least not decreasing over space.At a certain point in space -deined by the interaction between gradient steepness, dispersal rate and Allee efect strength -colonizer itness drops abruptly.Selection against dispersal due to high dispersal mortality leads to changes in the itness proiles (depicted in light gray colors).hese dynamics can only be observed in mortality gradients and to a smaller extent also in fertility gradients (Fig. 4D).All other gradients (Fig. 4E-J) do not show such an abrupt change in the spatial itness expectations and mostly exhibit decreasing colonizer itness early on.In fact, colonizer itness monotonically decreases across the range for gradients in both extinction risk (Fig. 4E, F) It is important to keep in mind that after range expansion, dispersal decreases evolutionarily at the margin in this gradient, implying a temporal change in the itness expectations according to the results presented in Fig. 4B.his dramatic impact of Allee efects and qualitative change of the spatial distribution of colonizer itness is strictly associated with scenarios in which (strong) range border elasticity can be observed (mortality, m, and fecundity gradients, example would be a species expanding into a landscape, which is inhabited by a predator.If one imagines that the predator follows a Holling type III functional response this would imply that the predator species ignores the range expanding prey species as long as the latter occurs at low densities.During the course of the range expansion, however, these densities will increase, ultimately leading to a shift in the predator's behavior, and triggering predation on the range expanding prey.his could lead to range contraction after expansion. Both abiotic and interspeciic mechanisms that can potentially result in range border elasticity are extrinsic processes -i.e. they induce an external change in the environment, which might force the focal species to contract its range.In this study, however, we focused on intrinsic, intra-speciic, reasons for range elasticity.Under this premise range border elasticity can only be achieved through Allee efects in concert with decreasing dispersal towards the range margin.It is important to keep in mind that the term 'Allee efect' is a phenomenological description of a variety of mechanisms ranging from sexual reproduction to social behavior (Courchamp et al. 2010).
he emergent elasticity of Kubisch et al. (2010) was also caused by an Allee efect. he authors modeled a species with sexual reproduction, thus implicitly assuming a mateinding Allee efect.Range elasticity might even be observed in more natural gradients of fragmentation, in which not only patch isolation increases, but patch size also decreases (Supplementary material Appendix 1, Fig. A5).Although lower patch sizes imply increased demographic stochasticity and thus selection for increased dispersal at the range margin, this selective force is outweighed by the strong selection for lower dispersal due to its increased costs.

Range expansion, Allee effects and dispersal kernels
he negative relationship we ind between range size and Allee efects as well as the speed of range expansion and Allee efects are in good accordance with previous theoretical studies.Using reaction-difusion models and ordinary-diferential-equation models Keitt et al. (2001) for example showed that for a wide range of biologically plausible conditions Allee efects may not only slow down invasions, but lead to the formation of stable range limits through invasion pinning (i.e.propagation failure).A good overview of the topic is provided by the literature review of Taylor and Hastings (2005), which summarizes known consequences of Allee efects for invasions and includes both theoretical and empirical work.
For reasons of simplicity and scale, we model dispersal as being restricted to surrounding populations.Yet, it is reasonable to assume that many species rather disperse varying distances, the distribution of which can be described by dispersal kernels (Clobert et al. 2012).To test the validity of our indings under the assumption of evolving dispersal distances we performed additional simulations using negative exponential dispersal kernels in analogy to Henry et al. (2013).Our results prove to be robust, as elasticity was again only found in scenarios with gradients in dispersal mortality and strong Allee efects, while being only weakly evident in fertility gradients (Supplementary material Appendix 2, Fig. A6-A10).and environmental stochasticity (Fig. 4I, J), as both factors lead to decreasing long-term population growth.he fact that colonizer itness is not afected by the gradient in carrying capacity might seem unexpected at irst.Yet, the reason is that deep in the gradient, where patch sizes are small and competition acts strongly already at small numbers of individuals, colonizer numbers are low, leveling the latter efect out.In general, the indings from our numerical analysis imply that range border elasticity cannot be found in gradients other than dispersal mortality and, to a much lower extent, growth rate.he result is in accordance with the indings by Kubisch et al. (2010), where also no elastic range behaviour could be found in gradients of patch size, growth rate and local extinction risk.While all investigated environmental gradients lead to the formation of a stable range limit, the eco-evolutionary feedback loop as described in the previous paragraph, which leads to intraspeciically caused range elasticity, is impossible in these scenarios.Especially decreasing patch size, increasing environmental stochasticity and increasing local extinction risk result in selection for increased emigration rates (Kubisch et al. 2014), which does not lead to source-sink conversion.

