Density dependence of seed dispersal and fecundity profoundly alters the spread dynamics of plant populations

Plant population spread has fundamental ecological and evolutionary importance. Both determinants of plant population spread, fecundity and dispersal, can be density‐dependent, which should cause feedback between population densities and spread dynamics. Yet it is poorly understood how density‐dependence affects key characteristics of spread: spread rate at which the location of the furthest forward individual moves, edge depth (the geographical area over which individuals contribute to spread) and population continuity (occupancy of the spreading population). We present a general modelling framework for analysing the effects of density‐dependent fecundity and dispersal on population spread and parameterize this framework with experimental data from a common‐garden experiment using five wind‐dispersed plant species grown at different densities. Our model shows that density‐dependent fecundity and dispersal strongly affect all three population spread characteristics for both exponential and lognormal dispersal kernels. Spread rate and edge depth are strongly correlated but show weaker correlations with population continuity. Positive density‐dependence of fecundity increases all three spread characteristics. Increasingly positive density‐dependence of dispersal increases spread rate and edge depth but generally decreases population continuity. Density‐dependent fecundity and dispersal are largely additive in their effect on spread characteristics. For population continuity, the joint effects of density‐dependent fecundity and dispersal are somewhat contingent on the dispersal kernel. The common‐garden experiment and the experimentally parameterized mechanistic dispersal model revealed density‐dependent fecundity and dispersal across study species. All study species exhibited negatively density‐dependent fecundity, but they differed qualitatively in the density‐dependence of dispersal distance and probability of long‐distance dispersal. The negative density‐dependence of fecundity and dispersal found for three species reinforced each other in reducing spread rate and edge depth. The positively density‐dependent dispersal found for two species markedly increased spread rate and edge depth. Population continuity was hardly affected by population density in all study species except Crepis sancta in which it was strongly reduced by negatively density‐dependent fecundity. Synthesis. Density‐dependent fecundity and seed dispersal profoundly alter population spread. In particular, positively density‐dependent dispersal should promote the spread and genetic diversity of plant populations migrating under climate change but also complicate the control of invasive species.


