William S. Hearn^{1}
CSIRO^{2} Division of Fisheries, Marine
Laboratories
G.P.O Box 1538, Hobart, Tasmania, 7001, Australia
Alexander Mazanov^{3}
School of Biological Science
University of New South Wales
PO Box 1, Kensington
New South Wales, 2033, Australia
^{1} Some of this work formed part of the principal author's doctorate awarded by University of New South Wales.^{2} Commonwealth Scientific and Industrial Research Organisation.
^{3} An earlier draft of this manuscript was prepared before Dr A. Mazanov's untimely death.
ABSTRACT
A tagrecapture experiment is suggested for determining the effect that a change in the fishing intensity of one fishery will have on its own catch and on the catch of another. The results are expressed as simple formulae in terms of one control variable: the proportional change in the fishing intensity of the altered fishery. There is no requirement to develop a complex parametric model of the dynamics of the population and fisheries or to collect comprehensive effort data. For the yearclass being studied, the ideal experimental design requires that a representative sample of the fish caught by one fishery be tagged, and all tagged fish recaptured by the same fishery be rereleased. In the second fishery (or fisheries), no fish need be tagged or rereleased. Alternative equivalent procedures to rereleasing tagged fish are described. The dynamics of the fish population and fisheries are assumed to be independent of the size of the population, i.e., a linear dynamic system is postulated. The formulae are proven for two population models for which, (i) two fisheries catch fish from a common, homogeneously  mixed population, and (ii) two fisheries catch fish from separate grounds (fish may migrate between grounds). In some situations the formulae are sizebased. Experimental design and formulae are developed to assess interactions when the fishing intensities of two or more fisheries are to be changed or a new fishery is proposed.
1. INTRODUCTION
Oceanic fish are mostly harvested as common resources. Many fish stocks are harvested by fisheries with different gear types, markets, seasons and fishing grounds. The degree to which the catch of one fishery decreases the catch of another at a later time is of concern to scientists, managers and fishers. With such knowledge, they may predict the effect that opening (or closing) a specific fishery may have on the catch of another fishery, which could assist managers prevent or resolve conflicts among fishers and optimise the exploitation of the resource.
The problem of fisheries interactions may be addressed in a number of ways (Kleiber, 1994). One approach is to develop parametric models of the dynamics of the fish population and associated fisheries. The parameter values are then estimated by fitting the model to detailed catch, effort and/or tagrecapture data (e.g., Sibert, 1984; Hilborn, 1990; Hampton, 1991; Kleiber and Fonteneau, 1994; Salvado, 1994; Sibert and Fournier, 1994; Hampton et al, 1995; Mullen, 1995; Sibert et al., 1995).
Another approach is to study statistical correlations, between different fisheries, of timeseries of fisheries statistics (e.g., Fonteneau, 1986; Suzuki, 1988; Bayliff, 1994; Medley, 1994). However, the correlations observed do not necessarily indicate a causal relationship, given the relatively short length of most fisheries catchandeffort timeseries.
Results from data analyses can also be incorporated into computer simulation models that estimate fisheries interactions under a variety of reasonable assumptions (e.g., Kleiber and Baker, 1987; Bertignac, 1995; Kleiber and Fonteneau, 1995). The range of possible results and interpretations of such an approach are often too wide, although the results are frequently useful for identifying the most critical areas for further study.
An experimental tagging approach developed by Majkowski et al. (1988) simulates the proposed change in one fishery by tagging and releasing fish in that fishery and estimating the effect on the catch of another from the biomass of the second fisheries recaptures. The basic concept behind this method is that each fish tagged in one fishery represents a fish which another fishery is potentially deprived of. If all fish caught by the first fishery are tagged, or the proportion tagged are known, one can determine the biomass of fish that the second fishery loses of due to the existence of the first fishery. This approach, unlike other methods, does not require the development of a population model, estimates of many highly correlated parameters, complex analyses or detailed catchandeffort data; however, its experimental requirements are demanding. Whilst our method differs from these in many aspects we nevertheless follow their basic approach.
Majkowski et al. (1988) used catch quota as the control variable which, for their method, implicitly requires that the age distribution of the change in catch of one fishery be the same as its present catch  an impractical condition for most fisheries, unless a total closure is contemplated. In other words it is assumed that changing the catch of that fishery does not affect the age distribution of its catches (the zero catch excepted). To avoid this problem, we use effort as the control variable. As a consequence our corresponding formulae are different from theirs. They also assumed the population to be steadystate before the change and for there to be a transition period (sometimes substantial) before the population reaches a new steadystate  a very difficult condition to meet. We consider a particular yearclass, so our approach is essentially a yieldperrecruit analysis, although we show how the formulae apply to the steadystate case.
The formulae of Majkowski et al. (1988) express only the change in one fishery in terms of the change in the other (which is unknown) and the tagging information. We also develop corresponding formulae for this, as well as those that express the change in catch of each fishery in terms of the present catch of one fishery and the tagging information.
The mathematical development of the method in Majkowski et al. (1988) assumes a homogeneouslymixed population, although they applied it to an example in which the two fisheries are spatially distinct. Fish migration and spatial differences between fisheries are important characteristics of many fisheries interactions  the analytic and experimental techniques for estimating them are still a matter of active research. We model the homogeneous case, as well as a more complex population structure in which two fisheries operate on separate fishing grounds. In the latter case tagging in one ground only is needed (unlike parametric models which require tagging in both grounds, as well as detailed catch and effort statistics).
