Mixing of propagules from discrete sources at long distance: comparing a dispersal tail to an exponential
 Etienne K Klein^{1}Email author,
 Claire Lavigne^{2} and
 PierreHenri Gouyon^{3}
DOI: 10.1186/1472678563
© Klein et al; licensee BioMed Central Ltd. 2006
Received: 23 November 2005
Accepted: 20 February 2006
Published: 20 February 2006
Abstract
Background
Rare long distance dispersal events impact the demography and the genetic structure of populations. When dispersal is modelled via a dispersal kernel, one possible characterisation of longdistance dispersal is given by the shape of the tail of the kernel, i.e. its type of decay. This characteristic is known to directly act on the speed and pattern of colonization, and on the spatial structure of genetic diversity during colonization. In particular, colonization waves behave differently depending on whether the kernel decreases faster or slower than an exponential (i.e. is thintailed vs. fattailed). To interpret and extend published results on the impact of longdistance dispersal on the genetic structure of populations, we examine a classification of dispersal kernels based on the shape of their tails and formally demonstrate qualitative differences among them that can influence the predicted diversity of a propagule pool sampled far from two distinct sources.
Results
We show that a fattailed kernel leads asymptotically to a diverse propagule pool containing a balanced mixing of the propagules from the two sources, whereas a thintailed kernel results in all propagules originating from the closest source. We further show that these results hold for biologically relevant distances under certain circumstances, and in particular if the number of propagules is large enough, as would be the case for pollen or seeds.
Conclusion
To understand the impact of longdistance dispersal on the structure and dynamics of a metapopulation, it might be less important to precisely estimate an average dispersal distance than to determine if the tail of the dispersal kernel is fatter or thinner than that of an exponential function. Depending solely on this characteristic, a metapopulation will behave similarly to an island model with a diverse immigrant pool or to a steppingstone model with migrants from closest populations. Our results further help to understand why thintailed dispersal kernels lead to a colonization wave of constant speed, whereas fattailed dispersal kernels lead to a wave of increasing speed. Our results also suggest that the diversity of the pollen cloud of a mother plant should increase with increasing isolation for fattailed kernels, whereas it should decrease for thintailed kernels.
Background
In plant species, the patterns of gene flow by way of seed and pollen dispersal determines the demographic behaviour of populations, the spatial distribution of neutral and selected genetic diversities, and their evolution. Longdistance dispersal (LDD) is an important characteristic of dispersal events affecting population ecology, species distribution, evolution, and conservation [1–4]. It can be quantified by the proportion of the dispersers present farther than a specified threshold or by a distance beyond which only a small fraction of dispersers are found, e.g. 1% [4]. An alternative formulation of LDD concerns the shape of the tail of the dispersal kernel. This characteristic is known to play a major role in the speed of colonization [5–7], spatial pattern during colonization [8] and genetic structure after spatial expansion [9]. Because of its major role, much effort is put in characterizing this shape of the dispersal tail, both through a better understanding of dispersal mechanisms (mechanistic models, see e.g. [3]) and through better empirical descriptions of observed patterns (empirical models, see e.g. [10, 11]). Dispersal kernels are themselves subject to evolution and LDD should generally be selected for under simple assumptions [12, 13].
There exists a large 'set of possibilities' when considering dispersal kernels. When addressing longdistance dispersal, it is common to distinguish leptokurtic (kurtosis of the distribution higher than that of a Gaussian with the same variance) versus platykurtic kernels (e.g. [9]). However, as we will show below, kurtosis is a property of the entire distribution and is an insufficient characterisation of the tail for the purpose of predicting longdistance dispersal. When dealing only with the shape of the tail, a first distinction made is whether the functions are exponentially bounded or not. Functions not exponentially bounded [7] are also named fattailed kernels [6]. They can be contrasted with thintailed kernels that decrease faster that an exponential function, with an intermediate behaviour for the exponentiallike kernels (those that have the same behaviour as the exponential function in ± ∞). Another way of classifying the tails of dispersal kernels is to distinguish whether they are regularly varying or rapidly varying [14]. Roughly speaking (Additional file 1 for a precise definition), powerlaw decreases, also named algebraic tails, are regularly varying [11] whereas all types of exponential decreases (including exponentialpower functions for instance) are rapidly varying.