| INTRODUC TI ON
Population spread is a fundamental process affecting the spatial and genetic structure, migration, and geographic distribution of plants (Harper, 1977;Sharma et al., 2022;Skellam, 1951;Smith et al., 2020).
The dynamics of plant population spread can be quantified through three major characteristics ( Figure 1). First, the spread rate is the rate at which the location of the furthest forward individual moves (Clark et al., 2001). Second, edge depth measures the geographical area over which individuals contribute to population spread. A larger edge depth increases the genetic diversity of the spreading population (Birzu et al., 2019;Paulose & Hallatschek, 2020;Travis et al., 2011). Consequently, edge depth is a continuous measure that characterizes spread dynamics along a gradient from 'pulled' (small edge depth) to increasingly 'pushed' (larger edge depth) dynamics (Lewis & Kareiva, 1993;Miller et al., 2020). Third, population continuity measures the spatial occupancy of the spreading population. Exponentially bounded dispersal kernels lead to high population continuity whereas fat-tailed dispersal kernels cause spread by isolated foci and reduced population continuity . Everything else being equal, lower continuity of populations reduces overall abundance and impact on the environment as well as our ability to measure spread  and to manage invasive species (Epanchin-Niell & Hastings, 2010).
Population spread depends on fecundity and dispersal (Buckley et al., 2005;Neubert & Caswell, 2000;Schurr et al., 2008;Skellam, 1951), both of which can respond to population density ( Figure S1). There is extensive literature on negative and positive density-dependence of fecundity (e.g. Buckley & Metcalf, 2006;Postma et al., 2021): plants tend to produce fewer seeds at high population densities due to intraspecific resource competition (Bleasdale & Nelder, 1960;Watkinson, 1980;Weiner & Freckleton, 2010), but they may also suffer from reduced reproduction at low densities that causes Allee effects (Allee, 1938;Courchamp et al., 1999Courchamp et al., , 2008. In contrast, there are comparatively few studies on density-dependent dispersal distance, emigration or immigration (Birzu et al., 2019), and these mostly focus on densitydependent emigration and immigration of animals (reviewed by Harman et al., 2020;Matthysen, 2005). Yet, a few studies document either positive or negative density-dependence of seed dispersal distance in wind-and animal-dispersed plants. Negative density dependence of dispersal distance was due to density reducing plant height in Arabidopsis thaliana (Wender et al., 2005), increasing competition for dispersers in the palm Attalea butyracea (Jansen et al., 2014), and decreasing the proportion of wind-dispersed achenes per seed head in the seed heteromorphic Catananche lutea (Ruiz de Clavijo & Jiménez, 1998) and Hypochaeris glabra (Baker & O'Dowd, 1982).
Positive density dependence of wind-driven seed dispersal distance can be due to density increasing seed release height and turbulence in Pinus halepensis (Schurr et al., 2008) or reducing seed mass in Dithyrea californica (Larios & Venable, 2015).
Since population spread inevitably causes spatiotemporal variation in population density, theory predicts that density-dependence of fecundity and dispersal will feed back on spread dynamics, altering the three characteristics of population spread . Spread rate is generally decreased by positive density-dependence of fecundity (Lewis & Kareiva, 1993;Maciel & Lutscher, 2015;Neubert & Caswell, 2000;, but can show complex fluctuating behaviour when positive density-dependence of fecundity interacts with either positive or negative density-dependence of dispersal propensity or distance (Schurr et al., 2008;Sullivan et al., 2017). Edge depth should increase with positive density-dependence of both fecundity and dispersal which enhances the contribution of high-density populations far from the edge to the spread of the population edge (Birzu et al., 2019;Dahirel et al., 2021). Population continuity may be increased by positive density-dependence of fecundity but decreased by positive density-dependence of emigration (Kubisch et al., 2011).
Still, we are not aware of any study exploring the effects of densitydependent dispersal distance on population continuity. However, continuity should change if density affects the fat-tailedness of dispersal kernels. This is because exponentially-bounded kernels lead to continuous spread whereas fat-tailed dispersal kernels cause spread by isolated foci and reduced population continuity ). Yet, we still lack a comprehensive analysis of how density-dependence of fecundity and dispersal distance affects the spread rate, edge depth and continuity of expanding populations.
The real-world relevance of density-dependent seed dispersal for plant population spread has seldom been investigated. This is mainly due to the challenge of quantifying seed dispersal distance Rogers et al., 2019), especially the rare long-distance dispersal events that drive population spread (Higgins et al., 2003).
This challenge can be eased with mechanistic analyses of the effects of density-dependent seed dispersal on population spread. In particular, seed dispersal by wind lends itself to such a mechanistic analysis because it is one of the best-understood dispersal mechanisms in plants and because validated mechanistic models are available to predict wind-driven dispersal distance using easily measurable dispersal traits and environments at various scales . Still, these mechanistic models have not yet been used to analyse the density-dependence of wind-driven seed dispersal.
To analyse the role of density-dependent fecundity and seed dispersal in population spread, we address two questions: (1) How does density-dependence of fecundity and dispersal distance affect spread rate, edge depth, and population continuity? (2) Does this density-dependence have noticeable effects on the spread of wind-dispersed plant species? To address the first question, we develop and analyse a general modelling framework that describes the effects of density-dependence on the three spread characteristics.
To address the second question, we apply this general framework to five wind-dispersed plant species by combining a mechanistic wind-dispersal model with experimental measurements of fecundity, dispersal traits and dispersal environments at different population densities. This combination of spread models with experimental data enables us to assess the real-world relevance of density-dependent fecundity and dispersal for population spread.

