Seed Dispersal as A Search Strategy: Dynamic and Fragmented Landscapes Select for Multi-Scale Seed Dispersal

Background – Plant dispersal is a critical factor driving ecological responses to global changes. Knowledge on the mechanisms of dispersal is rapidly advancing, but selective pressures responsible for the evolution of dispersal strategies remain elusive. Recent advances in animal movement ecology identied general strategies that may optimize eciency in animal searches for food or habitat. We here explore the potential for evolution of similar general movement strategies for plants. Methods – We propose that seed dispersal in plants can be viewed as a strategic search for suitable habitat, where the probability of nding such locations has been optimized through evolution of appropriate dispersal kernels. Using model simulations, we demonstrate how dispersal strategies can optimize key dispersal trade-offs between nding habitat, avoiding kin competition, and colonizing new patches. These trade-offs depend strongly on the landscape, resulting in a tight link between optimal dispersal strategy and spatiotemporal habitat distribution. Results – Our ndings reveal that multi-scale seed dispersal strategies that combine short-distance and long-distance dispersal, including Lévy-like dispersal, are optimal across a wide range of dynamic and patchy landscapes. Static patchy landscapes select for short-distance dominated dispersal strategies, while uniform and highly unpredictable landscapes both select for long-distance dominated dispersal strategies. Conclusions – By viewing plant seed dispersal as a strategic search for suitable habitat, we provide a reference framework for the analysis of plant dispersal data. This reference framework helps identify plant species’ dispersal strategies, the evolutionary forces determining these strategies and their ecological consequences, such as a potential mismatch between plant dispersal strategy and altered spatiotemporal dynamics due to land use change. Our perspective opens up directions for future studies, including exploration of composite search behaviour and ‘informed searches’ in plant species with directed dispersal.

reference framework for the analysis of plant dispersal data. This reference framework helps identify plant species' dispersal strategies, the evolutionary forces determining these strategies and their ecological consequences, such as a potential mismatch between plant dispersal strategy and altered spatiotemporal habitat dynamics due to land use change. Our perspective opens up directions for future studies, including exploration of composite search behaviour and 'informed searches' in plant species with directed dispersal.

Background
Dispersal plays a crucial role in the population dynamics and ecological interactions of plant species. In light of ongoing habitat fragmentation and climate change, dispersal is a particularly critical determinant of local, regional and global plant species survival (1)(2)(3)(4). This realisation has elevated plant dispersal to a research priority in the last decades. Quantitative information on plant species' dispersal distance distributions (or 'dispersal kernels') is needed for adequate species management, as dispersal kernels determine colonization probabilities and restoration success, as well as speeds of range expansions and invasions. Indeed, signi cant progress has been made in understanding how mechanisms of seed dispersal determine seed dispersal kernels (5)(6)(7)(8), and how these may be affected by global changes (1,(9)(10)(11). However, the selective pressures responsible for the evolution of dispersal strategies remain elusive.
Excellent overviews of studies on the evolution of dispersal in plants and animals are given in Ronce (12) and Duputié and Massol (13). For plants, a long line of research has developed on how (kin) competition, facilitation, inbreeding, and density dependent mortality translate to selective pressures for dispersal propensity (the tendency of an individual to disperse) (e.g., (14)(15)(16)(17)(18)(19)(20)(21)(22)(23)(24). In addition, theoretical studies have shown that increasing dispersal propensity can serve as an effective bet-hedging strategy to deal with spatiotemporal environmental variability (22).
Few studies to date included dispersal kernels in their evolutionary analysis (25)(26)(27). These studies usually focus on a single aspect of dispersal (e.g., long-distance dispersal) or compare dispersal strategies in a relative sense (e.g., how dispersal propensity increases or decreases in response to e.g. stress-related factors). However, many of the driving processes are acting at different spatial scales (13), so that exibility in the shape of dispersal kernels is required to balance trade-offs between driving processes at a range of scales. Here, we move beyond dispersal propensity or focus on a single aspect of the dispersal distance distribution, and argue that the entire distribution of seed dispersal distances matters and may optimally balance dispersal over shorter and longer distances.
