Test of the model with the plankton community
To apply eq. (4) to a natural community, some modifications are necessary. Here, we consider a plankton community composed of algae and zooplankton. Since carbon content of food relative to nutrient contents such as nitrogen and phosphorus, e.g. carbon to phosphorus ratio, is one of the most important food properties affecting growth efficiency in herbivore plankton ((Frost et al. 2006; Urabe et al. 2018), k can be expressed as:
k = q 1 ×α nutε1 , (5)
where α nut is carbon content relative to nutrient content of primary producers and q 1 is the conversion factor for adjusting to biomass unit. In this study, we applied a power function with coefficient of ε1 as a first order approximation since effects of this factor on H/Pbiomass ratio may not be linearly related to plant nutrient content. For example, if ε1 is much smaller than zero, it means that negative effects of carbon to phosphorus ratio of algal food on herbivore’s k are more remarkable when carbon to phosphorus ratio levels are high comparing to the case when carbon to phosphorus ratios are low. However, if this factor does not affect the H/P biomass ratio, ε1= 0 and k is constant.
Since herbivore plankton cannot efficiently graze on larger phytoplankton due to a gape limitation (Lampert and Sommer 2007), the feeding efficiency of herbivores or defense efficiency of producers’ resistance traits, β , would be related to the fraction of edible algae in terms of size as follows:
β = q 2 ×α ediε2, (6)
where α edi is a trait of primary producers determining edibility, q 2 is a factor for converting the traits to edible efficiency, and ε2 is how effective the trait is in defending against grazing. We expect ε2 = 0 if this factor does not matter in regulating theH/P biomass ratio but ε2 > 0 if it plays a role. Similarly, g can be described as
g = q 3 × µ ε3, (7)
where μ is the specific growth rate of producers,q3 is a conversion factor, and ε3is the effects of µ on growth rate. Again, we expect that ε3 ≠ 0 if g plays a role in determining theH/P ratio. Finally, assuming Holling type I functional response of carnivores, mortality rate of herbivores, m , is expressed as:
m = q 4 × θ ε4, (8)
where θ is abundance of carnivores, q 4 is specific predation rate, and ε4 is the effect size of carnivore abundance on m .
By inserting eq. (5) - (8) to eq. (4), effects of factors on theH/P biomass ratio is formulated as:
log(H/P ) = ε1log(α nut) + ε2log(α edi) + ε3log(µ ) − ε4log(θ ) + γ, (9)
where γ is log(q 1) + log(q 2) + log(q 3) − log(q 4). If differences in the H/P ratio among communities are regulated by growth rate (μ ), edibility (α edi), and nutrient contents (α nut) of producers as well as predation by carnivores (θ ), we expected non-zero values for ε1 to ε4. Thus, eq. (9) can be used to test the hypotheses if all of these factors simultaneously affect theH /P ratio and to examine the relative importance among these factors in determining the ratio in given communities by directly fitting it to data in natural communities. These were substantiated using data from natural plankton communities in experimental ponds where primary production rate was manipulated with different abundance of carnivore fish.