As mentioned in Chapter 1, a critical step in the development of a management plan is the elaboration of a mathematical model. A useful tool for fisheries modelling is based on the systems simulation approach (Manetsch & Park, 1982; Seijo 1987, 1989; Seijo & Defeo, 1994b). In this chapter we develop the main steps involved in the process of systems modelling. Particular emphasis is given to the application of this methodology to fisheries, detailing the main components or subsystems of a fishery, and the interactions among them.
The system simulation approach is a problem-solving process through which particular time solutions of a mathematical model are arrived at, based on specific assumptions regarding input variables, parameters and causal relationships between them (Manetsch & Park, 1982). It comprises the following steps (Fig. 4.1):
A clear definition of fishery information needs.
Fishery characterization, in terms of resource and fishing effort dynamics, ecological and technological interdependencies and management instruments.
Mathematical modelling of the fishery components or subsystems.
Data collection, from both primary and secondary sources, needed to estimate parameters and fit equations of the mathematical model.
Development of a computer model to solve numerically the mathematical model.
Stability and sensitivity analyses for the computer model.
Model validation.
Evaluation of the bioeconomic impacts of alternative management strategies.
Figure 4.1. A system simulation approach to fisheries management (after Seijo, 1989).
The approach can be extended to integrate the bioeconomic model into a linear/nonlinear optimization algorithm that could involve single or multiple management criteria. This strategy allows one to generate a management vector that minimizes the difference between the performance variable levels desired by the policy maker and those generated by the model (see Chapter 5).
A fishery research program begins with the recognition of some management problem related to bioeconomic observations on fishery performance (i.e., spatial and temporal patterns in stock abundance and intensity of fishing effort), and the search for mechanisms (management strategies) designed to acheive a sustainable fishing activity. The following approach could be useful to identify the relevant bioeconomic information required to address the management problem:
State clearly the management problem to be addressed, including observations about fishery performance and the person(s) involved, e.g. fishers, policy makers, or both (Ackoff, 1962; Seijo, in press).
Identify the management context of the fishery through the following types of parameters/ variables (Seijo, 1986):
Exogenous: environmental variables, species price.
Overt controllable: policy variables, which can be managed during system operation to alter the performance of the system in providing desired outputs.
Overt necessary: variables required in order for the system to function (e.g., gas-oil, fish bait, ice and food).
System design parameters: attributes of the system structure that have an impact upon the system desired output. Includes a selected classification of fishers groups according to the technology utilized, and production functions of fishing effort.
Specify the desired levels of fishery performance variables for the policy maker.
Determine at least two management alternatives with viable implementation, enforcement and execution from physical, political and economic points of view.
Specifying a state uncertainty concerning the management alternatives or combinations of different levels of these alternatives, is a better way of achieving the objectives stated in (c).
The fishery system can be decomposed into three subsystems that interact to give the overall system its unique behavior (fig. 4.2): (a) the resource; (b) resource users; and (c) resource management. Figure 4.2 also emphasizes the interface variables, which are the outputs of one subsystem acting as inputs to the others. The main system assumptions are considered as exogenous parameters not defined within the system, and are represented by arrows without an origin in the block diagram.
Resource subsystem. Includes: (1) aspects of the life cycle of the species, such as reproductive biology and recruitment, growth and mortality dynamics; (2) environmental factors affecting abundance and spatio-temporal distribution; and (3) ecological interdependencies.
| ALIMIi(t)=fish for (subsistence | Lijk= individual length | |
| Bijk(t)=biomass | Nijk(t)=number of individuals | |
| Cpunit cost of effort | xpi=ex-vessel price | |
| Cp=processing costs | wpi=wholesale price | |
| EMPm(t)=employment | qijm=catchability coefficient | |
| EXPm(t)= export earnings, harvesting | πm(t)=net revenues | |
| sector | SELijm=gear selectivity | |
| EXPp(t)=export earnings, proccessing | Vim(t)=number of vessels | |
| sector | Wij=individual weight | |
| fijkm(t)=fishing intensity | Yijkm=catch | |
| FRijkm(t)=fishing mortality rate | Rjk(t)=recruitment | |
| MRi=natural mortality rate | ||
| where: i= species; j=age; k=site; m=fleet; t=time | ||
Figure 4.2. Fishery characterization.
Resource users subsystem. Includes the explicit functions of fishing effort, decomposed by fleet type operating over different species or population components. Selectivity curves by species, sizes/ages and fleet type, and prices of the target and incidental species, are also included.
Resource management subsystem. In order to achieve the objectives and goals proposed in step (iii) of the management plan, one requires to consider possible approaches to State intervention, the selection criteria for management strategies, and a way of contending with situations of multiple criteria in the combination or selection of management instruments.
