multi-stressor extension of model from Mumby et al. 2007
We employ a simple multi-stressor extension to the well known Mumby et al.(Mumby et al. 2007) model. Here, we follow recent evidence from the literature that suggests coral calcification is inhibited by increased nutrients(Shantz and Burkepile 2014), and similarly making the intuitive assumption that nutrients increase the growth rate of macroalgae (a primary producer), as well as decreases coral’s resistance to macroalgae overgrowth and competitive ability(Lapointe 1999, McClanahan et al. 2003). Figure 1A shows a schematic summarizing our model and the interactions between state variables. As such, our new nutrient-dependent model is as follows:
dM⁄dt=M(aC( N_t⁄((N_0+N_t ) ))- g⁄((M+B) ) + yB) (1)
dC⁄dt=C(r(1-cN_t )B-m-aM( N_t⁄((N_0+N_t ) ))) (2)
Here, M represents macroalgae, C represents coral, and B represents other benthic r-strategists that instantaneously fill empty space left by dead coral and grazed macroalgae. Since B is assumed to instantaneously fill any space (r-strategists act on much faster time scales relative to the other state variables), B = 1 – M – C and the model can therefore be reduced to two dimensions. The original model by Mumby et al.(Mumby et al. 2007) referred to this variable as turf algae, a fast-growing r-strategist, however, we argue that nutrients and grazing rates would determine the composition of other components of benthic cover(Lapointe 1997, Lapointe et al. 2018) (Figure 1A). Towards this end, we include a simple extension that includes algal life history traits. Specifically, we include growth-palatability trade-offs in algal community composition to represent the role of r-strategists in response to disturbances and grazing pressure (Figure 1). High grazing rates suppress late-succession, palatable macroalgae while simultaneously selecting for fast-growth and less-palatable life strategies (through physical and chemical defenses) within early succession r-strategists. Here, we consider r-strategists as algae functional groups that colonize disturbed reefs quickly relative to thick, fleshy macroalgae (late succession K-strategists relative to other functional forms of algae, but of course fast-growing relative to corals). We argue that within the group of early succession r-strategists there is a variety of life history strategies, that will be differentially selected for based on grazing rates. High grazing rates favour less-palatable life strategies and fast growth rates that can withstand these grazing levels. At moderate-high grazing rates, fast-growing r-strategists can withstand the grazing rates, explaining why we see a lot of turf algae and some less-palatable algae like CCA in some reefs. However, at extremely high grazing rates (e.g., extremely high Diadema densities), only low palatability organisms will be able to survive. As a simple extension, we include a linear 1:1 relationship between grazing rates and the proportion of more/less palatable r-strategists composing the benthic r-strategist guild. We refer to highly palatable fast-growing r-strategists (e.g., filamentous turf) as r1-strategists, and slightly less palatable (but still fast-growing relative to K-strategists) r-strategists (e.g., CCA) as r2-strategists (Figure 1B). Therefore, B = r1 + r2, where r2 = gB and r1 = (1-g)B. These assumptions relate to well-known relationships between high grazing rates and less palatable algae(Sammarco 1982, Chiappone et al. 2006), as well as research showing that grazing rates (among other drivers such as nutrients) can alter the rates and stages of succession in coral reefs(Hixon and Brostoff 1996, Mcclanahan 1997).
Here, a is the rate that macroalgae overgrows coral, and thus can be thought of as an algal-coral competition rate. g represents the grazing rate from herbivores, which is scaled by densities (or % cover since M + C + B = 1) of both M and B. y is the rate that macroalgae overgrows benthic r-strategists (B; which could also include open space), r is the coral growth rate (again over B or open space), and m is the natural mortality rate of coral.
Nutrients (Nt) alter the algal-coral overgrowth rate (competition) in the Mumby model (a), based on the assumptions made above. We also assume that this competitive advantage eventually saturates and therefore N0 is a saturation constant that determines the effect of nutrients (Nt) on algal-coral competition (a). Additionally, c is the calcification rate – scaled by nutrient loading, Nt – which alters the coral growth rate (r) in response to nutrient loading. An important distinction between this model and Mumby et al.’s original model is that the parameter a can now be thought of as amax and r similarly as rmax due to their scaling by nutrients.