Models developed in Chapters 2 and 5 satisfy the dynamic pool assumption and thus
homogeneity in the spatial distribution of the stock and fishing effort exerted (*e.g.* pelagic
species). In sedentary resources (*e.g.*, mollusks), the above assumption is not always valid for
the following reasons (see Hancock, 1973; Caddy, 1975; Orensanz & Jamieson, 1998):

Populations are patchily distributed (Elliot, 1977).

The low or null mobility precludes species redistribution over a fishing ground, by filling gaps in patches resulting from a sequential distribution pattern produced by the heterogeneous allocation of fishing effort (Hancock, 1979; Caddy, 1989a, b; Conan, 1984; Orensanz

*et al*., 1991). Thus,*CPUE cannot be used*as an unbiased index of abundance.Growth, mortality and recruitment parameters are extremely dependent on environmental conditions even between small distances (Orensanz, 1986; Defeo, 1993a).

Two alternative approaches seem to be appropriate:

To relax the assumption of spatial homogeneity of stock distribution. Thus, extensive fishing grounds with variability in environmental conditions and related abundance heterogeneity, growth and mortality patterns, can be divided into smaller areas that can be considered as independent units (Caddy, 1975; 1989a; b; Hall, 1983; Sluczanowski, 1984; 1986). Hence, for a stock showing a continuous geographic range of population characteristics, a useful approach is to consider it as composed of several discrete subpopulations which can be studied independently, and the predictions integrated afterwards (Seijo

*et al.*, 1994b, c).To develop a comprehensive approach, integrating effects of different environmental regimes on the spatial structure of the population, spatial heterogeneity of fishing effort, biological interactions, and the implications of economic factors and human attitudes (behavior of resource managers and users) (Seijo & Defeo, 1994a; Seijo

*et al.*, 1994c). In view of the uncertainty involved in the above factors (Lewis, 1982; Anderson, 1984; Sissenwine, 1984a, b; Seijo, 1987), it is preferable to build stochastic models instead of deterministic ones, in which point predictions are replaced by probability functions (Botsford, 1986; Seijo, 1986; Fogarty, 1989; Seijo & Defeo, 1994a; Seijo*et al*., 1994b).

Studies on fishing effort dynamics have been focused on long-term decisions of fishers,
emphasizing the estimation of rates of entry and exit to the fishery and the characterization of
general patterns of allocation of fishing intensity (Smith, 1969; Clark, 1976; Mangel & Clark,
1983; Emerson & Anderson, 1989). However, it is in the short-term that fishers make their
spatial decisions: after deciding to go fishing and selecting the target species they decide
where to fish (Hilborn & Ledbetter, 1979; Bockstael & Opaluch, 1983; Defeo *et al*., 1991; Eales
& Wilen, 1986). The latter decisions are of utmost importance, because, in contrast with
traditional views which focus on excess of fleet capitalization, dissipation of economic rent can
arise primarily as a result of today's excess of fishermen's movements in response to
yesterday's spatial catch rate variations.

In this Chapter, spatial considerations in modelling fisheries are introduced, notably the distance from ports to fishing grounds, in order to further understand short-run decision-making of fishers in their allocation of fishing intensity. We also add a degree of complexity in the interdependencies discussed in Chapter 3, incorporating the spatial dimension in the analysis, with special reference to fisheries where the dynamic pool assumption is not satisfied.

Modelling the spatial dynamics of fisheries in the short-term allows us to better understand the intertemporal allocation behavior of fishing effort and thus to develop adequate management strategies. Some strategies of spatial allocation of effort have been indentified:

**Proportional**allocation according to the spatial abundance of the resource (Caddy, 1975).**Sequential**allocation to those patches of greatest abundance (Hilborn & Walters, 1987).**Random search**(Hilborn & Walters, 1987).**Free distribution**of allocation of fishing intensity (Gillis*et al.*, 1993).**Proportional allocation to**(Defeo*et al.*, 1991; Seijo*et al.*, 1994b):The quasi rent of variable costs (including transfer costs resulting from moving from port to alternative fishing grounds).

The

*friction of distance, i.e.*, non-monetary costs associated to distance.The probability of finding the target species at profitable levels.