| A general framework for modelling density-dependent population spread
We developed a general simulation model that describes the densitydependence of both fecundity and mean dispersal distance and predicts the spatial population dynamics of an annual plant either in one-dimensional linear space or on a two-dimensional grid. For the core analyses presented here, we choose the one-dimensional model because it is more computationally efficient. However, density effects are highly consistent between the one-and two-dimensional model versions (Table S3; compare Figure 2 with Figure S5 for the similarity between the output of the one-and two-dimensional models; see also Nathan et al., 2012;Peart, 1985). In the simulations, reproduction, dispersal and establishment occur sequentially in yearly time steps. Following Kubisch et al. (2011), fecundity per plant (R) is drawn from a Poisson distribution with density-dependent mean fecundity given by a power law function: where R 0 is the net reproductive rate, N is population density and β 1 is the strength of density-dependence for fecundity. Densitydependence of dispersal can be described in different ways. For the general sensitivity analysis, we modelled the density-dependent mean dispersal distance (D) also as a power-law function of density: (1) ln (R) = ln R 0 + 1 * ln (N), (2) ln (D) = ln D 0 + 2 * ln (N), F I G U R E 2 Effects of density-dependent fecundity and mean dispersal distance on (a, b) population spread rate, (c, d) edge depth, and (e, f) population continuity. Panels show results of a global sensitivity analysis with the one-dimensional model for negative exponential (left column) and lognormal (right column) dispersal kernels. For the simulations, net reproductive rate (R 0 ) was set to 2, and low-density dispersal distance (D 0 ) was set to 10 m. Note that contours in panels a-d have logarithmic spacing.

Exponential kernel Lognormal kernel
where D 0 is the mean dispersal distance at low density and β 2 is the strength of density-dependence for dispersal. For the species-specific simulations, density-dependence of dispersal distance instead arises from the mechanistic Wald analytical long-distance dispersal (WALD) model (Katul et al., 2005) (see below). Post-dispersal establishment of seeds is simulated as a lottery competition with carrying capacity K.
All simulations and subsequent analyses were implemented in R 4.2.1 (R Core Team, 2022). A full description of the simulation model and R code are provided in Supporting Information.

| Sensitivity analysis
We conducted a global sensitivity analysis to quantify how spread rate, edge depth and population continuity respond to the sign and strength of density-dependence of fecundity and seed dispersal. In a full-factorial design, we varied the net reproductive rate (R 0 ) from 1.1 to 2 in steps of 0.1, the density-dependence of fecundity (β 1 ) and mean dispersal distance (β 2 ) from −1 to 1 in steps of 0.1. The ranges of β 1 and β 2 cause mean fecundity R and dispersal distance D at carrying capacity (K = 1000 individuals/m 2 , corresponding to the typical density of central European grassland Roscher et al., 2004) to range between 12.5% and 800% of R 0 and D 0 , respectively. Cell size was set to 0.09 m × 0.09 m, corresponding to the central pot area in our experiment (see below). Low-density dispersal distance D 0 was set to 10 m.
To assess the effects of the shape of the dispersal kernel, we conducted this sensitivity analysis for both thin-tailed negative exponential and fat-tailed lognormal dispersal kernels (setting the log-scale standard deviation of the latter to 1). For each dispersal kernel and each parameter set, we conducted 20 replicate simulations starting with the first 100 grid cells filled with one individual per cell and followed the population spread over 20 time steps (generations). We then calculated spread rate as the distance between the furthest forward individual at time t and time t + 1, edge depth as the distance between the population edge at time t and the mother plant of the furthest forward individual at time t + 1, and population continuity as the proportion of occupied cells among all cells from the population core (the front end of the initial 100 cells) to the population edge ( Figure 1). For each simulation replicate, these three spread characteristics were averaged over the second half of the simulation period (t = 11-20; Figure S4).