A mathematical model is then built based on a series of statements, assumptions and equations, to provide possible explanations for the events observed. The model will contain the causal relationships between the three subsystems, and, if validated (see below), it will be a powerful tool to evaluate the bioeconomic impact of alternative management strategies and to perform simulation experiments. It is relevant to carry out a careful analysis of the underlying assumptions of the selected models/functions that describe the basic population dynamic processes. In Chapters 2 and 3, and also in this Chapter, models are built to represent fisheries whose behavior satisfies the dynamic pool assumption. In Chapter 6, this assumption is relaxed to model the heterogeneity in stock distribution and in the corresponding fishing intensity, especially for shellfisheries.
Population dynamics
This can be modelled by using the principles of a life table (Seijo, 1986; Begon et al.,1990), on the basis of the dynamic accounting of inflows and outflows of individuals to each age of the population structure. Changes in the number of individuals through time can be defined as in “Agestructured models”described in Section 2 (see equations 2.45 to 2.54).
The feedback mechanism for expression (2.49) can be generated by a stochastic recruitment function. Considering e.g., the Beverton & Holt (1957) function:
where Hmaxi represents the maximum number of eggs produced by the spawning stock of species i. Rmaxi is the maximum observed recruitment, and HSi(t) is the estimated number of eggs produced by the spawning stock i in time t. RNi(t) is a normally distributed random variable with 0 mean and known variance that represents the unexplained variability by equation (4.1). Another recruitment function (see Ricker, 1975) could be used according to the analyzed species (see Hilborn & Walters, 1992; De Anda et al., 1994).
HSi(t) can be estimated for each spawning period as the product of the spawning stock abundance in time t by the proportion of females (Hi) and the corresponding age-specific average fecundity (FECij):
where MAGEi is the maximum observed age and si is the age at first maturity of species i.
Biomass for each species is calculated as follows:
where the average individual weight of species at age Wij=aLbij (a and b are parameters). Lij can be estimated by the von Bertalanffy (1938) growth function:
Lij = L∞i(1 - e(-kpi(j-toi))) (4.4)
where Lij is the length at age, L∞i is the asymptotic length, kpi is the curvature parameter and oi the theoretical age at length=0.
As detailed described in Chapter 2, spawning and hatching periods can be modelled using the “distributed delay model”, based on the Gamma probability density functions:
where HSi(t) are the inputs to the delay process (number of eggs spawned in time t); γk(t) are the outputs of the delay (number of eggs hatched in time t); γ1(t), γ2(t),…, γg-1(t) are the intermediate rates; DELHS is the expected period of egg maturation and g is the order of the delay. The parameter g specifies a member of the Erlang family of density functions, which describes the transit times of individual entries as they pass through the delay process.
Fleet dynamics
In the long-run, variations in fishing effort through time can be modelled by applying Smith (1969) equation:
Short-run (seasonal) dynamics of the fleet can be modelled by the distributed delay model, in order to represent entry of vessels throughout the fishing season. Catch by species and sizes for each fleet type m in time t Yijm(t) can be described as:
Yijm=qijmfm(t)Bij(t)+RVm(t) (4.9)
where RVm(t) is a exponentially autocorrelated random variable which accounts for the uncertainty in the catch over time, and could be generated with other appropriate probability density functions.
A catchability coefficient by species, sizes and fleet type (qijm) is estimated by using Baranov (1918) concepts:
where AREA is the area occupied by the stock, RETijm is the gear retention of species i by age j of the gear of vessel type m, defined by a gear selection ogive, and aream is the area swept per day.
The specific gear retention by size is estimated by (Sparre et al., 1989):
where Lim50% and Lim75% are, respectively, the length at 50% and 75% gear retention by fleet type m. This catchability function is not always valid (e.g. q is inversely correlated with stock abundance for some pelagic stocks, MacCall, 1976), and thus it is recommended to use an age/fleet specific function according to the analyzed fishery.
The cumulative economic rent received by vessel type (πm(t)) is estimated by:
where TRm(t) y TCm(t) are, respectively, total revenues and total costs by vessel type. The former are estimated by considering prices and catches of the target and incidental species, whereas TCm(t) considers fixed and variable costs, as well as opportunity costs of labor and capital.
Direct employment generated by the fishery (EMP(t+DT) are expressed by:
where FISHm represents the average number of fishers per vessel type m.
The contribution of the fishery to seafood availability in rural coastal communities through time (ALIM,(t+DT)) is estimated by:
where ε is the proportion of the catch of species i and fleets m that will go to the export market, and thus 1-ε is the proportion that will be destined to the domestic market. Export earnings generated by the fishery throughout the year are estimated by:
where pij is the export price of the species idiscriminated by age/size j.
