Ecological Archives A21-064-A1

Jessica Melbourne-Thomas, Craig R. Johnson, Tak Fung, Robert M. Seymour, Laurent M. Chérubin, J. Ernesto Arias-González, and Elizabeth A. Fulton. 2011. Regional-scale scenario modeling for coral reefs: a decision support tool to inform management of a complex system. Ecological Applications 21:1380–1398.

Appendix A. The local-scale ecological model.

The regional-scale Coral Reef Scenario Evaluation Tool (CORSET) we describe and evaluate comprises multiple instantiations of a local-scale mean-field model of ecological dynamics (Fung 2009) that are connected by larval transport (Fig. 1 in main text). The local model is defined by seven differential equations which have been parameterized using available data for shallow coral reef habitats (~ 5 m – 20 m depth). In the CORSET implementation of the local-scale model, we distinguish between brooding and spawning corals, so that the model system comprises eight differential equations (Equations A.1 – A.8). CORSET uses discrete-time (difference equations) approximations of these differential equations, which were derived using Euler’s method. Details of the model development and clarification of assumptions are provided below. Mathematical analyses of the behavior of the local model and derivations of generic parameter ranges that apply to reefs worldwide are available in Fung (2009). The parameter ranges used in this paper, which apply to the western Atlantic (Table A1), are derived using the same approach as in Fung (2009).

Local-scale model development and assumptions

Benthic variables followed in the CORSET implementation of the local-scale model are the proportional covers of brooding corals (Cb), spawning corals (Cs), hard corals (C = Cb + Cs), macroturf (T), macroalgae (M) and grazed epilithic algal communities or EAC (E = 1 – CbCsTM). These proportional covers can be converted into percentage covers, as in the main text. The model represents key ecological processes that determine competition for space between corals, macroturf, macroalgae and grazed EAC. Other sessile organisms are not modeled and are assumed to have non-significant effects on coral-algal dynamics, and to take up a relatively small constant component of the benthos on average. The dynamics of brooding and spawning coral are modeled identically except for recruitment dynamics, as detailed in the main text (brooded coral larvae are assumed to settle locally while spawned larvae disperse in the plankton).

Corals experience background mortality that may be enhanced by human activities such as coastal development. Coral larvae may settle onto grazed EAC or macroturf (Birrell et al. 2005, Birrell et al. 2008), but the settlement rate onto macroturf is lower than the rate onto grazed EAC (Birrell et al. 2005). Settlement of coral larvae onto macroalgae is assumed to be non-significant (Nugues and Szmant 2006, Birrell et al. 2008). Corals can grow laterally over grazed EAC (which is assumed not to impede the growth of corals) and over macroturf (either through competitive superiority or shading and overgrowth by branching corals; McCook et al. 2001). The growth of macroturf is predominantly vertical, and hence there is no lateral growth of macroturf over grazed EAC. However, if EAC is left ungrazed then it grows into macroturf at a fixed rate. There is no overgrowth of corals by macroturf (based on findings from Jompa and McCook 2002, Jompa and McCook 2003, Birrell et al. 2008).

The local-scale model includes a term that represents overgrowth of corals by macroalgae (based on evidence from De Ruyter van Steveninck et al. 1988, McCook et al. 2001, and Jompa and McCook 2002), but this can be set to zero for cases where macroalgae are not competitively superior to corals (e.g. Nugues and Bak 2006). The growth rate of macroalgae over corals is less than that over grazed EAC, based on evidence that corals can reduce the growth rate of macroalgae at the coral-algal interaction fringe (De Ruyter van Steveninck et al. 1988, Jompa and McCook 2002). The model also includes a term that represents a reduction of coral growth rates over grazed EAC and macroturf due to the presence of nearby macroalgae (Lirman 2001, Jompa and McCook 2002). Macroturf recruitment is not modeled explicitly; instead, it is assumed that macroturf grow readily from grazed EAC because of basal portions that remain after grazing and/or because of a ready supply of macroturf propagules from the reef environment (Hay 1991, Diaz-Pulido and McCook 2002). Macroalgal recruitment is not modeled due to a lack of knowledge of supply-side dynamics (McCook 1999, Diaz-Pulido and McCook 2004). Similarly, background mortality of algal groups (macroalgae and macroturf) is not modeled because of a lack of available information regarding these dynamics on coral reefs. Thus, the main sources of mortality for macroturf are assumed to be grazing (the trophic flux from algae to herbivores is one of the largest on coral reefs, McCook 1999) and overgrowth by macroalgae (Mumby et al. 2006) and corals. The main source of mortality for macroalgae is assumed to be grazing (McCook 1999).