We propose that plant dispersal strategies evolve as search strategies for suitable habitat, in a way comparable to stochastic searches made by other moving organisms. Reynolds (28) already hypothesized that plants may maximize the likelihood of nding the nearest unoccupied site by adopting a Lévy ight-shaped inverse power-law seed dispersal kernel. We propose that plant species have a wide range of dispersal strategies, that each evolved in search of all suitable habitat of the species, and therefore depend strongly on the spatial and temporal distribution of their habitat. Using a theoretical framework inspired by animal movement ecology, we show how recent conceptual developments in analysing animal movement data can advance the eld of plant dispersal ecology towards identi cation of evolutionary drivers and ecological consequences of seed dispersal strategies.
Over the past decade, analyses of high-resolution movement data from a wide range of animals have broadly identi ed the signatures of complexity in movement patterns. Movement paths can be decomposed into consecutive movement steps of a speci c length (move length) that are separated by changes in direction (turns) (29,30). Move length distributions can be described by power-law relationships (p(x) ~ x −µ ; Fig. 1) with a scaling exponent µ ranging from ~ 1 (promoting long-distance movements, where long steps are as abundant as short steps) to > 3 (promoting Brownian motion, where short steps are abundant and long steps are very rare). At an intermediate µ ~ 2, many consecutive shortdistance movements are alternated with infrequent long-distance movements, producing complex multiscale movement patterns where many different move lengths occur. In such multi-scale movement strategies, the relationship between move length and frequency decays neither too quickly (so that one scale predominates) nor too slowly (so that all scales are equally frequent) (30)(31)(32)(33)(34). These complex multi-scale movement strategies are known as Lévy ights or walks (31).
In a random search, the move length distribution that maximizes search e ciency depends on the spatiotemporal distribution of targets being sought (e.g., food, mates; (32,35,36)). Theoretical studies have shown that various strategies may effectively balance short-distance and long-distance movements, depending on the spatiotemporal environmental conditions (34)(35)(36)(37)(38). Importantly, this balance is determined by the entire move length distribution. Experimental studies have shown that complex animal movement patterns indeed have intrinsic underlying patterns that optimize random search e ciency, thereby greatly enhancing individual tness (34,(39)(40)(41).
Although these recent developments have signi cantly advanced our understanding of the evolution and ecology of animal movement (41), the potential for evolution of similar random search strategies has not been explored for plants. We propose that the move length distribution (i.e., seed dispersal distance distribution or dispersal kernel) generated by all the seeds coming from a single plant can be viewed as a movement strategy to search for suitable habitat, or more speci cally, suitable sites for germination, establishment and reproduction. As adult plants are otherwise immobile, we expect that there is strong selective pressure on such dispersal kernels.
Here, we provide a theoretical framework that identi es null models for (evolution of) dispersal strategies in plant populations and explores the impact of spatiotemporal landscape structure (habitat fragmentation and patch turnover dynamics) on dispersal evolution, considering dispersal as a search strategy. We acknowledge that evolved plant dispersal strategies should match relevant biotic and abiotic factors determining eventual offspring success, such as the spatial structure of density-dependent mortality (42), competition, and facilitation (43,44). However, in this study, we rst explore the evolution of dispersal strategies in a neutral context where plants are identical except for dispersal strategy. By focussing primarily on dispersal strategy in relation to spatiotemporal habitat distribution, we derive fundamental null models that can serve as the basis for extension in the future to explore effects of additional relevant factors.