A more complete model should also include the processing sector (see Willmann & Garcia, 1985).
The information needed to estimate bioeconomic parameters and fit equations of the mathematical model can be summarized as follows (see also Seijo et al., 1991a, b, 1994a):
Information of biological parameters
Number of individuals by age/size.
Sex composition.
Natural and fishing mortalities by age/size.
Age/size specific estimates of the catchability coefficient q by gear or fleet type.
Individual growth function and sex specific length-weight relationship.
Age at first maturity, average fecundity, if possible, discriminated by age/size.
Information of fishing effort
Type of vessels and engine. Gear used by fishing fleet.
Average number of fishers per fleet type.
Number of vessels per port and fishing season.
Main fishing areas and estimated mean distance from port.
Average number of effective fishing days per trip, vessel type and fishery.
Average duration of a fishing trip.
Type(s) of bait utilized.
Catch information
Number of ports in which the target species is landed.
Total annual catch by species, fleet type, fishing season and trip.
Size composition of the catches.
When 2 or more countries share the stock, or if foreign vessels are authorized to fish,information about catch and fishing effort must be given by fleet.
Economic information: costs and revenues per vessel type
Returns by fishing trip.
- Average price paid to fishers for target and incidental species.
- Catch estimates of the target and incidental species throughout the year.
If the catch constitutes a significant proportion of the global supply of the target species, a price function (demand equation) must be estimated.
Costs of fishing effort: operation costs per trip, either on a nominal (e.g., fishing days) or effective (effective fishing hours) basis.
Variable costs
Oil and gas, ice, bait, food.
Vessel repair and maintenance, replacement of damaged gears.
Fixed costs
Depreciation of the vessel, engines and gear.
Interest payment from borrowed capital. Administration and insurance costs.
Opportunity costs of capital and labor. The former is defined by the net benefits that could have been achieved in the next best economic alternative of investment. Investment costs (boat, engine and gear) are required to estimate opportunity costs of capital, interest payment from borrowed capital and depreciation. Moreover, it is necessary to specify the average interest rate on loans and for equipment and harvesting expenses for the fishing sector and the bank interest rate on fixed deposits. The opportunity cost of labor is the income that could have been achieved in the next best economic activity available to fishers. Type of crew remuneration must be indicated to estimate opportunity costs, e.g.: (i) as a percentage of the catch per trip (shared system); (ii) as a daily salary; or (iii) as a daily salary rate in the next best option of employment available for the fishers.
Additional relevant information
Percentage of the annual catch that goes to: (a) the fishing community for subsistence purposes; (b) the local market; (c) further product processing; and (d) the foreign market.
Exportation prices of the target and incidental species.
The computer model can be developed in any structured programming language, as QuickBASIC, Visual BASIC or C. The general structure of the program includes the following steps:
Initialization phase
Values are assigned to model parameters and constants.
State and rate variables are initialized.
Time is initialized: T=0.
Specification of the simulation run
Length of the simulation experiment.
Time increment (DT).
Time for printing output.
Execution phase
Time is updated: T=T+DT
State variables are computed.
Random variables are generated.
Rate variables are computed for time T.
State and rate variables are printed.
Return to (g) if the simulation experiment is not completed.
Stability analysis consists in determining an appropriate value for the time increment DT, in order to carry out stable simulation runs of differential equations (e.g., distributed delay model). If the Euler numerical integration method is used, the necessary condition for stability requires DT to be in the interval:
2MIN[Dn]>DT>0 (4.18)
where Dn=DEL/g and MIN [Dn] is the delay constant (Manetsch&Park, 1982).
As the simulation model contains feedback processes in the population structure, it is required that:
1/c >DT>o (4.19)
where c = 1/Dn. Therefore, DT must be within the interval [1>DT>0]. In order to reduce the numerical integration error below 5%, DT might be 0.01.
Sensitivity analysis allows observing variations in model performance under marginal changes (20% increments/decrements) in model parameters (e.g., natural mortality) and initial conditions of rate and state variables (e.g.,initial biomass, fishing effort levels).
A simulation model is validated if it provides a correct representation of the real system. Validation procedures vary from qualitative to quantitative tests, depending on the amount of data available and system understanding (Seijo, 1986; Rykiel, 1996). Concerning the latter, parametric and non-parametric statistical tests (e.g., Kolmogorov-Smirnov test) could be used to test the null hypothesis of equality of distributions of actual and simulated variables, such as catch, effort and stock biomass (see Defeo et al., 1991). Discussion of model structure and results with resource specialists, system scientists and resource managers should also be undertaken (see also Payne, 1982; Graybeal & Pooch, 1980).