With the above considerations in mind, we present two models developed by Seijo *et al.*
(1994b), with different degree of complexity in spatial resolution and requirement of
bioeconomic information.

**ALLOC** is a short-run bioeconomic model that represents interdependencies between two fleet
types (*e.g.* artisanal and mechanized/industrial) coming from different ports of origin that
capture the same target species in alternative fishing grounds (Seijo *et al*., 1994b).

Spatial and temporal dimensions are incorporated in this discrete dynamic model by specifying:
(1) *CPUE*'s by fishing grounds; (2) distances from ports to different fishing grounds; and (3) a
function that determines the short-run spatial allocation of fishing intensity. The latter is
modelled in **ALLOC** as a function of the distances from ports to alternative fishing grounds, the
net revenues perceived in the previous trips, and the probability of finding the target species in
each ground at profitable levels.

The spatial dynamic allocation of fishing intensity (*i.e.*, effective effort per unit of area, sensu
Gulland, 1983: 44) of vessels type *m* from port *h* among fishing grounds *k* in time *t(f _{khm}(t))* is given
by:

f_{khm}(t)=SAE_{khm}(t)DAYS·V_{hm}(t) **(6.1)**

where *SAE _{khm}(t)* is a (0,1) optimum spatial distribution of fishing vessels;

where *P _{k}* is the probability of finding the target species at profitable levels in alternative fishing
grounds

Additional to the transfer costs from the port of origin to alternative fishing grounds, one element
considered in this model as a key explanatory factor for the spatial behavior of fleets is the *friction of
distance* (Isard & Liossatos, 1979, Seijo *et al.*, 1994b). Artisanal boats with limited autonomy are
assumed to give substantial weight to non monetary costs derived from e.g., insecurity associated
with fishing far away from port, as well as other environmental and cultural factors that assign non-monetary
costs to distance. Thus, **ALLOC** implicitly incorporates in *D _{kh}^{φm}* all those non-monetary
factors associated with distance. The weight of this parameter for a specific fleet type

To observe the short-term fleet performance, an estimation of the quasi rent received by an average
vessel type *m* from port *h* is needed:

*quasi _{πkhm}(t) = TR_{khm}(t) - VC_{khm}(t)*

where *TR _{khm}(t)* are the daily total revenues received by vessel type

*VC _{khm}(t)* are estimated considering: (a)

*VC _{khm}(t) = ωq_{m}B_{k}(t) + θD_{kh} + OVC_{m}*

In the short-run, vessels will remain in the fishery if *TR _{khm}(t)≥ VC_{khm}(t), i.e.*, the quasi rent of
variable costs ≥ 0. Considering the

When the distance is irrelevant in terms of transfer costs from port to alternative grounds (θ=0) and the friction of distance is zero (φ=0),*Case 1: artisanal littoral fisheries.**i.e.*, there are not non-monetary costs to fishing the different grounds, and the same probability of finding the species at profitable levels exists*(P*, the resulting_{1}=P_{2}=…=P_{k}, where k = 1…n)*SAE*distribution is proportional to the spatial variations in stock abundance._{khm}(t)Where transfer distances to alternative grounds from port are relevant (θ>0), but non-monetary costs are negligible because of the easy operation and navigability in protected fishing grounds (φ=0), and there is the same probability of finding the species at profitable levels*Case 2: fisheries in protected coastal zones.**(P*, where_{1}=P_{2}=…=P_{i}*i = 1…n)*, the*SAE*distribution is then proportional to spatial variations in the quasi-rent of variable costs._{khm}(t)When θ>0, the friction of distance is substantial (φ=0), and*Case 3: fisheries in exposed coastal zones.**P*, the_{1}=P_{2}=…=P_{k}*SAE*distribution is proportional to the spatial distribution of the quasi rent and inversely related to the friction of distance from port to the alternative grounds._{khm}(t)When transfer distances from port to alternative grounds are relevant in monetary terms (*Case 4: fisheries with high information costs.**D>0 y θ>0*), friction of distance (φ>0) occurs, and there is heterogeneous probability of finding the resource at profitable levels (*P*), the distribution of_{1}# P_{2}#…# P_{k}*SAE*is proportional to the spatial distribution of the quasi rent and to the probability of finding the resource at profitable levels, and inversely related to the friction of distance from port to alternative fishing grounds._{khm}(t)