| Real-world application of the modelling framework
To simulate species-specific seed dispersal for different population densities, we used the mechanistic WALD model (Bullock et al., 2012;Katul et al., 2005;Skarpaas & Shea, 2007;Zhu et al., 2021), which is a seed dispersal kernel model that models the probability that a seed travels over distance x as where the location parameter μ = H r * U/V t , the scale parameter λ = (H r /σ) 2 , U is the horizontal wind speed at seed release height H r , V t is seed terminal velocity and σ is a turbulent flow parameter reflecting wind speed variation (Katul et al., 2005).
To parameterise the WALD model, we To avoid pseudo-replication, each pool only contained one pot per plant species and population density. The water in the pools was refilled twice per week. The pots were regularly weeded, but neither directly watered nor fertilized. The study species had an average survival of 95% during the course of the experiment.
After the first capitulum matured in July 2019, seed release height was measured as the vertical distance between the soil surface and the matured capitula. When a target plant produced more than 10 capitula, the first 10 values of seed release height were measured. The first three measured capitula of each target plant in H. glabra, C. sancta, H. radicata, and B. perennis, as well as three haphazardly selected 5-cm-long branches in C. album were collected and stored separately in a cylindrical paper bag to keep the seed appendages intact. The paper bags were then transported to the laboratory and air-dried for at least 2 weeks before the measurement of seed terminal velocity. Three seeds were haphazardly selected from each capitulum, and their terminal velocity was measured following Zhu et al. (2022).
Except for C. album, the fecundity of each target plant was measured by multiplying the total number of capitula produced and the mean seed number per capitulum, which was estimated by counting the seed number in each of the first 10 matured capitula. For C.
album, the fecundity was measured by dividing the total seed mass of the target plant by the mass of haphazardly selected 50 seeds (Kleyer et al., 2008).
At the end of the experiment, the above-ground biomass in each pot was harvested and dried in a laboratory drying oven at 70°C for 72 h. The total above-ground biomass was determined by summing up the weight of the oven-dried plant material and the weight of airdried capitula or branches, as well as the weight of seeds dispersed during the experiment, which was estimated by multiplying the number of empty capitula and the average weight of a capitulum.
The mean horizontal wind speed, aerodynamic roughness length, and friction velocity were acquired from the Land-Atmosphere Feedback Observatory (https://lafo.uni-hohen heim.de/en/lafostart page; Späth et al., 2022) at the experimental site, covering the dispersal season from 1 July to 31 October 2019. We fitted a logarithmic wind velocity profile to the wind data, where u is the mean horizontal wind speed at the height z, u * is the friction velocity, k is von Kάrmάn's constant (0.41), d is zero-plane displacement distance, and z 0 is aerodynamic roughness length (Monteith & Unsworth, 2013). The above-ground biomass enters the dispersal model via its effect on leaf area index and zero-plane displacement distance (for details see the Supporting Information).

| Species-specific simulations of density-dependent population spread
To investigate the effect of density-dependence on population spread for the five study species, we embedded the density-dependent version of the WALD model into the general modelling framework (see above). In this density-dependent WALD model, dispersal traits (seed release height and seed terminal velocity; Equation 3) and dispersal environments (above-ground biomass affecting leaf area index and aerodynamic roughness length; Equation 4) were modelled as functions of population density (see Supporting Information; Figure S1). To quantify the effect of density-dependent fecundity, density-dependent seed dispersal and their interaction, densitydependence of fecundity and seed dispersal was either included (using the abovementioned mixed models to explicitly describe effects of log-transformed population density and its interaction with species identity on fecundity and/or dispersal traits and environments, see the preceding section) or excluded (ignoring density effects by using species-level means of fecundity and/or dispersal traits and environments). This resulted in a full-factorial simulation design with four scenarios of density-dependence per species.