Methods
We developed a straightforward spatiotemporal lattice model to explore the evolution of plant dispersal strategies in landscapes differing in their degree of fragmentation (patch size and inter-patch distance) and landscape dynamics (patch turnover). We excluded any variation in life history traits and competitive interactions between plants; all plants were completely similar entities that only differed in the shape of their dispersal kernels. To explore different dispersal strategies, we simulated seed dispersal kernels using a truncated two-dimensional (isotropic) Pareto distribution with discrete values of shape parameter µ ranging from 1.1 to 5 (Table 1; see Appendix S1 in Supporting Information for derivation of 2D form): where l min is the minimum distance (radius of a grid cell), l max is the maximum distance (equal to the domain size) and µ is the scaling exponent. The scaling exponent determines the power-law decay of the dispersal kernel. This scaling exponent makes the 2D-Pareto kernel a very convenient tool to explore different dispersal strategies; by changing only one parameter, the kernel can cover a full range of dispersal strategies ranging from very local (short-distance dispersal dominated, µ > 3) to non-local (relatively high probability of long-distance dispersal, µ → 1). For µ ~ 2 (i.e., canonical Lévy), this kernel produces a highly heterogeneous, multi-scale distribution of dispersal distances. We compared these 2D-Pareto kernels to 'benchmark' kernels or limiting cases on both ends of the spectrum: uniform dispersal across the entire domain (as benchmark for minimum µ) and dispersal only to the 8 nearest neighbours (Moore neighbourhood) in equal probabilities (as benchmark for maximum µ). In each model-run, two plant types with different dispersal kernels competed in a landscape with speci ed fragmentation and patch turnover characteristics. Plants were initially randomly placed in equal proportions throughout a landscape. For simplicity, plants only produced seeds once per generation and then died (i.e., we simulated semelparous plants). Between generations, the types were redistributed over the landscape following these event-driven steps: 1) dispersal, 2) death, 3) patch turnover, and 4) colonization. The type that remained after multiple generations was assumed to have the better dispersal strategy. We then related these dispersal strategies to three important dispersal metrics that quantify the success of dispersal in terms of (i) the success rate of nding habitat (hereafter referred to as 'habitat encounter'), (ii) the success rate of avoiding kin competition (hereafter 'kin avoidance'), and (iii) the success rate of colonizing new patches (hereafter 'colonization'). A conceptual gure illustrating the model and model details are provided in Appendix S2.

Simulations
We characterized landscapes by the parameters patch size, inter-patch distance, and patch turnover rate, and determined how evolved dispersal kernels depend on these landscape characteristics. Table 1 shows the parameter ranges used in our simulations.
For each set of parameter combinations, model simulations ran until one of the types was outcompeted, i.e. when it occupied less than 5% of all habitat grid cells, while the other type increased to at least 80% of all habitat grid cells or reached a stable equilibrium (no signi cant decrease over 200 generations). If both types were maintained after 1000 generations, we assigned no 'winner' and scored this as no strong selection on dispersal strategy. For each landscape, all possible combinations of dispersal kernels were used to determine the evolutionarily stable strategy (ESS). We repeated each simulation 12 times to test robustness of results. We summarized the outcome of these 12 replicate runs as follows: 1) a clear winner (one population won in at least 11 out of 12 repetitions), 2) extinction of both populations, or 3) no clear winner or no convergence. Per landscape, these results are presented in pairwise invasibility plots (PIPs, (45)). From each PIP, we extracted the ESS (expressed by parameter µ) and used this to identify changes of evolved strategy in relation to patch size, inter-patch distance, and patch turnover rate. We performed robustness tests of our model results to variations in plant seed number (Appendix S3) and our FFT approach (Appendix S4), which both had no signi cant impact on model output.

Dispersal metrics
We expected that, similar to random searches by animals, evolutionarily stable dispersal kernels would adequately balance a complex trade-off between short-distance and long-distance dispersal conditioned to the landscape con guration (habitat patch size, inter-patch distance and patch turnover). To facilitate interpretation of these underlying trade-offs, we calculated a number of dispersal metrics that relate to the success of dispersal for each landscape con guration. First, we calculated the success rate of nding habitat (hereafter referred to as 'habitat encounter'), as the fraction of seeds landing in suitable habitat. Second, we calculated the success rate of avoiding kin competition (hereafter 'kin avoidance'). Grid cells close to parent plants typically receive a high quantity of seeds (> 1), but only one individual can occupy a cell in the next time step. For each parent, we summed all fractions of seeds above one and then normalized these for the number of grid cells where kin competition took place to calculate the fraction of seeds involved in kin competition. We calculated kin avoidance as 1 -kin competition. Third, we determined the success rate of colonizing new patches (hereafter 'colonization') as the fraction of seeds landing in a new patch that emerged due to patch turnover.

Results
Evolutionarily stable dispersal strategies are tightly connected to the spatiotemporal distribution of plant habitat Our simulations show that all types of movement strategies (ranging from strategies dominated by shortdistance dispersal, to multi-scale dispersal, to long-distance dispersal) can be an ESS, depending on the spatiotemporal distribution of habitat in the landscape. The spatiotemporal distribution of habitat determines the optimal balance between short-distance and long-distance dispersal following trade-offs in habitat encounter, kin avoidance and colonization. In general, we found that in the most static and in the most unpredictable landscapes, the two extremes (Moore nearest-neighbour dispersal and uniform dispersal, respectively) are ESS's. When patch distributions are dynamic and fragmented, the trade-off between short-distance and long-distance dispersal results in a wide range of multi-scale dispersal strategies (including Lévy-like Pareto kernels) that are tightly connected to the spatiotemporal habitat distribution.