A multistage validation procedure should be applied to critical stages in the model building process, i.e., design, implementation and operation of the model (see Rykiel, 1996 for details). The model can be considered validated if it accurately predicts the actual data set (replicative validation) or some new or unexamined set of observations (predictive validation) (Power, 1993). Even though validation is often an essential testing process to evaluate model performance, for those models directed to “describe or systematize knowledge or to develop theory, validation is unnecessary and irrelevant” (Rykiel, 1996).
Once the bioeconomic model is built, it is necessary to conduct simulation experiments directed to evaluate the bioeconomic impact of different management strategies. This includes for example: (i) 20% increment in fishing effort of artisanal/mechanized fleets; (ii) extension (reduction) of a closed season; (iii) implementation of a minimum legal size; or combination of the above and other strategies.
A bioeconomic approach to fisheries management with multiple criteria, introducing a nonlinear optimization algorithm, could be integrated to reconcile economical and ecological criteria, in many cases conflicting with each other (see Chapter 6).
Suppose a sequential shrimp fishery is composed by an artisanal fleet that captures juveniles and young adults, and a mechanized fleet that essentially captures the adult component of the stock. A bioeconomic analyst has been requested to evaluate fishery performance under the following management instruments: minimum size at first capture, length of the fishing season and specification of the preferred months for a closed season. A limitation in the number of vessels is assumed. It is also assumed that the mechanized fleet is not versatile and therefore will pay for the fixed costs during the closed season. Since recruitment periods above the average (20% of the observed years) could occur, the manager is considering the possibility of increasing by 25% the number of licenses for both fleets. Main fishery parameters are shown in Table 4.1.
Short and long run dynamics. In the short-run, the model represents dynamically the seasonal distribution of recruitment (Fig. 4.3a) and fishing effort (Fig.4.3b). With a fishing season of 10 months, beginning in the third month, the effect on the spawning stock is reflected in the relative magnitude of both recruitment periods: in the former, the maximum number of individuals is 13.2 millions, whereas in the second it is 7.8 millions. If the start of the fishing season is changed, the success of both recruitment periods could be affected. Analogously, the recruitment magnitude for both periods would be affected if the length of the fishing season varied. The above results allow us to combine and evaluate the effects of different starting dates and durations of the fishing season, given a known distribution of recruitment peaks, in order to maintain and, if possible, to increase the economic rent that the fishery could generate through time.
| Parameter/variable | Value |
|---|---|
| Maximum age of the species | 24mo |
| Age at first maturity | 6mo |
| Age at first capture for fleet 1 (mechanized) | 3mo |
| Age at first capture for fleet 2 (artisanal) | 2mo |
| Average fecundity | 1072000eggs |
| Growth curvature parameter | 0.27/mo |
| Growth parameter to | -0.03/mo |
| Asymptotic length | 245 mm |
| Asymptotic weight | 140g |
| Area swept per day fleet 1 | 1.41Km2 |
| Area swept per day fleet 2 | 0.25Km2 |
| Length at 50% gear retention, fleet 1 | 170mm |
| Length at 50% gear retention, fleet 2 | 150mm |
| Length at 75% gear retention, fleet 1 | 175mm |
| Length at 75% gear retention, fleet 2 | 170mm |
| Stock distribution area | 6405Km2 |
| Fleet types | 2 |
| Fleet 1 dynamic parameter | 0.000001 |
| Fleet 2 dynamic parameter | 0.0000015 |
| Maximum observed recruitment | 76900000ind |
| Length of the fishing season | 10 mo |
| Start of the fishing season | Mo3 |
| Month of first recruitment | Mo4 |
| Month of second recruitment | Mo11 |
| Length of simulation run | 3yr |
| Order of the distributed delay | 2mo |
| Fishing days/mo fleet 1 | 20 |
| Fishing days/mo fleet2 | 12 |
| Time delay of boats entry | O d |
| Average time delay of recruits transit | 2mo |
| Unit cost of fishing effort, fleet 1 | 65000 ($/vessel/yr) |
| Unit cost of fishing effort, fleet 2 | 3000($/vessel/yr) |
| Minimum price (90–100/ib) | 1500 ($/tonne) |
| Slope of the curve of prices/length | 700($) |
| Incidental weighted income | 318 ($/tonne) |
Fishery yield decreases through time for both fleets (Figs. 4.3c, d). However, the relatively low unit cost of artisanal fishing effort allows it to generate economic rent. Figures 4.3e and 4.3f show dynamic fluctuations of costs and benefits as perceived by each fleet. The accumulated rent throughout the fishing season will determine, under open-access, potential long-term changes in the number of vessels. For both fleets, the number of vessels begins to decrease from the second year of the simulation run as a result of negative profits. This shrimp fishery reflects a clear trend towards overexploitation and overinvestment.