CASES | Monetary cost of distance D | Friction of distance (φ) | Probability of finding the resource (P) | Expected distribution |
---|---|---|---|---|

Case 1 | D=_{1}D=…=_{2}D θ=0_{k} | φ=0 | P_{1}=P_{2}=…=P_{k} | Proportional to the abundance |

Case 2 | D∞_{1}D…_{2}D θ>0_{k} | φ=0 | P=_{1}P=…=_{2}P_{k} | Proportional to the quasi rent |

Case 3 | D∞_{1}D…_{2}D θ>0_{k} | θ>0 | φ=0 | P=_{1}P=…=_{2}P Proportional to the quasi rent and 1/D_{k}^{φ} |

Case 4 | D∞_{1}D…_{2}D θ>0_{k} | φ=0 | P=_{1}∞P=…=_{2}∞P_{k} | Proportional to the quasi rent, 1/D and ^{φ}P_{k} |

The cumulative economic rent received by vessel type *m* from port *h* over the fishing season is
estimated by:

where *TR _{khm}(t)* are the total revenues received by vessel type

*TR _{khm}(t)* and

*TR _{khm}(t) = (q_{m}B_{k})(t)Ptar + Cine Pinc) f _{khm}(t)*

*TC _{khm}(t) = FC_{m}V_{hm} + VC _{khm}(t) f _{khm}(t)*

where: *Ptar* and *Pinc* are, respectively, the average price paid per kg of target and incidental
species; *Cinc* is the average incidental catch per fishing trip of vessel *m*; *q _{m}* is the catchability
coefficient of vessel type

The dynamic catch rate is estimated as:

*Y _{khm}(t)=(q_{m}B_{k}(t) + RV_{khm}(t)) f _{khm}(t)*

where the random variable *RV _{khm}(t)* is generated with an exponentially autocorrelated
probability density function, using the observed variance of daily catch for vessels type

The catch per unit of effort of vessel type *m* in ground *k* in time *t* (*CPUE _{km}(t))* is estimated by:

*CPUE _{km}(t)=q_{m}B_{k}(t)*

Biomass for each fishing ground *B _{k}(t)* is estimated by spatially disaggregating the logistic
function:

where *r _{k}* and

A time lag frequency distribution representing the rate at which small-scale fishing fleets enter the
seasonal fishery or shift from one fishery to another, is frequently observed (Seijo *et al*., 1987).
Hence, time lags in the short-run dynamics of fishing fleets are incorporated in **ALLOC** to relax the
assumption that every boat that eventually will fish for the seasonal target species, will enter the
fishery exactly the first day of the season. This aspect is incorporated through the distributed DELAY
model (Manetsch, 1976) based on the Erlang (Gamma) probability density function (Seijo, 1987).
Thus, the seasonal dynamics of vessel type *m* from different ports (*V _{hm}(t))* can be described by a
distributed delay function of order

where *V _{hm}* is the input to the delay process (number of vessels from port

The cumulative spatial allocation of fishing intensity along the fishing season of fleets *m* from port *h*
in ground *k (fa _{khm}(t))* is given by:

The spatial allocation of fishing intensity among alternative fishing grounds is a function of the probability of finding the target species, the net revenues obtained from different fishing grounds in previous trips, and the distances to alternative fishing grounds from different ports of origin.

*CPUE*is a function of q and the resource biomass in each fishing ground over time, and will tend to decrease as fishing intensity and the corresponding exploitation rate (*sensu*Gulland, 1983) increase at ahigher rate than the natural biomass growth rate in each site.There is no cross migration or immigration among fishing grounds.

Uncertainty in the spatial allocation of fishing effort over time is represented by random variables (0, VAR) with an exponentially autocorrelated probability density function.

Fishers price-taking behavior (Clark, 1985) was assumed for prices of target species and incidental catch. This assumption could be relaxed if a robust time series of fish price, quantity harvested and real income of resource consumers were available.

Costs of inputs needed per fishing trip remains constant along the fishing season.