The way the tail of the dispersal kernel and the number and positions of individuals emitting propagules interact to determine the number of propagules arriving at a long distance is well formalized and documented because it plays a major role in determining the colonization speed (e.g. [5]). The effect of this interaction on the genetic diversity of distant propagule pools and the consequences for the structure of genetic diversity has only been approached through simulations (e.g. [9, 15–17]). We investigate here the relative contributions of sources at different distances to the pool of propagules by asking an apparently simple question: at long distance from two isolated propagule sources, does one source dominate in the propagule pool or is there an approximately even mixture of propagules from both sources?
We show that, asymptotically, the answer depends critically on whether the dispersal kernel is fattailed or not. Fattailed kernels lead to a balanced mixing of the propagules from the two sources, and thus to a diverse propagule pool. Contrarily, thintailed kernels result in a propagule pool of low diversity, with nearly all propagules originating from the closest source. We further show that this asymptotic property is valid for biologically relevant distances and numbers of propagules.
Results and discussion
Asymptotic results
1dimension

0 for a thintailed kernel

a value π_{lim} strictly between 0 and 1/2 for an exponential dispersal kernel

1/2 for a fattailed kernel.
This means that at long distances the propagules from the two sources are well mixed with a ratio approximating 1:1 only if the dispersal kernel is fattailed. On the other hand, if the dispersal kernel has a thinner tail than the exponential the fast majority of propagules received originate from the closer source and almost all those originating from the source located farther are absent in the propagule shadow. The exponential kernel thus appears as a critical point where the composition of the propagule shadow changes qualitatively.
Six families of 1dimensional dispersal kernels used in this study, together with their characteristics. The mean distance travelled is obtained from $\delta ={\displaystyle \underset{\infty}{\overset{+\infty}{\int}}\leftx\right\gamma (x)dx}$. Expression Γ() stands for the Gamma function.
Kernel families  Expression  Parameters values  Weight of the tail  Mean distance travelled, δ 

Exponential  ${\gamma}_{1}(x)=\frac{1}{2\alpha}\mathrm{exp}\left(\left\frac{x}{\alpha}\right\right)$  α>0  Rapidly varying Exponential  α 
Gaussian  ${\gamma}_{2}(x)=\frac{1}{\sqrt{\pi}\alpha}\mathrm{exp}\left(\frac{{x}^{2}}{{\alpha}^{2}}\right)$  α>0  Rapidly varying Thintailed  $\frac{\alpha}{\sqrt{\pi}}$ 
Powerlaw  ${\gamma}_{3}(x)=\frac{a1}{2\alpha}{\left(1+\left\frac{x}{\alpha}\right\right)}^{a}$  α>0 a>1  Regularly varying Fattailed  $\frac{\alpha}{a2}$ 
Exponential power  ${\gamma}_{4}(x)=\frac{c}{2\alpha \Gamma (1/c)}\mathrm{exp}\left({\left\frac{x}{\alpha}\right}^{c}\right)$  α,c>0  Rapidly varying Thintailed for c>1 Fattailed for c<1 Exponential for c = 1  $\alpha \frac{\Gamma (2/c)}{\Gamma (1/c)}$ 
2Dt  ${\gamma}_{5}(x)=\frac{\Gamma (a)}{\alpha \sqrt{\pi}\Gamma \left(a\frac{1}{2}\right)}{\left(1+\frac{{x}^{2}}{{\alpha}^{2}}\right)}^{a}$  α>0 a>1/2  Regularly varying Fattailed  $\frac{\alpha}{\sqrt{\pi}(a1)}\frac{\Gamma (a)}{\Gamma \left(a\frac{1}{2}\right)}$ 
Mixture of two Gaussians  ${\gamma}_{6}(x)=p\left(\frac{1}{\sqrt{\pi}{\alpha}_{1}}{e}^{{x}^{2}/{\alpha}_{1}^{2}}\right)+\left(1p\right)\left(\frac{1}{\sqrt{\pi}{\alpha}_{2}}{e}^{{x}^{2}/{\alpha}_{2}^{2}}\right)$  α_{1}≠α_{2} p>0  Rapidly varying Thintailed (but leptokurtic)  $p\frac{{\alpha}_{1}}{\sqrt{\pi}}+\left(1p\right)\frac{{\alpha}_{2}}{\sqrt{\pi}}$ 
2dimensions

tends either towards 0 or 1 depending on the direction followed if the kernel is thintailed

tends towards values strictly between 0 and 1, depending on the direction followed, if the kernel is exponential

tends towards 1/2 independently of the direction followed if the kernel is fattailed.