Abiotic and biotic factors influencing range border elasticity
he crucial aspect of a landscape which allows for range elasticity is not that itness expectations of colonizers are spatially non-linear.To demonstrate this fact, we tested convex, concave and sigmoid gradient shapes with varying cline strength (Supplementary material Appendix 3, Fig. A11-A14).Our results are qualitatively not afected by the shape of the gradient -in the absence of Allee efects no elasticity can be observed.Hence, itness expectations of colonizing individuals need not only to change in space, but also in time.As mentioned in the introduction this temporal change could in principle be caused by more mechanisms than the one we focus on in this study.
Abiotic changes, e.g.temporally changing climatic conditions, can also lead to elastic range dynamics.We performed additional simulations, in which we continuously worsened conditions by shifting the baselines of linear gradients.Depending on the strength of change we were able to create range elasticity in every gradient, without Allee efects (Supplementary material Appendix 3, Fig. A15).An example for this type of elasticity is reported in the study of Henry et al. (2013), who modeled a range shift driven by temporally shifting gradients.Similar to what we describe in this study, the populations in that scenario evolved higher dispersal during that range shift.Once climate change stopped the individuals continued to disperse further into unsuitable habitat.Subsequently, natural selection led to a reduction in dispersal at the range margin, but with a time lag between the end of climate change and the optimal adaptation of dispersal strategies.During this time lag individuals continue to disperse outside their range which leads to strong source/ sink-dynamics.As Henry et al. concluded, in such a scenario the nature of the gradient is irrelevant for elasticity to occur.
Besides abiotic factors, it is also plausible that interspeciic interactions can result in range elasticity.A hypothetical

Figure 1 .
Figure 1.Range border position as a function of simulation time for a gradient in dispersal mortality (m).Allee efect strength increases from (A) to (H).For parameter values see main text. he black lines show the median values of 50 replicate simulations, the shaded grey areas denote 25% and 75% quantiles.

Figure 2 .
Figure 2. Marginal emigration rate as a function of simulation time for a gradient in dispersal mortality (m).Allee efect strength increases from (A) to (H).For parameter values see main text. he black lines show the median values of 50 replicate simulations, the shaded grey areas denote 25% and 75% quantiles.

Figure 3 .
Figure 3. Results of the quanitative analysis of elasticity depending on Allee efect strength.Shown are (A) the speed of range expansion, (B) the relative range size, (C) the speed of range contraction and (D) the relative amplitude of the elastic range efect.Shown are median values of 50 replicate simulations, error bars denote 25% and 75%-quantiles.(E) A sketch of the -for elasticity typical -relationship between relative range border position and time to illustrate the measures used in (A-D). he curve was created by using Eq. 3 with the following parameters: v e  0.0015, v c  0.0004, ∆ t  2000,  r  0.35). he meaning of the four parameters used in the analysis is denoted.

Figure 4 .
Figure 4. Results of the numerical analyses of mean colonizer itness as a function of the Allee efect strength (a), landscape characteristics and emigration rate.Shown is the approximated geometric mean per capita ofspring number to be expected for colonizers arriving at every potential x-location under the assumption that the species' range ends right before each position (see main text for details).Line coloring refers to the rate of dispersal, which decreases from 0.5 (black) to 0.05 (lightest gray) in 10 equidistant steps.

Table 1 .
Applied dimensions of gradients including references to the according results figures.Figures A1 to A10 can be found in the Supplemental material Appendix 1-2.