| Effects of density-dependent fecundity and dispersal distance on dynamics of population spread
Density-dependent fecundity and dispersal strongly affect all three population spread characteristics (spread rate, edge depth, and population continuity) for both exponential and lognormal dispersal kernels ( Figure 2). Across all combinations of net reproductive rate, density-dependence (of fecundity and/or mean dispersal distance), and dispersal kernels, spread rate and edge depth were strongly correlated (Pearson correlation coefficient: ρ = 0.95) but showed weaker correlations with population continuity (ρ = 0.21 and 0.06, respectively; Table S1).
Positive density-dependence of fecundity increases all three spread characteristics ( Figure 2). This is because-for a given net reproductive rate R 0 -positively density-dependent fecundity increases seed production at non-zero density (Equation 1). This increased seed production has positive effects on all three characteristics. In alternative parameterizations, which hold carrying capacity K rather than R 0 constant (e.g. Lewis & Kareiva, 1993), positively density-dependent fecundity instead decreases seed production at densities below K. Our finding that Allee effects increase spread rate for a given R 0 is thus consistent with the finding that Allee effects reduce spread rate for a given K .
Increasingly positive density-dependence of dispersal increases spread rate and edge depth but generally decreases population continuity ( Figure 2). This is because positive density-dependence of dispersal distance causes seeds from the high-density population core to contribute to the advancement of the population edge. Such "leapfrogging" (Becheler et al., 2016;Fontaine et al., 2013) promotes spread rate and increases edge depth by enlarging the number of seeds that contribute to spread. Simultaneously, leapfrogging decreases population continuity by increasing the space between individuals in a spreading population (Figure 2e,f).
Density-dependent fecundity and dispersal are largely additive in their effect on spread characteristics, with additive effects explaining 25%-84% of the variance in characteristics, whereas interactions between density-dependent fecundity and dispersal only account for 0%-12% of the variance (Table S2). Still, for population continuity, the joint effects of density-dependent fecundity and dispersal are somewhat contingent on the dispersal kernel: for negative exponential kernels, population continuity is highest when fecundity is positively density-dependent and dispersal is densityindependent (β 2 ≈ 0; Figure 2e). For fat-tailed lognormal kernels, however, highest population continuity is reached for positively density-dependent fecundity and negatively density-dependent dispersal distance (Figure 2f).

| Real-world application of the modelling framework
The common garden experiment and the experimentally parameterized mechanistic WALD dispersal model revealed density-dependent fecundity ( Figure 3a) and density-dependent dispersal (Figure 3be) across the five wind-dispersed study species. All study species exhibited negatively density-dependent fecundity (Figure 3a), although the strength of this density-dependence varied between species ( 2 4df = 13.65, p = 0.008). In contrast, we found more qualitative interspecific differences in the density-dependence of dispersal traits and environmental variables (Figure 3b-d). Seed release height decreased with population density in all study species except B. perennis in which it increased with population density (Likelihoodratio test, 2 4df = 13.36, p = 0.010; Figure 3b). Seed terminal velocity increased with population density in H. glabra, C. sancta and B.
perennis, but it decreased with density in C. album and H. radicata ( 2 4df = 11.00, p = 0.027; Figure 3c). Total above-ground biomass increased with population density in H. glabra, H. radicata and B.
Species-specific density-dependence of dispersal traits and dispersal environments causes species to differ qualitatively in the density-dependence of wind-driven dispersal distance ( Figure 3e) and probability of long-distance dispersal (Figure 3f). For instance, mean dispersal distance decreased with population density in H. glabra, C. album, and C. sancta, but increased with density in H. radicata and B. perennis ( 2 4df = 743.02, p < 0.001; Figure 3e). The ratio between mean dispersal distance at high and low density ranged from 42% (H. glabra) to 118% (H. radicata; Figure 3e). The probability of long-distance dispersal varied substantially with population density ( Figure 3f). For instance, the ratio of the probability of dispersal distance ≥10 m at high and low density ranged from 1.0 × 10 −12 (C. Density-dependent fecundity and seed dispersal also influenced simulated population spread of the study species (Figure 4). The negative density-dependence of fecundity decreased spread rate and edge depth across all five study species (Figure 4a,b). The negative density-dependence of seed dispersal decreased spread rate and edge depth in H. glabra, C. album and C. sancta (Figure 4a,b). In these species negative density-dependence of fecundity and dispersal furthermore reinforced each other in their negative effect on spread rate and edge depth (Figure 4a,b). Ignoring density effects could thus lead to an overestimation of spread rate in H. glabra, C. album and C. sancta.
The positively density-dependent dispersal found for H. radicata and B. perennis had qualitatively different effects on population spread, markedly increasing spread rate and edge depth (Figure 4a,b). In H. radicata, positively density-dependent dispersal and negatively density-dependent fecundity largely cancelled each other out in their effect on spread characteristics. For B. perennis, however, spread rate and edge depth were dominated by densitydependent dispersal with little effect of density-dependent fecundity (Figure 4a,b). Thus ignoring density effects could lead to an underestimated spread rate in H. radicata and B. perennis.
Population continuity was hardly affected by population density in all study species except C. sancta in which it was strongly reduced by negatively density-dependent fecundity (Figure 4c).