Short-distance and long-distance dominated dispersal strategies When the distribution of habitat is patchy in space and static in time, the landscape is highly predictable and movement strategies dominated by short-distance dispersal are evolutionarily stable (Fig. 2a-c). The shape of these short-distance dispersal kernels varies with patch size, with decreasing patch sizes corresponding to shorter dispersal distances (larger µ) (Fig. 2c). These movement strategies are essentially driven by the optimization of both habitat encounter and kin avoidance (Fig. 2d-f). With increasing patch size, the edge-to-area ratio decreases and habitat encounter increases accordingly, allowing for strategies with somewhat longer dispersal, which improves kin avoidance.
When the distribution of habitat is continuous in space (the most predictable scenario) or when the distribution of habitat is unpredictable, either in space (patch size = 1, i.e. patches can hold one individual only) or in time (patch turnover rate = 1), dispersal strategies dominated by long-distances were evolutionarily stable (Fig. 3a-c). In these situations, dispersal strategies are driven exclusively by avoidance of kin competition (Fig. 3d-f). Maximizing habitat encounter does not contribute to selecting the optimal strategy, because habitat encounter is similar for all possible values of µ; either because the landscape consists of homogeneous habitat, or because the habitat distribution is so unpredictable that no µ optimizes habitat encounter better than another (Fig. 3d-f).

Multi-scale dispersal strategies
In many landscapes, habitat distribution is patchy and dynamic to some extent. In these situations, evolved dispersal strategies are dominated by multi-scale dispersal strategies, i.e. broadly heterogeneous and heavy tailed kernels (e.g. Lévy-like Pareto distributions). These strategies balance local, within-patch dispersal to provide high habitat encounter and non-local dispersal to avoid kin competition and colonize new patches. This balance is driven by all patch distribution characteristics: patch size, inter-patch distance and patch turnover rate (Fig. 4, Appendix S5). The ESS's for these landscapes, as re ected by µ, are most strongly determined by patch turnover rates (Fig. 4, 5, S5), which very strongly increase the need to colonize new patches. Higher patch turnover rates correspond to lower values of µ, with µ ~ 2 for a wide range of landscapes with patch turnover rates between 0.1 and 1 (Fig. 5). Secondarily, evolutionarily stable dispersal strategies are also driven by the relative importance of colonization of new patches in relation to habitat encounter and kin competition, which decreases with patch size and inter-patch distance (Fig. 4, S5). Under low dynamic conditions, this translates to multi-scale dispersal strategies with more local dispersal when patch sizes are small and inter-patch distances short (as the role of habitat encounter becomes more important, µ → 3). Under highly dynamic conditions, this translates to more long-distance dispersal when inter-patch distances are large (as the effects on colonization are strengthened by the role of kin avoidance, µ → 1.5; Fig. 4, S5). The short-distance dominated dispersal strategies that we found to be ESS's in static landscapes with small patch sizes (2-8) and large interpatch distances (> 50) changed immediately when patch turnover rate was even slightly larger than zero, as winning the local competition within patches was no longer a stable strategy in the long term.
In some dynamic landscapes with large inter-patch distances, no dispersal strategy could ensure population survival. In these cases, the probability of seeds ending up in new habitat was too low to overcome the loss of habitat due to patch turnover.

Discussion
Studies of animal movement behaviour have identi ed general optimal movement strategies based upon the spatial distribution of resources (32,36). Using a similar analytical approach, we show that general optimal dispersal strategies can be identi ed for plants based purely on the shape of the entire seed dispersal kernel in relation to the spatiotemporal distribution of the plant habitat. While earlier studies have shown how dispersal propensity may evolve in response to landscape structure, cost of dispersal and other (density-dependent) processes (13,(25)(26)(27)46), we here show that the entire shape of the dispersal kernel can be seen as a multi-scale search strategy that needs to balance incentives to disperse over short-distances, long distances and everything in between.
