APPENDIX I

Single-Species Model
Ecosystem model

Single-Species Model

Fishing strategies were evaluated with a Monte Carlo simulation model based on delay-difference equations. The model predicts next year’s biomass (B_t+1) and numbers (N_t+1) according to the equations (Hilborn and Walters, 1992):

	(1)
	(2)

where w_k is the weight at recruitment age k (years), s_t = l(1-h_t) is the total survival rate, l is the natural survival rate (l = e^-M), M is the instantaneous natural mortality rate (=0.95 year^-1), and h_t is the exploitation rate. The exploitation rate can be calculated as the ratio catch/biomass in a given year, or as a function of fishing effort, i.e., , where E is effort and q the catchability parameter.

The delay-differential model assumes growth in mean body weight at age (w_a) can be described using a Ford-Walford plot and linear model , where a and r are constants. Sardine parameter values for a (= 0.025) and r (= 0.896) were obtained by regressing data on weight at consecutive ages from Cergole (1995). Recruitment (R_t+1) is included in the model as functions representing three different stock-recruitment hypotheses (Vasconcellos, 2000; Table I.1; Figures I.1 and I.2). In the analysis of fishing strategies all three hypotheses were assigned equal degree of belief, i.e., they are assumed to fit stock and recruitment data equally well.

Table I.1. Hypotheses, models and parameters used to predict recruitment rates in the delay-difference model. Hypotheses 1 and 2 are represented by a Beverton-Holt stock recruitment function modified to include depensatory effects (Myers et al., 1995). Hypothesis 3 is represented by a modification of the Beverton-Holt function according to Walters and Parma (1996). In the latter, the density-independent mortality risk (M₁) follows a sinusoidal trend with period of 10 years, thus representing decadal regimes in marine carrying capacity.

Hypothesis	Model	Parameter values
1. Recruitment is a function of stock size		a = 135 K = 0.0888 x = 1 v = 0.4
2. Recruitment is a function of stock size with depensation at low stock sizes.		a = 867 K = 0.0138 x = 2 v = 0.4
3. Recruitment is a function of stock size and low frequency environmental regimes.	a = exp-M1	M2 = 0.152 v = 0.4

Figure I.1. Graphic representation of two hypothesis used to describe the relationship between spawning stock biomass and recruitment. The replacement line represent the number of recruits needed to replace the corresponding spawning stock biomass. Depensation occur when the average stock-recruitment relationship crosses the replacement line at low stock sizes.

Figure I.2. Graphic representation of the “dome-shaped” regime hypothesis (hypothesis 3). Upper panel shows the two extreme stock-recruitment relationships modeled to represent a “good” and a “bad” environmental regime. The model is used to generate a sinusoidal trend in the marine carrying capacity which results in a dome-shaped relationship of recruitment with time (lower panel).

Errors in the estimation of stock biomass by direct methods (e.g., acoustic surveys, egg production, etc.) were introduced in the simulations by including a normally distributed error around the true stock biomass (Frederick and Peterman, 1997), where

B_est = B_t + (B_t · CV · w)

Ecosystem model

Construction of an Ecopath mass-balance model
Simulation with Ecosim

Construction of an Ecopath mass-balance model

Ecopath (Christensen and Pauly, 1992) provides a static picture of the ecosystem trophic structure by estimating trophic flows and biomasses which satisfy growth and mortality constraints. The model relies on the truism that for each group (i) in the system, and to any time period:

Production by (i) = All predation on (i) - Fisheries catches - Other mortality - Losses to adjacent systems

(1)

where in a system of i=1,...,n functional groups; P/B_i is the production/biomass ratio of (i) (equal to the total mortality rate Z_i under the assumption of equilibrium); EE_i is the ecotrophic efficiency, i.e. the fraction of the production that is accounted for by consumption within the system (predation) or harvested; Y_i is the yield of (i), in weight, with Y_i = F_i.B_i, where F_i is the fishing mortality; EX_i is other exports of (i) from the system; B_j is the biomass of the consumers or predators; (Q/B)_j is food consumption per unit of biomass for consumer j, and DC_ji is the fraction of i in the diet of j. DBi is biomass accumulation rate per time in cases where the analysis does not use data from an initial equilibrium situation. Model parameters (tables I.2 and I.3) were estimated by Rocha et al. (1998) and Vasconcellos (2000).