| DISCUSS ION
This study demonstrates the central role of density-dependence of fecundity and dispersal in plant population spread. We show that density-dependence of fecundity and dispersal distance strongly alters three major spread characteristics: spread rate, edge depth and population continuity (Figure 2). We provide experimental evidence that wind-dispersed plant species differ qualitatively in the densitydependence of fecundity and of seed dispersal distance (Figure 3), and predict that this density-dependence profoundly alters their spread rate and edge depth (Figure 4). Our findings are thus important for understanding the consequences of density-dependent fecundity and seed dispersal for plant population spread in general and the mechanisms and consequences of density-dependent seed dispersal by wind in particular.

| Mechanisms of density-dependent seed dispersal by wind
Although it is well understood that seed dispersal by wind depends on seed release height (Soons et al., 2004;Zhu et al., 2016) and terminal velocity (Caplat et al., 2012;Nathan, Horvitz, et al., 2011), in this study we elucidate how the density-dependence of dispersal traits leads to density-dependent dispersal distance (Figure 3; Figure S3). Negatively density-dependent seed dispersal in H. glabra, C. album, and C. sancta mainly arises from decreased seed release height at high population density ( Figure S3), which we conjecture to be a consequence of reduced resource availability caused by intraspecific competition (Wender et al., 2005). In contrast, positively density-dependent seed dispersal arises from either reduced terminal velocity (H. radicata) or increased seed release height (B. perennis) at high density ( Figure S3). Increased seed release height is likely a shade avoidance response (Ruberti et al., 2012), whereas reduced seed terminal velocity could reflect a plastic response of seed size to population density (Postma et al., 2021). Since plastic response and competition-induced resource depletion are ubiquitous in plants (Hutchings & de Kroon, 1994;Sultan, 1995), densitydependent wind dispersal is likely very common. Although the absolute change in mean dispersal distance in our study species is rather small, the changes in long-distance dispersal probability are F I G U R E 3 Effects of population density on fecundity, dispersal traits, dispersal environments and predictions of the WALD dispersal model for five wind-dispersed plant species. Rows show the density-dependence of (a) fecundity, (b) seed release height (H r ), (c) seed terminal velocity (V t ), (d) above-ground biomass, (e) simulated mean dispersal distance and (f) dispersal kernels. In rows (b-e), solid lines represent model predictions, and polygons represent the predicted values ± the standard errors of predictions. In row (f), solid curves indicate predicted dispersal kernels at various population densities, and dashed lines indicate a negative exponential dispersal kernel for comparison. Note that Y-axes are plotted on a logarithmic scale. large. We speculate that the strength of density effects may be even larger in trees (Schurr et al., 2008) which disperse their seeds over longer distances (Thomson et al., 2018) and have stronger effects on wind flows in their vicinity.