Here again, at longdistances from the two sources, the closest source is the only one that contributes significantly to the propagule shadow for a thintailed kernel, whereas both sources evenly contribute for a fattailed kernel. The exponential kernel is a critical point between these two behaviours. However, notice that on the line of equidistance between A and B the proportion of propagules from A is logically 1/2, whatever the kernel, and whatever the distance to the sources.
Finite distances
Effect of the range of distances
The asymptotic results presented above are true whatever the distance between the two sources and the mean dispersal distance but only at an infinite distance from the sources. The distances at which this asymptotic behaviour is a good approximation depend on the distance between the sources relative to the mean dispersal distance. The limit value for the exponential kernel, which is between 0 and 1/2, also depends on these two measures.
When the distance between the two sources is small with regard to the mean distance travelled (that is when (x_{ B }x_{ A })/δ is small), the three models tend to behave similarly (Figure 4, top, black lines): the exponential model remains equal to a value close to 1/2, the fattailed models increase to 1/2 but starting from a high value at point x_{ B }(thus the increase is weak) and the thintailed models decrease to 0 very slowly.
Range of distances actually travelled in natural conditions
Among a given amount of propagules emitted (R), the one that travels the longest distance is known as the furthest forward propagule (FFP). It is of particular importance during a colonisation since its position will define the extent of the population at the next generation. The position of the FFP also indicates the range of distances to consider when evaluating the biological relevance of the asymptotic results given above. As already known [5], the position of this FFP largely depends on the amount of propagules emitted (R). This is particularly true for fattailed dispersal kernels (Figure 4, bottom).
For the exponential kernel, the proportion of propagules from A in the propagule shadow does not depend on position. Thus, whatever the number of propagules emitted, the level of mixing of propagules only depends on the distances between the sources (relative to the mean distance travelled).
For a thintailed kernel the position of the FFP from B shows a weaker dependence to the number of propagules emitted. For the Gaussian we have used the asymptotic behaviour of the propagule shadow (absence of propagules from A) is (i) always reached for far sources, (ii) only reached for large number of propagules emitted (above 10^{3}) if the distance between the sources equals the mean dispersal distance and (iii) never reached for nearby sources (Figure 4).
For the powerlaw kernel we have used, the position of the FFP from B shows huge variations depending on R. The asymptotic behaviour of the propagule shadow (one half of the propagules coming from A) is (i) almost always true for close sources, (ii) reached for large number of propagules emitted (above 10^{3}) if the distance between the sources equals the mean dispersal distance, and (iii) still observable for far sources if the number of propagules is very large (10^{6}).