The model used in this study is very simple, with individual plants being identical except for differences in dispersal strategies. Competition is not explicitly parameterized in the model, but implicitly there is some form of density dependence due to the role of kin avoidance. Multi-scale dispersal strategies emerge as optimal strategy even within this simple and straightforward framework, suggesting that they are of importance as a baseline in the natural setting where more processes play a role in spatial population dynamics. The reference framework following from our results is visualised conceptually in Fig. 6. The main hypotheses for real plant data generated from our ndings are: (1) In static, but patchy habitats, short-distance dispersal (e.g. µ > 3) dominates multi-scale dispersal strategies, due to the importance of optimizing habitat encounter. Particularly when patches are small and inter-patch distances are large, there is a strong selection in favour of extremely short-distance dispersal. (2) In contrast, extreme longdistance dispersal (µ → 1, or even uniform dispersal kernels) is favoured in both stable, continuous habitats as well as in unpredictable and dynamic landscapes. These dispersal strategies are driven by avoidance of kin competition and need to colonize newly formed patches. (3) In patchy and dynamic environments, a complex trade-off between nding habitat, avoiding kin competition and colonizing new patches results in multi-scale dispersal strategies with µ correlated to average patch size, inter-patch distance and, most importantly, patch turnover rate. Our results suggest that multi-scale kernels similar to Lévy ights (µ ~ 2) would be selected for in patchy landscapes with intermediate patch sizes (~ 2 to 100 times the plant size), intermediate inter-patch distances (~ 5 to 100 times the plant size) and relatively high patch turnover rates of around 50% per generation.
Some aspects of our ndings are in line with well-known patterns observed in plant communities: Plant species in patchy and highly dynamic habitats typically have dispersal strategies dominated by longdistance dispersal and species from patchy but highly static landscapes tend to display predominantly short-distance dispersal that promotes the chance of success in 'winning the home patch' (22,(47)(48)(49). Yet, such hypotheses are not trivial. For example, in static but patchy landscapes short-distance dispersal strategies may rapidly evolve. Colonization has been followed by rapid loss of long-distance dispersal in plants on islands and patches in urban environments (50)(51)(52). Such species are extremely vulnerable to habitat loss and fragmentation, as their dispersal strategy is not adapted to colonizing new areas (53).
With ongoing global change, such dispersal-limited species are under great threat of extinction -an example of such a case is the endemic and highly threatened Centaurea corymbosa which is adapted to long term persistent, but isolated rocky outcrops (54).
Some hypotheses generated within our study may appear counterintuitive. For example, species in homogeneous habitats are suggested to have uniform dispersal kernels. This hypothesis would explain why, indeed, many species of large-scale, more or less continuous habitats, such as primary forest (55) and heathlands (56), have adaptations for very long-distance dispersal. Previous studies may have suggested that these adaptations serve to avoid density-dependent mortality close to the parent (16,57,58), but this would not explain dispersal over more than a few tens of m (the decay rate of pest-induced mortality, (24)). Our results suggest that selection for kin avoidance may explain these long-distance dispersal syndromes, although escaping density-dependent mortality may be an additional, enforcing factor.
Our analyses also lead to interesting untested hypotheses: species subjected to patchy environments should have multi-scale dispersal strategies that vary in the fatness of their tail in relation to patch size and inter-patch distances, but primarily in relation to patch turnover rates. Analyses of measured plant dispersal kernels across real landscapes should reveal whether these hypotheses indeed re ect reality. It is, however, di cult to obtain complete dispersal kernels from eld measurements, as long-distance dispersal events are extremely di cult to measure and at the same time form a vital component of the dispersal strategy. For wind dispersal, mechanistic models have been developed that simulate complete dispersal kernels (including long-distance dispersal events), and these have withstood tests against eld tracking and trapping data (e.g., CELC, (59); and WALD, (60)). Simulations of tree dispersal kernels using WALD indicate that forest trees such as Liriodendron tulipifera in oak-hickory forests, one of the largest and most continuous forest habitats in temperate regions, could have tails with power laws of µ ~ 1.5 (60) and species such as Pinus taeda are likely to have even fatter tails (55). These kernels are close to the long-distance dominated dispersal kernels that would be expected for species in continuous habitats. Simulations of wind dispersal using the CELC model for herbs characteristic of patchy and temporary wet grasslands ((59); data from (61)) generate dispersal kernels that are best tted by 2D-Pareto distributions with µ ~ 2 (1.9 for Cirsium dissectum, 2.0 for Hypochaeris radicata). These values match the Lévy-like multi-scale dispersal kernels expected for species in successional, patchy habitats. For species typical of highly disturbed sites, such as Tussilago farfara in disturbed open sites, extreme long-distance dispersal has been reported -up to 4000 m in one generation (62), with a roughly estimated µ of 0.59 (63). Another species typical of disturbed sites is Cecropia obtusifolia, a pioneer tree colonizing forest gaps. Seed trap data of this species in young forests are best t by a 2D-Pareto distribution with µ = 1.1 (data from (64). We summarize these rst lines of evidence in Fig. 6.