| Density-dependent fecundity and seed dispersal strongly alter the spread dynamics of plant populations
Our study reveals that spread rate and edge depth are strongly dependent on the additive effects of density-dependent fecundity and density-dependent dispersal. Increasingly positive densitydependence of fecundity and dispersal strongly increases the spread rate. This holds for both exponentially bounded and fat-tailed dispersal kernels. These findings have several important implications.
First, it is essential to account for density-dependence when analysing data on fecundity and seed dispersal and using these data to simulate population spread. Ignoring density-dependence could result in an over-or underestimated spread rate, which is especially relevant in the context of biological invasions and conservation . Second, our findings could shed some new light on Reid's paradox, the finding that tree spread in the Holocene occurred more rapidly than predicted by diffusion models parameterised with field estimates of R 0 and D 0 (Clark, 1998;Reid, 1899). While the paradox can be resolved with fat-tailed dispersal kernels (Clark, 1998), positive density-dependence of fecundity and/or dispersal distance may also play a role. If such positive density-dependence occurs, spread models that are parameterized with estimates of R 0 and D 0 at low population density and that ignore positive density-dependence will underestimate spread rate. In this case, spread models would experience the reduction of fecundity and dispersal at low density but not the increase of fecundity and dispersal with density. Third, quantifying the density-dependent spread rate may more realistically evaluate plant migration facing future climate and land-use changes.
Increasingly positive density-dependence of fecundity and seed dispersal also increases edge depth, causing spread dynamics to be increasingly 'pushed' by dispersal from high-density regions far from the edge (cf. Birzu et al., 2019;Dahirel et al., 2021;Miller et al., 2020). This increasing importance of the population core arises because positive density-dependence causes high-density core regions to produce and/or export disproportionally more seeds than the low-density edge. By enlarging edge depth, positive density dependence of fecundity and dispersal should cause increased gene flow from the core to the edge of a population, thus increasing genetic diversity and suppressing genetic drift at the spreading population edge (Birzu et al., 2019;Miller et al., 2020;Roques et al., 2012).
Accordingly, density-dependence of seed dispersal should also influence the spatial genetic structure of the spreading population on F I G U R E 4 Effects of density-dependent fecundity and dispersal on (a) population spread rate, (b) edge depth, and (c) population continuity of five wind-dispersed plant species. Net reproductive rate (R 0 ) was set to 2, and other model parameters were measured experimentally. No density−dependence D ensity−dependent fecundity D ensity−dependent dispersal B oth large scales (Hargreaves & Eckert, 2014). This knowledge is crucial for conserving the genetic diversity of endangered plant species (McKay et al., 2005).
By increasing edge depth and genetic variance at the edge, positive density-dependence of fecundity and dispersal counteracts the fixation of maladapted genotypes at the expanding edge ('expansion load') and increases the potential of subpopulations at the edge to respond to selection (Miller et al., 2020;Peischl et al., 2013). However, if the selective regime differs between the edge and the core, a high edge depth can cause edge subpopulations to be 'swamped' by maladapted genotypes from the core, causing a 'migration load' that counteracts the beneficial effects of edge depth on genetic variance (Lenormand, 2002). Moreover, density-dependence itself can cause selection to differ between the edge and core (Miller et al., 2020).
For instance, negative density-dependence of fecundity causes local selection for increased dispersal and reproductive output at the edge (Phillips et al., 2010;Travis & Dytham, 2002). The potential of edge subpopulations to respond to this local selection may be higher when edge depth is reduced by negatively density-dependent dispersal. The spectrum of theoretically possible eco-evolutionary dynamics may become even broader if density-dependence itself evolves during spread (e.g. Travis et al., 2009). An understanding of the proximate mechanisms of density-dependence (as developed here for wind-driven seed dispersal) should help to constrain this broad spectrum of theoretically possible dynamics to the biologically plausible.
Although population continuity is generally high in our simulations, it is affected by density-dependent fecundity and seed dispersal. As a result, density-dependence may cause discrete or aggregated subpopulations even in homogenous landscapes. In particular, increasingly positive density-dependence of seed dispersal decreases population continuity by leapfrogging dispersal events.
This may reduce the detectability of the individuals at the population edge, increasing the difficulty of controlling plant invasion (Epanchin-Niell & Hastings, 2010). In such cases, priority should be given to the eradication of plants from the population core at the early stage of invasion to reduce the probability of long-distance dispersal (Buckley et al., 2005).