Between the sources
Conclusion
Our results contradict a very intuitive idea. One would expect that the propagule shadow received at long distance from two close and distinct sources should be similar to that received from one source emitting half of each propagule type. However, we show that this is only the case if propagules are dispersed following a fattailed dispersal kernel: at long distances, a mixing of the propagules from the two sources happens with a ratio 1:1 of propagules from each source. On the other hand, in species with thintailed dispersal kernels all the propagules collected far from the sources originate from the closest source, independently of the distance between the sources. Propagules from the farther source are absent in the propagule shadow.
This property has several implications in terms of the dynamics of the genetic diversity over a landscape with several distinct propagule sources. The genetic composition of the propagule pool will qualitatively depend on the type of dispersal kernel. In general, pools are expected to be diverse and little differentiated with fattailed dispersal kernels, where all the sources contribute significantly to the pool. The opposite is expected with thintailed kernels, where only the closest source has a significant contribution to each pool. In a metapopulation context, our result suggests that a metapopulation will follow an island model or gene pool model [20, 21] with fattailed kernels, whereas thintailed dispersal kernels lead to steppingstone or onedonor models [20].
Our results further help us to understand why thintailed dispersal kernels lead to a colonization wave of constant speed, whereas fattailed dispersal kernels lead to a wave of increasing speed [5, 22, 23]. With thintailed dispersal kernels the only individuals/sources that contribute to the advance of the front of the wave are those already located on the front; and the number of these individuals/sources remains constant. On the contrary, with fattailed dispersal kernels, all the individuals/sources in the population contribute to the colonization events. Thus, as the population grows so does the number of long distance dispersal events as well as the longest distance travelled. The speed of advance of the wave thus increases. Investigating the position of the parent of the furthest forward propagule could help to confirm this idea. The consequences of our analysis are less clear concerning the genetic structure during colonization. Fattailed kernels lead to longdistance dispersers founding very isolated satellite populations, with large founder effects (e.g. [24]). Our results indicate that all the individuals of the population are putative parents of these longdistance dispersers. Thus concerning the global genetic structure we expect a high spatial differentiation, but with no general isolation by distance pattern, a same genotype being present in very distant positions. For thintailed kernels, no longdistance founding events are expected, and only the few individuals in the front of the colonization contribute to the next settlers (which is a typical property of the diffusion models, obtained from Gaussian kernels [8]). This could imply a progressive loss of diversity during the advance of the front, leading to weaker differentiation [15] and an isolation by distance pattern. This thintailed scenario is particularly well illustrated by [25] who show that a new variant appearing on the front of a colonization either stays where it appeared or moves with the colonization front but in both cases variant individuals remain clustered. Remark finally that the significance of the foundation events in reducing diversity, particularly for fattailed kernels, could be decreased if the further propagules are more clumped than modelled usually. This is indeed a pattern observed in some experiments (e.g. [26]). It though depends on whether the clumped propagules originate from the same source or not.
Obviously, no dispersal kernel will extend forever stricto sensu but this does not discredit asymptotic results, as illustrated in many scientific domains. Yet, the validity of the asymptotic results should be checked for ecologically relevant dispersal distances, which we investigated here by considering the position of the further forward propagule. We show here that, in practice, the difference between thin and fattailed kernels depends on the interactions among (i) the distance between sources, (ii) the mean dispersal distance or any scale parameter of the dispersal kernel (actually, the ratio of distance between sources to the mean dispersal distance gathers i and ii) and (iii) the number of propagules emitted by the sources. For instance, when sources are very close, the absence of mixture for thintailed kernels is only effective at very long distances, whereas it is effective at realistic distances when sources are distant enough. Since the position of the furthest forward propagule increases with the number of propagules dispersed, these realistic distances are more likely to be reached for larger numbers of propagules.
