Schoolmaster and Snyder

Schoolmaster and Snyder

Digital Appendix

Here we derive expressions for the resident and invader density distributions and use these to derive expressions for, and. Our strategy for calculating these quantities is to take the Fourier transform of the expression. This turns convolutions into products and allows us to apply the Wiener-Khinchin theorem, which states that if f(x) and g(x) are functions with mean 0, then the Fourier transform of Cov(f, g)(x) is , where and the asterisk denotes the complex conjugate. We then take the inverse Fourier transform, and evaluate it at.

Population dynamics

Define the relative density as the local population density normalized by its spatial average: We can convert the dynamics of,

, 1)

into dynamics of by dividing both sides by and replacing with on the right hand side:

. 2)

Let represent small deviations of the population from its spatiotemporal average:

, 3)

and let and represent deviations from the average germination/establishment and growth rate:

4)

5)

The perturbations,,, and are , where by , we mean that can be made less than or equal to some positive constant K for small enough. Assuming that there is no globally synchronized component to the variation, so that is constant, the spatial averages of, and are equal to their spatiotemporal averages (e.g.) to, so that

6)

Using eqs. A3, A4, and A5 to substitute for, , and in eq. A2, we get

7)

As noted in Chesson (2000b), . The covariance is and so to, we can replace by yielding,

8)

noting that . To proceed, and need to be rewritten in terms of population density and germination. Returning to the definitions of, we find

9)

10)

The are found by expressingin terms of and and keeping only the terms:

11)

12)

At this point we will focus on the dynamics of the resident. Derivation of invader dynamics will follow. Taking the spatiotemporal average of the equation for the dynamics of, eq. A1, we find that to, , so that

13)

. 14)

Substituting back into eq. A8, we get

15)

We take the spatial Fourier transform, where the spatial transform and its inverse are given by

16)

the temporal transform is given by

17)

and the spatiotemporal transform by

18)

(In the main body of the paper, we express spatial and temporal frequencies in terms of spatial and temporal periods, and.) The Fourier transform of a convolution is the product of the Fourier transforms of the convolved functions, and so

. 19)

Note that through its dependence on , depends on germination rates in year t-1 and earlier. Assuming that ”forgets” its initial spatial conditions, , we can write

20)

where

21)

. 22)

Eq. A20 can be expressed as a discrete convolution in time:

23)

where

24)

Discrete Fourier transforms turn discrete convolutions into products just as continuous Fourier transforms do with continuous convolution. Our convention for the discrete Fourier transform and its inverse is, , where.

Taking the discrete temporal Fourier transform, we reach the simple form

25)

where

26)

Switching to polar notation, we rewrite as, where

27)

28)

and where we extend the range of tan-1 to by declaring to be in the first quadrant if both the numerator and the denominator of tan-1’s argument are positive, the second quadrant if the numerator is positive and the denominator is negative, the third quadrant if both the numerator and the denominator are negative, and the fourth quadrant if the numerator is negative and the denominator is positive. Thus,

29)

Invader Dynamics

Derivation of invader dynamics follows the same steps as the resident. From eqs. A9-A10 we get

30)

. 31)

Substituting into eq. A8 we get

. 32)

Taking the spatial Fourier transform gives

. 33)

Rewriting eq. A33 as a sum

34)

where

35)

36)

. 37)

Writing eq. A34 as a convolution in time gives

38)

where

39)

. 40)

Taking the temporal Fourier transform gives the form

41)

where, analogous to eq. A26,

. 42)

Writing for , noting that and rewriting in polar notation gives the final expression of invader dynamics:

43)

where

44)

45)

and where we extend the range of tan-1 to as described for the resident dynamics.

Deviation of expression for

We calculate the long-run regional growth rate as the geometric average of the regional growth rate:, where

. 46)

We take the natural logarithm of both sides of eq. 46 to get

. 47)

Rewriting gives and Taylor expanding gives

. 48)

Thus, to

49)

To proceed we must find the spatial mean of . Recall,

. 50)

Rewriting in terms of and gives

51)

which can be approximated to as

. 52)

Taking the spatial mean gives

53)

where notation indicating dependence on x and t has been suppressed for brevity.

The covariance of local population growth and population density in eq. A48 is can be written in terms of small perturbations from spatial averages:

54)

which can be rewritten as

. 55)

From eq. A4,

56)

Taking the Fourier transform gives

57)

where is the Dirac delta function, for which. We now have the ingredients to calculate, and. In each case, we proceed by taking the Fourier transform of the function, substituting the previously derived expressions for , andinto it and calculating the inverse Fourier transform. As an example, we present the calculation for.

Calculation of

We start by taking advantage of the fact that covariance of two variables equals the mean of their product minus the product of their means. Because andhave spatial means of zero,

58)

A multidimensional corollary of the Weiner-Khinchin theorem states that if equals the inverse Fourier transform of evaluated at , t=0. Thus,

59)

Using eq. A29 to substitute for ,

60)

We use eq. A57 to substitute for. Noting that is even in space and time, while is even in space but odd in time, we take the sums, resulting in,

. 61)

. 62)

The calculation of and the terms in proceed in a similar fashion.

1