Consumer variables followed in the local-scale model are the biomasses of herbivorous fish (H), small-to-intermediate piscivorous fish (Ps), large piscivorous fish (Pl) and sea urchins (U). The process of growth is modeled differently for herbivorous fish and piscivorous fish.  Herbivorous fish grow by consuming macroturf and macroalgae (the grazing rate on macroalgae can be lower than on macroturf since macroalgae can be chemically defended against grazing by secondary metabolites; Hay 1991). The realized grazing pressure exerted by herbivorous fish (see Equation A.3b) determines the amount of available algal cover which is consumed and converted into energy (which is used for growth).  Realized grazing pressure depends on the accessibility of the algal food source to herbivorous fish, where accessibility is a proxy for foraging ability and reef rugosity. At high rugosity, more individuals per unit planar area are needed to achieve a given grazing pressure, due to the greater benthic surface area per unit planar area (i.e. algal resources are effectively less accessible). The realized grazing pressure for fish also includes exploitative competition from herbivorous sea urchins (Hay 1984, Carpenter 1990). Somatic conversion rates for herbivorous fish grazing on macroalgae, macroturf and EAC are assumed to be equal given a lack of information on relative rates at which the different types of modeled algal covers are converted by grazers into somatic biomass.

Piscivorous fish grow by consuming herbivorous fish and smaller piscivorous fish. In the local-scale model, the growth rate of piscivorous fish is a function of prey biomass and is assumed to follow a Holling Type-III response with an exponent of two. The Holling Type-III response means that at low and high prey biomasses the rate of increase of the realized growth rate is slow. At low prey biomasses this reflects the difficulty of prey capture due to low encounter rates, while at high prey biomasses the Type-III functional response reflects prey saturation, such that predation (and therefore growth) is no longer significantly limited by the availability of prey. The model assumes that a certain proportion of small-to-intermediate piscivorous fish transition to the large piscivorous fish functional group at each timestep, due to growth.

All fish groups suffer morality due to fishing, predation, and natural processes excluding predation (such as senescence). Fishing decreases the fish biomass of each group through time and the amount of fish caught depends on fishing pressure, the standing stock fish biomass and the accessibility of each type of fish to fishermen (Kramer 2007). Fishing pressure is represented as the maximum fish biomass which can be caught, and is related to the number of fishing hours and the technology (type of fishing gear) used. Accessibility is a proxy for the technology used and the structural complexity of the coral reef habitat.

Sea urchins, like herbivorous fish, grow by consuming macroturf and macroalgae. The available algal cover grazed is determined by the realized grazing pressure exerted by sea urchins (see Eq. A.3c), which includes exploitative competition from herbivorous fish (Hay 1984, Carpenter 1990). It is assumed as a simplification that growth accumulation parameters for sea urchins are proportional to those for herbivorous fish. For sea urchins, mortality occurs due to predation by invertivorous fish (not modeled explicitly) and other natural processes.

Local-scale model equations and parameters

The change in proportional cover of brooding corals () is given by

(A.1)

where all parameters (, ,  etc.) are defined in Table A1, as are all parameters used in the following equations. Similarly, for the proportional cover of spawning corals ()

(A.2)

and macroturf ()

(A.3)

where q  is the grazing term given by

(A.3a)

Grazing by herbivorous fish () and sea urchins () is scaled by competition between these two groups such that

(A.3b)
(A.3c)

The change in proportional cover of macroalgae () is given by

(A.4)

and in the biomass of herbivorous fish () by

(A.5)

Here, ‘other mortality’ refers to mortality from processes other than predation and fishing. Similarly for the biomass of small-to-intermediate piscivorous fish ()