Pollen dispersal is a typical case where large numbers of propagules are dispersed. Our results show that the outcrossing part of the pollen pool of isolated plants will be diverse only if the dispersal kernel is fattailed. At the opposite, the contribution of the closest pollen donors will be largely dominant if the kernel is thintailed. It thus seems that a correct estimation of the shape of dispersal kernels is even more crucial for pollen than for seed dispersal if we wish to predict the impact of population fragmentation or low density on the maintenance and spatial structure of genetic diversity. Empirical studies that compare thin and fattailed kernels estimated with a variety of methods tend to find that pollen dispersal kernels are fattailed in tree species [27–29] as well as in grasses and forbs ([30] using paternity analysis; [31] for a review on crops, [32, 33] using phenotypic markers). Interestingly, as expected from our results, if pollen dispersal kernels are generally fattailed, reasonable low densities or levels of isolation tend to promote diversity of pollen clouds, as estimated by an efficient number of fathers in tree species ([34] for a review) or correlated paternity within sibships [35]. Diverse pollen clouds were observed, for example, over isolated or low density individuals of Prunus mahaleb L. [36], Pinus sylvestris [35] or Sorbus torminalis (OddouMuratorio et al. submitted). Similar results were obtained on malesterile plants of oilseedrape [17]. Among these species, the last three had been shown to disperse pollen following fattailed dispersal kernels [17, 27, 33, 37]. Empirical results showing a diversity of pollen pools over reasonably isolated individuals thus tend to indicate that pollen dispersal kernels are generally fattailed. A thorough investigation of the impact of the number and spatial arrangement of sources on the results presented in this study would be necessary before drawing definite conclusions.
Methods
1dimensional asymptotic results
Let us consider two sources of propagules A and B (seeds, pollen or spores), located in a 1dimensional space at positions x_{ A }and x_{ B }>x_{ A }. Those two sources are assumed to emit the same quantity of propagules and to disperse them around themselves following the same kernel γ(x). The proportion of propagules from A received at point x is independent of the total number of propagules emitted by each source. This can be written
${\pi}_{A}(x)=\frac{\gamma (x{x}_{A})}{\gamma (x{x}_{A})+\gamma (x{x}_{B})}=\frac{1}{1+{\rho}_{B}(x)},$
where ${\rho}_{B}(x)=\frac{\gamma (x{x}_{B})}{\gamma (x{x}_{A})}$ is the ratio of the number of propagules from B over the number of propagules from A. If ρ_{ B }tends to 0, π_{ A }tends to 1, if ρ_{ B }tends to 1, π_{ A }tends to 1/2 and if ρ_{ B }tends to +∞, π_{ A }tends to 0. We shall now calculate this ratio for three different types of dispersal kernels (Table 1) chosen because (i) they are the most commonly used in the literature, (ii) they are the simplest parametric families for dispersal kernels and (iii) they can be associated through sums and products to encompass almost all models of dispersal kernels (detailed below).
Exponential kernels
For the exponential kernels, the ratio of propagules from source B over propagules from source A has the following form between x_{ B }and +∞:
${\rho}_{B}(x)={e}^{({x}_{B}{x}_{A})/\alpha}>1.$
This means that ρ_{ B }and π_{ A }do not depend on x for x>x_{ B }and that π_{ A }takes a constant value between 0 and 0.5.
Powerlaw kernels
For powerlaw kernels, the ratio of propagules from source B over propagules from source A between x_{ B }and +∞ equals:
${\rho}_{B}(x)={\left(\frac{1+\frac{x{x}_{B}}{\alpha}}{1+\frac{x{x}_{A}}{\alpha}}\right)}^{\text{a}}>1.$
When x tends to +∞, ρ_{ B }tends to 1, and thus π_{ A }tends to 1/2. A second property is that ρ_{ B }decreases for x>x_{ B }(this can be checked by writing it in the classical form ${\left(\frac{x\xi}{x\zeta}\right)}^{a}$ with ξ<ζ<0), and consequently, π_{ A }increases for x>x_{ B }.