When bioeconomic information is available, some of the assumptions made in **ALLOC** can be
relaxed to achieve a better representation of the population dynamics and of the spatial
resolution of the model. **CHART** is an age structured, dynamic spatial model that incorporates
the main features of the **YRAREA** model (Caddy, 1975) with the short and long run dynamics of
the fishing fleets in a bioeconomic and geographic context (Seijo *et al.*, 1994b).

The initial population structure is estimated by the maximum observed *CPUE* on each ground
(*CPUEmax _{kl}*), where

where *B _{kl}(t)* is the biomass in cell

Total number of recruits for each year is seasonally distributed through the use of a distributed delay function, which specifies the peak recruitment month and its period (in months) of occurrence. The spatial distribution of recruitment patches is generated by randomly selecting the recruitment patch center coordinates, and then calculating recruitment density out from patch center by using the bivariate normal distribution (Caddy, 1975):

where: *R _{kl}(t)* is the number of recruits located in the patch center of the cell

where *TSG* and *TTG* are, respectively, the length and width of the cell, estimated by a
previously specified spatial resolution (*e.g.* degrees, minutes and seconds). Moreover:

i = 1, 2, 3,…, SG | and | SG = number of rows in the geographic grid. |

j = 1, 2, 3,…, TG | and | TG = number of columns in the geographic grid. |

y = SG/2 | and | z = TG/2. |

The biomass allocated in each cell *kl* is estimated by:

where *N _{jkl}(t)* is the number of individuals at age

*L _{j}* = :

*W _{i} = aL^{b}_{j}*

where a and b are constants.

The spatio-temporal distribution of population structure (*N _{jkl}(t)*) is estimated by:

where *M* is the instantaneous natural mortality rate and *F _{jkl}(t)* is the age specific fishing mortality
rate in site

*CPUE* in site *kl* in time t (*CPUE _{kl}(t)*) is estimated by:

where the age specific catchability per unit of area (*qua _{j}*) is estimated using the area swept
equation of Baranov (1918) and the logistic selectivity curve (Sparre

where: *pcap* is the probability of capture by the gear; area is the area swept per day; *SEL _{j}* is
the age-specific gear retention (see Chapter 4); and

Short-run spatial allocation of fishing intensity (*e.g.* within the fishing season) is estimated as in
**ALLOC**, with one modification: parameter *P _{k}* is substituted by the perceived risk of fishing on
the fishing ground with coordinates

Spatial allocation of fishing intensity of vessels from port *h* in site *kl* in time *t*, (*f _{klh}(t)*) is given by:

*f _{klh}(t) = SAE_{khm}(t)DAYS V_{h}(t)*

where: *V _{p}(t)* is the number of vessels from port

where *Risk _{kl}* is the perceived risk of fishing in alternative grounds

The number of boats from port *h* present in the fishery over time (years), is modelled by
spatially applying Smith (1969) equation:

where is a positive constant. Equation (6.25) states that in the long-run and under open access conditions, if the sum of net revenues obtained from all fishing grounds is: (i) positive, there will be entry to the fishery; (ii) negative, exit is expected to occur; and (iii) zero, the fishery is at bioeconomic equilibrium, and there will be no stimulus for exit/entry.

Recruitment does not occur at a single point in time, but within a period whose distribution is modelled by the distributed delay model.

Extension of patches is defined by the density variance of a bivariate normal distribution.

Migration, dispersal and movement are not considered.

Natural mortality is constant for all ages.

Spatial allocation of fishing intensity is a function of ground-specific variations in the: (a) quasi-rent, (b) risk of fishing, and (c) distances from different ports.

Distances between ports and fishing grounds are estimated geographically without consideration of possible obstacles.

For the long-term fleet dynamics, an estimated time lag of 6 months is assumed for vessel entry/exit to the fishery, whereas in the short-run, the user is requested to provide the observed average time lag of entering the seasonal fishery.

Uncertainty of daily catches in each ground is modelled by an exponentially autocorrelated probability density function.

Price of target species and costs of fishing effort inputs are constant and exogenously determined.

The interested reader can conduct simulation experiments with these spatial models, which are
contained in the package SPATIAL published by FAO (Seijo *et al.*, 1994b).