| Applying the general framework to other study systems
The general framework presented here can be altered in several ways to fit other study systems. First, it can be applied at coarser spatial scales at which Allee effects caused by limited reproduction are more likely to operate than in our small-scale experiment (Kunin, 1993). At these scales, transplant experiments between subpopulations of different densities can serve to quantify the density-dependence of fecundity (Lachmuth et al., 2018). The framework can also be extended to describe density-dependence at more than one scale (e.g. resource competition acting at finer and Allee effects operating at coarser scales; Nottebrock et al., 2013. Secondly, the grid-based nature of our model makes it easy to incorporate empirical information on impacts of environmental heterogeneity on plant fecundity and dispersal. For instance, our study species tend to produce fewer seeds and disperse to shorter distances when water-stressed (Zhu et al., 2021), which may reinforce the effects of positive density-dependence on population spread detected here (Figures 2 and 4). By incorporating environmental heterogeneity, it would also be possible to study the real-world relevance of density-dependence for plant population spread through patchy landscapes (Dahirel et al., 2021;Maciel & Lutscher, 2015;Pachepsky & Levine, 2011).

| CON CLUS IONS
By embedding a mechanistic wind-dispersal model and experimental data in a general modelling framework, we have demonstrated that density-dependence of fecundity and seed dispersal distance can strongly impact the dynamics of plant population spread. Densitydependent changes in wind dispersal traits due to intraspecific competition and shade avoidance response (Ruberti et al., 2012) are ubiquitous in plants. Hence, density-dependence should be very common, not only for fecundity but also for seed dispersal by wind.

ACK N O WLE D G E M ENTS
We thank Volker Wulfmeyer for providing the wind data and F.R.
Castro, S. Appiah and C.P. Monteiro for field and laboratory assistance. We are grateful to all members of the "Publication club"

CO N FLI C T O F I NTE R E S T S TATE M E NT
The authors have no conflict of interest.

PEER R E V I E W
The peer review history for this article is available at https:  Population spread depends on the population growth rate and dispersal distance which depends on dispersal traits and dispersal environments that consist of both biological and physical environments. Population growth rate, dispersal traits and dispersal environments further depend on population density.
In this study, we apply this framework in the context of wind dispersal which is determined by seed release height, terminal velocity, as well as dispersal environments such as above-ground biomass and roughness length.    replicate, the three spread characteristics were averaged over the second half of the simulation period (t = 11-20). Since the twodimensional simulations were more computationally intensive than the one-dimensional simulations, we set the low-density dispersal distance (D 0 ) to 2 m (rather than 10 m) and did 5 (rather than 20) simulation replicates. The remaining simulation parameters were identical to the one-dimensional simulations. Note that contours in panels a-d have logarithmic spacing. Table S1. Correlation between the three spread characteristics in the general sensitivity analysis. Table S2. R 2 of the linear regression models for relationships between characteristics of population spread and density dependence of fecundity (b1) and mean dispersal distance (b2) for sensitivity analysis. Table S3. Correlation between population spread characteristics predicted by the one-dimensional and two-dimensional models.
All correlation coefficients are highly significantly different from 0 (t 439 > 42, p < 0.001) for all combinations of spread characteristics and dispersal kernels.