Exponential power kernels
In the case of a kernel from the exponential power family, (including the Gaussian kernels), the ratio of propagules from source B over propagules from source A when x>x_{ B }is given by
${\rho}_{B}(x)=\mathrm{exp}\left({\left(\frac{x{x}_{A}}{\alpha}\right)}^{c}{\left(\frac{x{x}_{B}}{\alpha}\right)}^{c}\right),$
and this function can be written as
${\rho}_{B}(x)=\mathrm{exp}\left(\frac{{x}^{c}}{{\alpha}^{c}}\left(c\frac{{x}_{B}{x}_{A}}{x}+\text{o}\left({x}^{1}\right)\right)\right),$
where o(x^{1}) is any function tending to 0 strictly faster than x^{1} in +∞. So, in +∞,
${\rho}_{B}(x)\sim \mathrm{exp}\left(\left({x}_{B}{x}_{A}\right)c\frac{{x}^{c1}}{{\alpha}^{c}}\right).$
Thus, ρ_{ B }tends to +∞ if c>1 and to 1 if c<1. Consequently, π_{ A }tends to 0 if c>1 and tends to 1/2 if c<1. The particular case c = 1, corresponding to the exponential kernels leads to the same result as above, i.e. a limit strictly between 0 and 1/2.
Moreover, for x>x_{ B }, the derivative of ρ_{ B }is
${{\rho}^{\prime}}_{B}(x)=\frac{c}{\alpha}\left({\left(\frac{x{x}_{A}}{\alpha}\right)}^{c1}{\left(\frac{x{x}_{B}}{\alpha}\right)}^{c1}\right){\rho}_{B}(x),$
which is positive when c>1 and negative when c<1. Consequently, for c>1, ρ_{ B }increases towards +∞ when x increases if x>x_{ B }and π_{ A }thus decreases towards 0. When c<1, ρ_{ B }decreases towards 1 when x increases if x>x_{ B }and π_{ A }thus increases towards 1/2.
Rules for compounds of simple functions
If a kernel γ(x) can be written as the weighted sum of two simpler functions, γ(x) = w_{1}g_{1}(x) + w_{2}g_{2} (x), where g_{2} has a heavier tail than g_{1} (i.e. $\frac{{g}_{2}(x)}{{g}_{1}(x)}\to \infty $ when x tends to ∞) then the ratio of propagules from source B tends to the same value as that obtained for the function g_{2} when x tends to ∞.
This can be seen from the ratio ${\rho}_{B}(x)=\frac{\gamma (x{x}_{B})}{\gamma (x{x}_{A})}=\frac{{w}_{1}{g}_{1}(x{x}_{B})+{w}_{2}{g}_{2}(x{x}_{B})}{{w}_{1}{g}_{1}(x{x}_{A})+{w}_{2}{g}_{2}(x{x}_{A})}$, in which the first terms of both the numerator and the denominator will be negligible in regards to the second terms when x tends to ∞.
If a kernel γ(x) can be written as the product of two simpler functions, γ(x) = g_{1}(x) × g_{2}(x), where g_{2} has a heavier tail than g_{1} (i.e. $\frac{{g}_{2}(x)}{{g}_{1}(x)}\to \infty $ when x tends to ∞), then the ratio of propagules from source B behaves, when x tends to ∞, as that obtained for the function g_{1}.
Indeed the ratio ρ_{ B }can be written as a product of two ratios:
${\rho}_{B}(x)=\frac{\gamma (x{x}_{B})}{\gamma (x{x}_{A})}=\frac{{g}_{1}(x{x}_{B})}{{g}_{1}(x{x}_{A})}\times \frac{{g}_{2}(x{x}_{B})}{{g}_{2}(x{x}_{A})}$. If the first ratio tends to ∞, then ρ_{ B }tends to +∞, whatever the limit of the second ratio (because this second ratio is larger than 1). If the first ratio tends to 1, so does the second ratio since g_{2} has a heavier tail than g_{1} and thus ρ_{ B }tends to 1. If the first ratio tends to a limit strictly over 1, then the second ratio tends either to 1 or to a limit strictly over 1 and in both cases, ρ_{ B }tends to a limit strictly over 1.
