Linking Dispersal and Immigration in Multidimensional Environments

Abstract

Many problems in ecology require the estimation of rates of dispersal of individuals or propagules across physical boundaries. Such problems arise in invasion ecology, forest dynamics, and the neutral theory of biodiversity. In a forest plot, for example, one might ask what proportion of the seed rain originates from outside the plot. A recent study presented analytical approximations that relate the rate of immigration across a boundary to plot geometry and to the parameters of a dispersal kernel in one- and two-dimensional environments. In this study, we provide a more rigorous derivation of these expressions and we derive a more general expression that applies in environments of arbitrary dimension. We discuss potential applications of the one-, two-, and three-dimensional results to ecological problems.

Appendices

Appendix A

Here, we present a derivation of an exact expression for $\hat{m}(r)$ for the special case where the region G is a disc in ℝ², so that the boundary X is a circle. Recall that $\hat{m}(r)$ is defined by Eq. (2) and represents the immigration rate for a fixed dispersal distance r.

Let R be the radius of the circle X and assume that r<R. First, consider a point P in G at a distance x≤r from the circle (we can ignore other points in G because no dispersal events can reach them). Consider an isosceles triangle defined by three points: P, and the two points on X that are at a distance r from P. Let 2θ _x be the angle at P. From the law of cosines, we have

$$\theta_x = \cos^{-1} \frac{R^2-r^2-(R-x)^2}{2(R-x)r} $$

An expression for $\hat{m}(r)$ can be obtained by integrating over all possible points P in x:

This integral can be evaluated using the following substitution:

$$v = \frac{R^2-r^2-(R-x)^2}{2(R-x)r} $$

which gives

Integrating this by parts and simplifying leads to

$$\hat{m} (r) = \frac{1}{2\pi} \biggl\{ \frac{r}{R^2} \sqrt{4R^2-r^2} - 2 \tan^{-1} \frac{r\sqrt{4R^2-r^2}}{r^2-2 R^2} \biggr\} $$

We can also write this as a Taylor series:

$$\hat{m} (r) = \frac{2}{\pi} \frac{r}{R} - \frac{1}{12\pi} \biggl(\frac{r}{R} \biggr)^3 - \frac{1}{320\pi} \biggl(\frac{r}{R} \biggr)^5 - O \biggl( \biggl(\frac{r}{R} \biggr)^7 \biggr) \sim\frac{2}{\pi} \frac{r}{R} $$

It is clear that this last result is consistent with Eq. (6) in the main text. The key point to note here is that the curvature of the circle introduces higher-order polynomial terms in r/R, but that these terms vanish asymptotically as r/Φ=r/(2πR)→0.

Appendix B

To evaluate the inner integral in Eq. (5) we require the general formula for the hypersurface area of an n-dimensional hypersphere of radius r:

$$A\bigl(S_n (r)\bigr)=\frac{2\pi^{n/2} r^{n-1}}{\Gamma (n/2)} $$

and the general formula for the hypersurface area of a hyperspherical cap of height h on this hypersphere:

where ₂ F ₁(a,b,c;z) is the hypergeometric function (Abramowitz and Stegun 1972). Evaluating (5) now just requires some technical integration:

The integral of the hypergeometric function can be evaluated using the identity:

$$\frac{d\,_2F_1(a,b,;c;z)}{dz} = \frac{ab}{c}\,_2F_1(a+1,b+1;c+1;z) $$

with the substitution z=(u/r)² giving

$$\int_0^r 2u\,_2F_1 \biggl(\frac{1}{2},\frac{3-n}{2};\frac{3}{2}; \biggl( \frac{u}{r} \biggr)^2 \biggr) \,du = \frac{2r^2}{n-1} \biggl[ \frac{\sqrt{\pi}\, \Gamma (\frac {n+1}{2})}{\Gamma (\frac{n}{2})} - 1 \biggr] $$

Substituting this back into the integral of A(C _n (r,r−u)) and simplifying gives

$$\int_0^r A\bigl(C_n (r,r-u)\bigr) \,du = \frac{\pi^{\frac{n}{2}-\frac{1}{2}} r^n}{\Gamma (\frac{n+1}{2})} $$

Then returning to Eq. (5), we have

$$\int_0^r \frac{A(C_n (r,r-u))}{A(S_n (r) ) } \,du = \frac{1}{A(S_n (r)) } \int_0^r A \bigl(C_n (r,r-u)\bigr) \,du = \frac{r}{2 \sqrt{\pi}} \frac{\Gamma (\frac{n}{2})}{\Gamma (\frac{n+1}{2})} $$