By proposing to analyse plant seed dispersal as a search strategy for nding suitable habitat and using kernels with different scaling behaviour to compare dispersal strategies across different landscape dynamics, we break with the tradition of investigating dispersal propensity or only a single aspect of plant dispersal kernels such as the tail or modal distance. With methodological hurdles to the study of long-distance dispersal being overcome (55,65), much research has focused on quantifying the tail of the dispersal kernel (5,(65)(66)(67). This has resulted in rapid progress in our understanding of, and ability to predict, the connectivity of plant populations in fragmented landscapes (68,69) and has helped to explain species' abilities to track climate change (3,70) or become invasive (71). At the same time, other studies have focused on the mode of the dispersal distribution to facilitate cross-species comparisons (72), as modal distance is an attractive parameter to study, representing the distance where most seeds end up and being far easier to measure. We however stress that the entire dispersal kernel de nes the movement strategy of plants, and as such is relevant for local, landscape-scale and global species survival. Such an integrated approach to plant dispersal has also been advocated in the general 'movement ecology paradigm' (73), and an important rst step in making large cross-species comparisons of entire dispersal kernels has recently been taken (11). The simplicity of our approach, which uses a exible dispersal function parameterized by a single parameter, µ, facilitates further comparisons across large numbers of species with widely differing dispersal strategies, while also allowing for the exploration of relations between species' dispersal strategies and their traits, life history strategy or habitat characteristics.
As a nal point, we hope our framework facilitates plant ecological research to bene t from conceptual advances in animal movement ecology. Promising future directions for plant ecological research include exploring how different costs of dispersal (e.g. due to investments in traits) modify the optimal search strategy (cf. (36)) and examining plant dispersal kernels for the existence of 'composite walks', which combine multiple movement types into one dispersal strategy (cf. (74)). The latter would be relevant in species with dispersal dimorphisms or species using multiple dispersal vectors. Another interesting direction would be to explore to what extent plant searches can be considered as 'informed searches'. There is a growing body of evidence that plants dispersed by animals, water, and wind utilize 'directed dispersal' strategies, in which they use environmental cues or select speci c vectors that result in disproportionate arrival of seeds at more suitable sites (75)(76)(77). In a recent study, 'informed dispersal' has been suggested as a strategy to escape competition and environmental stress (78). Thus, future research could explore these strategies in the light of 'informed searches' in plants, similar to how animals use cues to guide their search towards suitable sites (cf. (79)). Insights in how these factors shape the evolution of dispersal strategies, and progress in knowledge of dispersal mechanisms can mutually inspire each other, and thereby improve the understanding and quanti cation of dispersal in plants.

Conclusions
Our results clearly show that the full range of dispersal kernels, from extreme long-distance to short distance dispersal, can be adaptive, depending on the spatiotemporal habitat distribution across the landscape. Intermediate landscape dynamics and fragmentation would lead to the most complex and heterogeneous (multi-scale) kernels in terms of seed distributions. These multiple scales re ect an intensive-extensive search trade-off that determines the success rates of habitat encounter, kin avoidance, and colonization of new patches. Our analysis can serve as a framework that generates null hypotheses for dispersal strategies of plant species based on the spatiotemporal distribution of their habitat that can be used to analyse and compare plant dispersal data. Availability of data and materials -The model code will be made available on Github; the model data supporting the results will be archived in the public repository Dryad.
Competing interests -The authors declare that they have no competing interests. Move length distributions can be described using a Pareto distribution (p(x) ~ x-μ). Shape parameter μ determines whether dispersal strategies are characterized by a high probability of long-distance dispersal, by balancing long-and short-distance dispersal strategies, or dominated by short-distance dispersal only.   Heat maps showing the evolutionary stable dispersal strategies of all possible pairwise invasibility plots for landscapes with increasing patch turnover rates (from 0, top left, to 1, bottom right), as a function of both patch size and inter-patch distance.