2dimensional asymptotic results
The analyses can readily be extended to the twodimensional case by considering two point sources of propagules A and B, located at positions (x_{ A }, 0) and (x_{ B }, 0) with x_{ A }= x_{ B }. Those two sources are assumed to emit the same number of propagules and to disperse them around themselves following the same 2dimensional kernel γ(x, y). We consider only isotropic kernels, that is kernels satisfying $\gamma (x,y)={\gamma}_{r}\left(\sqrt{{x}^{2}+{y}^{2}}\right)$. Each point (x, y) where we calculate the proportion of propagules coming from source A can be expressed in polar coordinates by a distance r and an angle θ. Its distance to point B is then given by
${r}^{\prime}=\sqrt{r{(\mathrm{cos}\theta {x}_{B})}^{2}+{(r\mathrm{sin}\theta )}^{2}}=\sqrt{{r}^{2}2r\mathrm{cos}\theta {x}_{B}+{x}_{B}^{2}},$
and its distance to point A is given similarly by
${r}^{\u2033}=\sqrt{{r}^{2}2r\mathrm{cos}\theta {x}_{A}+{x}_{A}^{2}}=\sqrt{{r}^{2}+2r\mathrm{cos}\theta {x}_{B}+{x}_{B}^{2}}.$
We are interested in letting r go to infinity while keeping θ constant. This means that we are considering a point moving away from the two sources in a given direction. The Taylor expansions of expressions r' and r" can then be obtained as:
${r}^{\prime}=r\left(1\frac{\mathrm{cos}\theta {x}_{B}}{r}+\text{o}\left({r}^{1}\right)\right)\text{and}{r}^{\u2033}=r\left(1+\frac{\mathrm{cos}\theta {x}_{B}}{r}+\text{o}\left({r}^{1}\right)\right),$
where o(r^{1}) stands for any function negligible compared to r^{1} when r tends to +∞.
The ratio of propagules from B over propagules from A at point (x, y) = (r, θ) can then be written as
${\rho}_{B}(r,\theta )=\frac{{\gamma}_{r}({r}^{\prime})}{{\gamma}_{r}({r}^{\u2033})}=\frac{{\gamma}_{r}\left(r\mathrm{cos}\theta {x}_{B}+\text{o}(1)\right)}{{\gamma}_{r}\left(r+\mathrm{cos}\theta {x}_{B}+\text{o}(1)\right)},$
where o(1) stands for any function tending to 0 when r tends to +∞.
By analogy with the 1dimensional equation, in the direction θ when r tends to +∞, the ratio ρ_{ B }(r, θ) behaves just as the ratio for a 1dimensional kernel γ_{ r }, and a distance between the two sources of 2cosθ x_{ B }. The following results are thus direct consequences of the 1dimensional results.
Exponential kernels
Four families of 2dimensional dispersal kernels used in this study, together with their characteristics. The mean distance travelled is obtained from $\delta ={\displaystyle \underset{0}{\overset{+\infty}{\int}}\gamma (r)2\pi rdr}$. Expression Γ() stands for the Gamma function.