Seijo & Defeo (1994a) conducted a short and long run analysis of the yellow clam *Mesodesma
mactroides* fishery of Uruguay. For this end, a dynamic, non-linear and stochastic model was
built to represent the behavior of resource and fishers through time. The short term space-time
distribution of fishing effort was represented as a function of the dynamic catch per unit effort in 4
alternative grounds, the distance from different localities, the perceived probability of finding the
species at profitable levels, and the corresponding net revenues obtained in the previous fishing trip
(see also Defeo *et al.*, 1991). The long-term dynamics of the fishery considered time lags in
spawning, hatching and recruitment processes, as well as in fishermen's entry and exit to the
fishery. The distributed delay model was applied for this purpose. A stock-dependent,
overcompensatory (Ricker) function characterized the dynamics of the stock-recruitment relationship
(Defeo, 1993a). Recruitment was modelled by using the distributed delay model (gamma function
of order 3) in accordance with a recruitment period of 4 months. A dynamic and density-dependent
natural mortality was estimated as a function of the initial density of recruits and salinity (which is
substantially affected by the outflow of freshwater creeks) on each fishing ground. Effort dynamics
in the long-run (number of fishers from locality *h* present in the fishery over time) was modelled by
spatially applying the Smith (1969) equation and the distributed delay model (see Seijo & Defeo,
1994b). Price of species, opportunity costs of labor and operation costs were also included in the
model.

Simulation experiments were conducted with three management strategies to observe the dynamic impact on performance variables. The following three policy vectors were simulated and compared with the base run representing the current management scheme (minimum size restriction of 50 mm, and 50 fishing licenses):

The current management scheme and a catch quota of 50 tonnes/year.

A minimum harvestable size of 43mm (size at sexual maturity) and 150 fishing licenses.

Minimum size restriction of 50–mm and 150 fishing licenses.

The model reproduced well the temporal variability in abundance of recruits (Fig. 6.1a) and adults (Fig. 6.1b), and was statistically validated. Adults markedly increased from 1988 on, as a result of a human exclusion experiment conducted between March 1987 and November 1989 (see Defeo, 1993a; 1996, 1998; Defeo & de Alava, 1995 for details). Recruitment was inhibited in 1989 and 1990, as a result of overcompensation due to high adult densities. This fact has important management connotations: simulation showed that, as a result of a prolonged closed season (more than 2 years), high adult density could inhibit recruitment success and hence the magnitude of the stock available for fishing two years later (see Fig. 6.2).

**Figure 6.1** *Mesodesma mactroides*. Observed vs. estimated densities of (a) recruits and (b) adults.

The fishery showed a strong seasonal component. The model (Fig. 6.2) adequately reproduced the
marked increase in catch and fishing intensity during summer and spring. A decline in the demand
for local market consumption and a reduced availability of the resource because of downward
migration towards the infralittoral explained the diminished catches in autumn and winter (Defeo,
1989). The model also satisfactorily explained the marked spatial variability in effective fishing
days, catches and fishing intensity (Fig. 6.3a; see also Defeo *et al*., 1991), as well as spatial
variations in the economic rent. Fishermen spatially allocated fishing effort proportionally to
spatial variations in stock abundance, *i.e.*, the central zones 2 and 3, with greater stock
abundance presented highest yield values (Fig. 6.3b) and fishing intensity. The model correctly
represented variations within and between years. As a result of the above mentioned trends and
the marked seasonality of the fishery, net revenues were also greater from the central grounds,
peaking in spring and summer (see also Defeo *et al*., 1991 for a detailed analysis).

**Figure 6.2** Seasonal variations in simulated fishing intensity by fishing ground from 1983 to 1992 for the
yellow clam fishery of Uruguay.

**Figure 6.3** *Mesodesma mactroides*. Comparison of observed vs. simulated: (a) seasonal fishing effort;
and (b) catches by fishing ground (summer 1985).

According to the results obtained, a spatial management scheme based upon a rotation of areas was proposed, taking into account periods of unequal demand for the product and temporal variations in stock accessibility and abundance. Because of relatively low total costs, the bioeconomic equilibrium is estimated at high levels of fishing effort, which in turn increases the risks of exhausting this highly vulnerable sedentary species.