Table I.2. Parameters of the Ecopath trophic model of the Southeastern shelf ecosystem for the late 1980s - early 1990s. Underlined values, trophic levels and omnivory index were estimated by the model. Values in brackets are parameters used to describe the late 1970s conditions.

Species/Group	Trophic level	Omnivory index	Biomass tons · Km-2	P/B year-1	Q/B year-1	EE	Catches (tons·Km-2· year-1)
							Bottom trawlers	Purse seiners	Shrimp trawlers	Pole and line
Phytoplankton	1.0	0.000	24.00	70.00	-	0.93	-	-	-	-
Detritus	1.0	0.190	10.00	-	-	0.90	-	-	-	-
Salps	2.0	0.000	20.00	5.40	18.00	0.00	-	-	-	-
Zooplankton	2.1	0.053	4.12	60.00	288.00	0.84	-	-	-	-
Benthos omniv.	2.1	0.003	13.14	0.40	2.84	0.55	-	-	-	-
Marine shrimps	2.1	0.003	0.31 (0.35)	3.93	18.00	0.95	-	-	0.081 (0.101)	-
Benthos detrit.	2.3	0.169	30.00	3.00	27.27	0.88	-	-	-	-
Anchovy	2.8	0.233	2.33 (1.00)	1.29	11.20	0.16 (0.78)	-	-	-	-
Benthos carniv.	2.9	0.374	35.00	0.96	3.28	0.30	-	-	-	-
Sardine	2.9	0.177	0.63 (1.49)	1.92	11.20	0.47 (0.33)	-	0.317 (0.784)	-	-
Juvenile sardine	2.8	0.177	0.27 (0.61)	7.00	23.33	0.25 (0.13)	-	-	-	-
Other forage fish	2.9	0.177	5.00	1.29	11.20	0.07	-	-	-	-
Juv. Weakfish	3.1	0.001	0.12 (0.13)	2.00	10.00	0.90	-	-	-	-
Juv. Triggerfish	3.2	0.019	0.02 (0.02)	2.00	10.00	0.90	-	-	-	-
Croaker	3.4	0.246	0.32 (0.39)	0.40	3.88	0.90	0.027 (0.042)	-	-	-
Rays/Skates	3.5	0.162	0.01 (0.003)	0.40	4.00	0.90	0.003 (0.001)	-	-	-
Triggerfish	3.5	0.098	0.10 (0.05)	0.90	6.13	0.90	0.013 (0.000)	-	-	-
Other Bent. fish	3.5	0.163	0.29	0.92	5.20	0.52	-	-	-	-
Other Pel. fish	3.7	0.114	0.34	0.48	5.60	0.85	-	-	-	-
Bonito	3.7	0.053	0.41	0.98	4.51	0.10	-	-	-	0.040 (0.008)
King Weakfish	3.8	0.445	0.12 (0.13)	0.90	6.16	0.90	0.011 (0.012)	-	-	-
Adult Weakfish	3.9	0.102	0.02 (0.01)	0.90	6.70	0.90	0.013 (0.011)	-	-	-

Table I.3. Diet matrix of the trophic model of the Southeastern shelf ecosystem. Values represent the proportion of the diet of a predator (column) made of a given prey (row). Dots are preys with less that 1% importance to a given predator.