(A.6)

large piscivorous fish ()

(A.7)

and sea urchins ()

(A.8)

TABLE A1. Local model parameter definitions and values derived for the western Atlantic. Parameter values are derived using the same parameterization methodology as detailed in Fung (2009). Where possible, parameters were derived using data just from the western Atlantic. In cases where western Atlantic data were not available, data were taken from the Indo-Pacific region. Mathematically derived parameter restrictions detailed in the footnotes are required to keep state variables within a realistic biological range, i.e., in the range 0 – 1 for benthic covers, and ≥ 0 (but not tending to infinity) for consumer biomasses (see Fung 2009 for detailed derivations).

Parameter

Definition

Derived value(s)

Sources*

Benthic Parameters

The background mortality rate of brooding and spawning corals.

0.02 – 0.1 yr-1

1, 2, 3

The growth rate of existing coral over grazed EAC.

0.04 – 0.2 yr-1

4, 5, 6, 7, 8

The growth rate of coral over macroturf, relative to its growth over grazed EAC.

0 – 1

 

The recruitment rate of coral onto macroturf, relative to the rate onto grazed EAC.

0.05 – 0.15

9

П

The recruitment rates of brooding and spawning corals.

Modeled externally to the local model

 

The maximum rate (per unit of grazing pressure) at which existing macroturf is grazed down.

5 – 15 yr-1

10, 11

The rate at which grazed EAC grows into macroturf.

2 – 20 yr-1

11, 12

The maximum rate (per unit of grazing pressure) at which existing macroalgae is grazed down.

0.01

10, 11, 13, 14, 15

The growth rate of macroalgae over grazed EAC.

0.05 – 0.4 yr-1

16

Coral growth is inhibited by the presence of nearby macroalgae and this is represented as depression of  by the factor .

0.4 – 0.9 yr-1

4, 5, 6, 7, 8, 17, 18

The growth rate of macroalgae over coral, relative to its growth over grazed EAC.

0 – 0.9

4, 8, 16, 17, 19

The growth rate of macroalgae over macroturf, relative to its growth over grazed EAC.

0 – 0.9

4, 8, 16, 17, 19

Consumer parameters

A parameter that measures the accessibility of algae (turf and macroalgae) to herbivorous fish grazing.

3 × 103 – 5 × 105 kg/km2

16, 20, 21, 22, 23

A parameter that measures the accessibility of herbivorous fish to predation by piscivorous fish.

7 × 103 – 1 × 104 kg/km2

8, 20, 24–40

A parameter that measures the accessibility of herbivorous fish to fishermen.

7 × 102 – 1 × 103 kg/km2

8, 20, 24–40

The mortality rate of herbivorous fish from all factors other than predation by piscivorous fish and fishing.

 – 2 yr-1

41

§

The herbivorous fish biomass accumulated from grazing on 100% cover of macroalgae, macroturf and EAC respectively, and which contributes to somatic growth of herbivorous fish.

60 –  kg km-2 yr-1

30, 31, 42, 43, 44

The proportion of the total fishing pressure which acts on herbivorous fish.

0 – 1, with

 

П

The recruitment rate of herbivorous fish.

Modeled externally to the local model

 

A parameter that measures the accessibility of small-to-intermediate piscivorous fish to predation by large piscivorous fish.

7 × 103 – 1 × 104 kg/km2

8, 20, 24–40

A parameter that measures the accessibility of small-to-intermediate piscivorous fish to fishermen.

7 × 102 – 1 × 103 kg/km2

8, 20, 24–40

The mortality rate of small-to-intermediate piscivorous fish from all factors other than predation and fishing.

– 2 yr-1

41

The maximum predation rate of small-to-intermediate piscivorous fish on herbivorous fish.

1 – 2 yr-1

44

The proportion of consumed biomass which is used for somatic growth, for small-to-intermediate piscivorous fish.

0.035 – 0.105

31, 43, 44

A parameter which measures the rate at which small-to-intermediate piscivorous fish biomass becomes large piscivorous fish biomass due to predation and subsequent growth.

0 – 10

 

The proportion of the total fishing pressure which acts on small-to-intermediate piscivorous fish.