Kernel families  Expression  Parameters values  Weight of the tail  Mean distance travelled, δ 

Exponential  ${\gamma}_{1}(x,y)=\frac{1}{2\pi {\alpha}^{2}}\mathrm{exp}\left(\frac{r}{\alpha}\right)$  α >0  Rapidly varying Exponential  2α 
Gaussian  ${\gamma}_{2}(x,y)=\frac{1}{\pi {\alpha}^{2}}\mathrm{exp}\left(\frac{{x}^{2}+{y}^{2}}{{\alpha}^{2}}\right)$  α >0  Rapidly varying Thintailed  $\frac{\alpha \sqrt{\pi}}{2}$ 
Powerlaw  ${\gamma}_{3}(x,y)=\frac{(a2)(a1)}{2\pi {\alpha}^{2}}{\left(1+\frac{r}{\alpha}\right)}^{a}$  α>0 a>2  Regularly varying Fattailed  $\frac{2\alpha}{a3}$ 
Exponential power  ${\gamma}_{4}(x,y)=\frac{c}{2\pi {\alpha}^{2}\Gamma (2/c)}\mathrm{exp}\left({\left(\frac{r}{\alpha}\right)}^{c}\right)$  α,c>0  Rapidly varying Thintailed for c>1 Fattailed for c<1 Exponential for c = 1  $\alpha \frac{\Gamma (3/c)}{\Gamma (2/c)}$ 
Exponential power kernels
For an exponential power dispersal kernel (Table 2), if c>1,ρ_{ B }(r, θ) tends to +∞ when π/2<θ<3π/2 and tends to ∞ when π/2<θ<3π/2. If c<1, ρ_{ B }(r, θ) tends to 1 whatever the value of θ. Finally, if c = 1, corresponding to the exponential kernel, we find again the previous result: ρ_{ B }(r, θ) tends to $\mathrm{exp}\left(2\frac{{x}_{B}}{\alpha}\mathrm{cos}\theta \right)$. Consequently, the proportion of propagules from A, π_{ A }(r, θ) tends to 0 for π/2<θ<3π/2 (direction of B) and to 1 for π/2<θ<3π/2 (direction of B) when c>1; it tends to 1/2 whatever the direction θ when c<1. For the exponential kernel (c= 1), π_{ A }(r, θ) tends to values strictly between 0 and 1, such as already mentioned above.
Powerlaw kernels
For the powerlaw kernels, ρ_{ B }(r, θ) tends to 1 for all a>2, and all θ. This means that the proportion π_{ A }(r, θ) tends to 1/2 in all directions for any powerlaw kernel.
Validity of the asymptotic results for natural scales
We computed distances actually travelled by some propagules to evaluate whether the asymptotic results analytically derived are effective at finite distances of biological interest. The range of distances of interest depends on the dispersal kernel, by virtue of the mean distance travelled and the dispersal tail; in addition the range of distances is also very sensitive to the number of propagules R emitted by each source [5].
We derived numerically for different kernels γ and number of propagules R, the distribution of the position of the furthest forward propagule (FFP) coming from B [5]. The probability density function (PDF) of this random variable is given by
FFP_{ B }(x) = Rγ(x  x_{ B })F(x  x_{ B })^{R1}
where F is the cumulative distribution function (CDF) associated with γ. This expression means (i) that the further forward propagule from B is one of the R propagules emitted by B, (ii) that it falls at a distance xx_{ B }from its source (B) and (iii) that the R1 other propagules from B fall at distances smaller than (xx_{ B }).
As a simplification, we have chosen to focus on the position of the FFP from B (and not on the position of the FFP from both sources A and B), because its distribution does not depend on the position of source A. This is a conservative choice because the extreme dispersal event of interest for biological questions (such as speed of colonization or contamination between fields...) is the FFP, which is either the FFP from B, or the FFP from A if it travelled further than the FFP from B [5].
We considered three wideranged values for R equal to 10; 1000 and 1000000. These orders of magnitude correspond for instance to the numbers of dispersed rodents offspring, tree seeds and crop pollen grains.
The computation of the proportions of propagules from the source A at distances close to that of the FFP from B allowed asserting if the asymptotic properties obtained when the distance is tending toward infinity are good approximations for what happens to the further forward individuals.
Declarations
Acknowledgements
We thank C. Gliddon, B. Albert, C. Devaux and R.E. Michod for helpful comments on the manuscript and S. Huet for the apparently naive question that initiated this work. This work was partly supported by the programs "OGM et environnement" (INRA/Ministère de la Recherche). We are also grateful to Thomas Hovestadt and two anonymous reviewers for their suggestions on the final version of the manuscript.
Authors’ Affiliations
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