Supplementary Information
Ecological patterns of diversity along gradients of productivity and disturbance
The observation of a bivariate peak of diversity at intermediate levels of both productivity and disturbance (Figs. 1 and 2) is strongly reminiscent of well-known patterns of diversity occurring on ecological time scales. These results are consistent with models of competition among niche specialists subject to global disturbances
1-3
.
Here we explore one of these models 2,3 in detail and contrast our results with those that extend the classic Intermediate Disturbance Hypothesis to incorporate productivity
4-6
.
The original model by Levene
7
considers the conditions for coexistence of two (or more) types distributed into two (or more) niches. Briefly, a mixed pool of i types is distributed at random into j niches where selection occurs, followed by reproduction and local population regulation, denoted by c j
, the proportion of offspring contributed to the common pool by each niche (note that ∑ c j
= 1). Let the frequency of type 1 be p and the proportion of offspring contributed from niche 1 be c. With two types and two niches, the frequency of type 1 in the next generation, p’
, is p '
 c

 pw
11
 pw

1
11
 p
 w
21




1
 c


 pw
12
 pw

1
12
 p
 w
22

 (1)
, where w ij is the absolute fitness of the i th
type in the j th
niche. The combination of differences in absolute fitness coupled with local population regulation generates competition among types within niches for access to dispersal sites. So long as different types are fittest in different niches (that is, w
11
> w
21
and w
22
> w
12
, or the reverse) and

population regulation occurs within each niche rather than at the level of the total population, diversity can be stably maintained through negative frequency dependent selection
2, 3, 7 , 8
.
This model has previously been extended to incorporate productivity
2 and disturbance
3
separately. To incorporate the joint effect of both, assume the number of individuals of each type grows logistically after disturbance at time T towards a carrying capacity set by the resource supply rate, R . Let the carrying capacity of type i in niche j ,
K ij
, be a Michaelis-Menton or Monod function,
K ij

 max
Km ij
 ij 

R
R


R
R min min ij ij 

(2) where
 max ij
is the maximum numbers of individuals produced at high resource supply rates, Km ij
is the resource supply rate at which the number if individuals is halfmaximum and expresses the efficiency with which a type converts resources into offspring, and R min ij
is the minimal resource supply needed to support a viable population. The absolute fitness of the i th type in the j th
environment under variable disturbance and productivity is, then, w ij

1

K ij
 ij e
 r ij
( T
 t ij
)
(3)

where
 ij
is a constant defined as
K
 ij
N ij
N ij
 
 
with N ij
being the number of individuals at time T = 0 following a disturbance event, r ij
is the intrinsic rate of increase, and t ij
is the acclimation period required before growth can begin. These fitness values are then used to calculate diversity along gradients of productivity and disturbance by substituting the fitness values into equation 1 and solving for the equilibrium frequency of type 1
13
. A similar logic can be used to extend the familiar Lotka-Volterra models of competition under resource competition
9
.
Supplementary figures 1a and 1b illustrate two plausible scenarios leading to the maintenance of diversity at intermediate rates of both productivity and disturbance for two types occupying two niches. Note that parameter values are chosen to reflect the underlying biology of the relationship between SM and WS. WS requires more resources and time before a coherent mat is formed
10
, which is reflected in our model by type 2 having a higher R min and longer acclimation period ( t ) than type 1; changing the absolute value of the parameters changes the absolute fitness of the types concerned but not the conditions for the maintenance of diversity (which are given by equation [1] 2 ). In supplementary figure 1a, the carrying capacity curves of the two fittest genotypes cross whereas in supplementary figure 1b, they come very close but do not cross (data not shown). Note that in both cases there is a peak of diversity as one moves along the diagonal from the lower left (low resource supply rate and frequent disturbances) to the upper right (high resource supply rates and infrequent disturbances). In the former, however, diversity forms a ridge extending indefinitely along both the productivity and disturbance axes and one type replaces the other along the diagonal from the lower left to the upper right. In supplementary figure 1b, by contrast, a unimodal peak of diversity

occurs at intermediate rates of both resource supply and disturbance and the same type predominates at the extremes along the lower left-to-upper right diagonal. Note that qualitatively similar results are obtained if the intrinsic rate of growth, rather than the carrying capacity, is allowed to vary in relation to productivity.
Supplementary figure 1c depicts the pattern of diversity that results when the fitness of type 1 in niche 2 is allowed to depend on the frequency of the fitter type (type
2) in that niche. In the simpler version of the model described above, the fitness of type 1 in niche 2 is simply 90% that of the absolute fitness of type 2 in niche 2. If we allow type
1 to obtain a small fitness advantage in niche 2 due to the growth of type 2 in niche 2, then its fitness becomes, w
12

( 0 .
9
 b ) w
22
[4] where b is fractional advantage (or disadvantage) gained by type 1 in niche 2 for every extra type 2 individual in the same niche. Supplementary figure 1c shows the effect on diversity of incorporating a value of b = 0.004 into the version of the model depicted in supplementary figure 1b. Note that the qualitative pattern of diversity does not change, although the range of productivities and disturbances within which diversity is maintained is substantially narrower than before.
Two alternative models describing the joint effects of productivity and disturbance on diversity have been proposed
4, 5
, both extensions of the classic
Intermediate Disturbance Hypothesis
11
. These models also predict a peak of diversity at intermediate productivity and disturbance, but the mechanism involves a trade-off

between colonization rate and competitive ability in a metapopulation subject to local disturbances. Both our model and the Lotka-Volterra resource competition models, by contrast, involve global disturbances to the entire community and do not require a colonization-competition trade-off. Our model, moreover, does not predict an interaction between productivity and disturbance in the location of the peak of diversity, as has been suggested previously
5,6
. It seems plausible that global disturbances may be typical of the harvesting of marine resources by dredging and trawling; indeed, two studies have shown that diversity in high biomass areas, which are likely to be heavily fished and so frequently disturbed, are often lower than in un-fished, lower biomass areas
12, 13
. Other communities that may be similarly affected are terrestrial plant communities disturbed periodically by grazing or fire
14
.
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