0 – 1, with

 

П

The recruitment rate of small piscivorous fish.

Modeled externally to the local model

 

A parameter that measures the accessibility of large piscivorous fish to fishermen.

7 × 102 – 1 × 103 kg/km2

8, 20, 24–40

The mortality rate of large piscivorous fish from all factors other than predation and fishing.

– 0.9 yr-1

41

The maximum predation rate of large piscivorous fish on herbivorous fish.

0.25 – 0.75 yr-1

44

The proportion of consumed biomass which is used for somatic growth, for large piscivorous fish.

0.03 – 0.09

31, 43, 44

The predation rate on small-to-intermediate piscivorous fish by large piscivorous fish, relative to that on herbivorous fish.

2 – 4

44

The proportion of the total fishing pressure which acts on large piscivorous fish.

0 – 1, with

 

¥

The maximum fish biomass which can be caught.

0 – 5 × 103 kg km-2 yr-1

45, 46, 47, 48

The linear mortality rate of urchins.

– 1 yr-1

49, 50, 51

Δ

The quadratic mortality rate of urchins.

1 × 10-6 yr-1

 

#

A parameter that measures the accessibility of algae (turf and macroalgae) to urchin grazing.

 – 3 × 106 kg/km2

16, 21, 22, 23, 52

A parameter that measures the biomass accumulated by urchin grazing and which contributes to somatic growth relative to that for herbivorous fish grazing.

1 – 10

31, 43, 44, 50, 53

П

The recruitment rate of urchins.

Modeled externally to the local model

 

A parameter that determines the competitiveness of herbivorous fish relative to urchins.

0.6 – 1

54

A parameter that determines the competitiveness of urchins relative to herbivorous fish.

Equal to

54

* Sources: 1Bythell et al. (1993), 2Lirman (2003), 3Nugues and Roberts (2003), 4Edmunds (2007), 5Langmead and Sheppard (2004), 6Huston (1985), 7Chornesky and Peters (1987), 8García-Salgado et al. (2006), 9Birrell et al. (2005), 10Hughes et al. (2007), 11Mumby et al. (2006), 12McClanahan (2002), 13Bellwood et al. (2006), 14Russ and St. John (1988), 15Hay (1991), 16Mumby et al. (2005), 17Lirman (2001), 18Box and Mumby (2007), 19Nugues and Bak (2006), 20Williams et al. (2001), 21Ostrander et al. (2000), 22Edmunds (2002), 23Miller and Gerstner (2002), 24Sandin et al. (2008), 25Mumby and Dytham (2006), 26Williams and Hatcher (1983), 27Letourneur et al. (1998), 28McClanahan et al. (1996), 29Arias-González et al. (1997), 30Arias-González (1998), 31Van Rooij et al. (1998), 32Friedlander and DeMartini (2002), 33Gribble (2003), 34Kramer (2003), 35Arias-González and Morand (2006), 36Newman et al. (2006), 37Craig et al. (2008), 38Tsehaye and Nagelkerke (2008), 39Aliño et al. (1993), 40McClanahan et al. (1999), 41Froese and Pauly (2004), 42Gardner et al. (2003), 43Johnson et al. (1995), 44Opitz (1996), 45Stevenson and Marshall (1974), 46Koslow et al. (1994), 47Halls et al. (Halls et al. 2002), 48Arias-González et al. (2004), 49Karlson and Levitan (1990), 50Levitan (1989), 51Ebert (1975), 52Carpenter (1986), 53Levitan (1988), 54Hay (1984).

П Recruitment terms () are modeled externally to the local model, based on matrices defining connectivity between reef cells as detailed in the main text. Parameters used for modeling larval production and post-settlement processes are detailed in Appendix B.

. Here, .

§ , where .

.

.

¥ Estimated values for  presented in Table A1 are based on published values for the western Atlantic. In the two scenarios presented in our study,  was zero or varied over time.

.

Δ Quadratic mortality of sea urchins, , is included in the regional implementation of the local ecological model to prevent uncontrolled population explosions in this functional group. This term is a proxy for consumption of sea urchins by fish and invertebrate predators (Randall et al. 1964, Harborne et al. 2009).

#.

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