Phytoplankton size-diversity mediates an emergent trade-off in ecosystem functioning for rare versus frequent disturbances

Biodiversity is known to be an important determinant of ecosystem-level functions and processes. Although theories have been proposed to explain the generally positive relationship between, for example, biodiversity and productivity, it remains unclear which mechanisms underlie the observed variations in Biodiversity-Ecosystem Function (BEF) relationships. Using a continuous trait-distribution model for a phytoplankton community of gleaners competing with opportunists, and subjecting it to differing frequencies of disturbance, we find that species selection tends to enhance temporal species complementarity, which is maximised at high disturbance frequency and intermediate functional diversity. This leads to the emergence of a trade-off whereby increasing diversity tends to enhance short-term adaptive capacity under frequent disturbance while diminishing long-term productivity under infrequent disturbance. BEF relationships therefore depend on both disturbance frequency and the timescale of observation.


Total biomass and nutrient concentration
The community average growth rate of phytoplankton (over all size classes, l) can be approximated based on the assumed log-normal size distribution. Thus, the rate of change of the total biomass of the community, P T , is approximated based on a Taylor expansion about the mean size,l, assuming a Gaussian distribution [Merico et al., 2009]: in which G T = g max ZQ(P T ) is the total grazing rate in terms of the feeding probability, Q, from equation (S-13). Here we have exploited the fact that the total grazing rate depends only on the total biomass, not on its distribution nor on the value of the prey switching parameter α [Vallina et al., 2014]. The second partial derivative of the specific growth rate (µ) will be negative at the mean value of log size,l, assuming that the latter is near the optimal value of l, at which µ is maximal. That is, the community as a whole will grow more slowly than phytoplankton of precisely the mean (and optimal) size, because of the presence of other (sub-optimal) sizes.
The rate of change of the zooplankton biomass, Z, is: where µ Z is the specific growth rate of zooplankton (defined below), and m Z is the mortality rate coefficient for zooplankton. The specific growth rate of zooplankton is: where β Z is the assimilation efficiency of zooplankton, and Q(P T ) is the feeding probability from equation (S-13).
The rate of change of the nutrient concentration, N , is: where fraction Z of the un-assimilated grazing (first term) and fraction Ω of the zooplankton mortality (second term) are assumed to be remineralized instantaneously to N . The mass balance for detrital nitrogen, D, is: where k D is the specific remineralization rate of detritus.

Size-scaled Kill-the-Winner grazing
The generalized grazing expression [Vallina et al., 2014], for the rate of grazing (by zooplankton) on discrete prey class i, having biomass P i (mmol N m −3 ), is: where g max (d −1 ) is the maximum grazing rate, Z (mmol N m −3 ) is the biomass of (the implicit community of) zooplankton, ρ i is the fixed preference for prey of discrete class i, parameter α determines the prey switching behavior, and parameters k sat (mmol m −3 ) and β determine the shape of the overall (total) grazing response in terms of total prey biomass, The latter is the sum over all n prey classes of fixed prey preference times biomass: Prey switching is determined by the ratio: with α = 1 giving 'passive' switching, resulting in competitive exclusion for prey, and α > 1 giving active switching, resulting in kill-the-winner response [Vallina et al., 2014]. Dividing by P i gives the specific loss rate of prey class i to grazing: This can be re-written in terms of the feeding probability Q, which depends on P T , but not on P i : For a continuous size distribution of prey, defined by probability density P (l), the specific loss to grazing for size l can be written: where the total palatable prey is: with ρ(l) defined as some continuous function of l. Here we assume ρ(l) = 1 for all l, so that the specific grazing rate simplifies to: The size distribution of phytoplankton (prey) will be approximated as log-normal [Schartau et al., 2010, Wirtz, 2013 so that its probability density function, P (l), in terms of log-size, l, is Gaussian: wherel is the (biomass weighted) mean log cell size and σ l is the standard deviation of log cell size. Then, the normalizing integral in the denominator of equation (S-16) is: Substituting into equation (S-16) gives: Derivatives with respect to size Derivatives of growth rate The derivatives of the growth rate with respect to size (l) are needed to calculate the rates of change of the total biomass and the mean and variance of the size distribution. The first derivative of the specific growth rate, equation 5 (main text), with respect to l is: -20) and its second derivative is: Taking the derivative of each term of the above, in turn, gives the third derivative: -22) and combining the third and fifth terms gives: Again taking the derivative of each term, respectively, gives the fourth derivative of µ: which, after substituting equation (S-23) and collecting terms, simplifies to:

Derivatives of grazing rate
Here we take the derivatives with respect to l of the specific (to phytoplankton) grazing rate,

equation (S-19), for use in equation (S-3).
The first derivative of the specific grazing rate with respect to l is: The derivative of P (l) (α−1) , based on equation (S-17), is then: At the mean size,l, ∂g(l)/∂l = 0, which based on equation (S-1) means that the grazing response will not directly cause changes inl (although indirect effects are possible through changes in nutrient concentration).
Taking the derivative of equation (S-28) gives the second derivative: which can be expressed in terms of g(l) using equation (S-19): For the second derivative evaluated at the mean size, as in equations (S-2) and (S-3), only the first term remains, and it can be expressed in terms of the specific grazing rate from equation (S-19): Table S1.

Values of model parameters.
Size-scaled parameters for phytoplankton and size-independent parameters for the implicit zooplankton community [Vallina et al., 2014], which exhibits passive prey switching for α = 1 and active switching, i.e., 'Kill-the-Winner' response, for α > 1.
Parameter value units description   Figure S1: Size diversity index, h, averaged over 7 d following the first disturbance versus the a) KTW parameter α, and b) TD parameter ν. Vertical arrows specify frequencies of disturbance. Short-term Adaptive Capacity (AC) is quantified by avg. values over the same 7 d of: mean specific growth rate, µ P , for the phytoplankton community (c, d), nutrient concentration, N (e, f), and specific growth rate of zooplankton, µ Z (g, h), each plotted vs. h averaged over 7 d following the first disturbance.   Figure S2: Size diversity index, h, averaged over 90 d following the first disturbance versus the a) KTW parameter α, and b) TD parameter ν. Vertical arrows specify frequencies of disturbance. Long-term Productivity (LP) is quantified by avg. values over the same 90 d of: mean specific growth rate, µ P , for the phytoplankton community (c, d), nutrient concentration, N (e, f), and specific growth rate of zooplankton, µ Z (g, h), each plotted vs. h averaged over 90 d following the first disturbance.