Prey \ Predator	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20	21	22
1. Phytoplankton	1.00	0.85			0.10	0.30		0.20		0.20
2. Detritus		0.10	0.99	0.99	0.73		0.38
3. Salps
4. Zooplankton		0.05	·	·	0.12	0.70	0.20	0.80		0.80	0.80	0.90			0.10		0.02	0.10		0.02
5. Benthos omniv.							0.02				0.05	0.02	0.15	0.15	0.10	0.10	0.06			0.01
6. Marine shrimps											0.10					0.05	0.1		0.25	0.05
7. Benthos detrit.			·	·	0.05		0.32				0.05	0.04	0.17	0.50	0.40	0.50	0.05		0.25	0.01
8. Anchovy																	0.23	0.30		0.26
9. Benthos carniv.							0.08					0.04	0.28	0.28	0.40	0.28	0.01			0.01
10. Sardine																	0.03	0.05		0.07
11. Juv. sardine																	0.20	0.05		0.20
12. Other forage fish																	0.23	0.50		0.26
13. Juv. weakfish													0.02	·		0.01	0.05		0.10
14. Juv. triggerfish														0.01					0.02	·
15. Croaker														·		0.01			0.10	0.01
16. Rays/Skates
17. Triggerfish													0.02						0.05	0.01
18. Other benth. fish													0.02	0.04		0.02			0.10	0.02
19. Other pel. fish													0.02	0.02		0.02			0.10	0.02
20. Bonito
21. King weakfish													0.02	·		0.01			0.05	0.01
22. Adult weakfish

Simulation with Ecosim

By re-expressing the system of linear equations (1) as differential equations, Ecosim provides a dynamic model suitable for simulation of the effects of F varying in time on the biomass of each group in the system. The model provides dynamic biomass predictions of each (i) as affected directly by fishing and predation on (i), changes in food available to (i), and indirectly by fishing or predation on other groups with which (i) interacts (Walters et al., 1997). Constructing a dynamic model from equation (1) involves three changes; a) replace the left side with a rate of change of biomass; b) provide a functional relationship to predict changes in P/B_i with biomass B_i and consumption, and c) provide functional relationships predicting how the consumption will change with changes in the biomasses of B_i and B_j (Walters et al., 1997). Thus equation (1) is re-expressed as

(2)

where f (B_i) is a function of B_i if (i) is a primary producer or f(B_i) = g_iS c_ji (B_i,B_j) if (i) is a consumer, where g_i is the net growth efficiency, and c_ij(B_i.B_j) is the function used to predict consumption rates from B_i to B_j. Ecosim uses a function for c_ij derived from assuming possible spatial/behavioral limitations in predation rates

cij = vij aij Bi Bj/(vij+v’ij+aij Bj)

(3)

where

c_ij is the trophic flow, biomass per time, between prey (i) and predator (j) pools;
B_i and B_j are the biomasses of prey and predators, respectively;
a_ij is the rate of effective search for prey i by predator j; and
v_ij and v’_ij are prey vulnerability parameters

Ecosim represents linkages between split pool pairs (juvenile and adult stages), through flow of biomass and number of individuals, using a delay-difference model for each split pool case in Ecopath (Walter et al., in press; Christensen and Walters, 2000). Parameterization of the delay-difference model requires, besides the normal Ecopath input parameters, data on growth and age/weight at transition from juvenile to adult stage (Table I.4).

Table I.4. Parameters of the split pools in Ecopath used by the delay difference model in Ecosim. K is the von Bertalanffy growth parameter (year^-1), w_k is the weight (g) at the age t_k (years) fish graduate to the adult pool. Parameters for sardine were obtained from Cergole (1995). Parameter values for weakfish and triggerfish are from FishBase (1998).

Split pool	K	wk	tk
Weakfish	0.3	100	2.0
Triggerfish	0.5	128	1.0
Sardine	0.5	44	1.5

In fitting Ecosim predictions to sardine and anchovy time series data, a built-in search routine was used to generate time series values of annual relative primary productivity and sardine egg production that could represent historical productivity ‘regime shifts’ affecting biomasses. For more details on the searching routines employed by Ecosim see Walters and Christensen (2000).