The experimental design described in the present paper requires that a representative sample of fish harvested in one fishery is tagged and released. (Note most analyses of tagging experiments assume a representative sample of the population is tagged.)
The approach requires that some fishers rerelease tagged fish they recapture. Compliance with this requirement is unlikely, but analytic methods and experimental approaches are proposed as alternatives.
2. THEORY
2.1 General Requirements
Consider two separate fisheries, X and Y, catching fish from the same or different fishing grounds. Management may be considering reducing (or increasing) the fishing intensity of X by a proportion e and wish to assess the resultant change in Y's catch of a particular yearclass. The fishing intensity of Y is assumed to remain unchanged. The more general objective is to estimate the effect on the catches of X and Y of changing the fishing intensities of both fisheries.
It is deemed that the following assumptions or conditions apply to the fishery, population and tagging experiment:
1. The rates of growth, natural and fishing mortalities, and migration are agedependent and independent of population size.Conditions 1 and 2 are implicit in most yieldperrecruit and interaction analyses. If the population and fisheries were significantly influenced by environmental fluctuations, then tagging programs would need to be conducted over a number of years to average over the range of normal variation.2. Management can uniformly reduce the fishing intensity of fishery X by the desired proportion e, while keeping the fishing intensity of Y unchanged.
3. The normal conditions of tagging experiments apply: that tagging does not harm fish or change their growth or behaviour; that tags are durable, are not shed from fish and are all reported by finders (or the level of tag loss and nonreporting can be estimated); and that enough fish are tagged to allow statisticallyreliable results.
4. The agedistribution of the present catch of X is almost certainly required (see discussion on next condition). In some cases the total weight, or number, of fish caught by fishery X is needed.
The requirements of Condition 3 are difficult to meet, particularly that concerning the reporting of tags. The assumptions in tagging experiments and procedures for meeting them, or coping with noncompliance, are examined by Beverton and Holt (1957), Paulik (1961), Beverton and Bedford (1963), Hearn et al. (1991) and Pollock et al., 1994; 1995), and the literature on tuna and billfish tagging is reviewed by Bayliff and Holland (1986).
2.2 Experimental Design
The experimental method presented here is based on an idea originally proposed by Alien (1953) for estimating the optimum sizelimit for a fishery by equating the marginal yield to zero. Knight (1970) extended it to estimate the optimum size of a fishing fleet. More recently, Majkowski et al. (1988) further developed this approach to estimate the effects of interactions among fisheries.
We design a tagging experiment to estimate DK_{x} and DK_{y}, the corresponding potential changes in the biomasses of catches (or yields) from a specified yearclass by fisheries X and Y due to the proposed change in the fishing intensity of X. Under the existing fishing regime, the total number of fish caught by fishery X from the specified yearclass is C_{x}, having a biomass of K_{x} and a total biomass K_{y} of fish is caught by fishery Y. Apart from the need to comply with Condition 3, the experimental design requires two additional conditions:
5. Randomly tag N fish, having biomass W, of the selected yearclass from the catch of fishery X. Tagging should be conducted over the entire period and geographical range that fish are susceptible to being caught by X, which may be for several years.Sometimes it may be difficult to determine which yearclass a fish belongs to, particularly if many yearclasses are harvested by fishery X. As illustrated in Appendix C, mathematical adjustments would need to be made if the age distribution of fish tagged was different from those caught by fishery X. Condition (5), in effect, requires that the age (or length) distribution of the catch of X be estimated by some means, i.e., implies that the first part of Condition (4) be met.
6. Fishers in fishery X are required to rerelease (unharmed) a proportion e (the proposed proportional reduction in the fishing intensity of X) tagged fish they recapture (akin to the requirement of Majkowski et al. (1988) which requires that all fish recaptured by X be rereleased) and report recapture weights (or lengths) to scientists. (This condition will be later modified.) Fishers in fishery Y are to retain all their recaptured fish and return tags with records of recapture weights (and/or lengths) to scientists.Condition 6 is an experimental analogue of the thinning technique that is used in computer simulation studies, which generates a required mortality rate from another moreeasilygenerated mortality rate (see Lewis and Shedler, 1979; Devroye, 1986, p264). Therefore, the expected fishing intensity of X on tagged fish is reduced by a proportion e from its level on untagged fish. Without Condition 6 only interactions, due to marginal changes of fishery X, can be estimated by this approach. The fishing intensity by Y on the tagged fish is assumed to remain unchanged.
If all vessels and their attributes are identical, then an equivalent requirement to Condition 6 is for a fraction e of vessels belonging to X to rerelease all recaptures, while fishery X's other vessels retain all theirs. The assumptions allow the tagged fish to represent the entire fish population, both in number and biomass, for which the change in fishing regime is proposed. The biomasses of tagged fish caught and retained by X and Y are R_{x} and R_{y}, respectively.
Given these conditions, we will prove the following relationships first obtained in Hearn (1986) for a homogeneouslymixed population:





(1) 





(2) 
and 




(3a) 
or 




(3b) 
In order to intuitively justify the validity of Equations (1) and (2), consider the effect of removing a proportion e of vessels from fishery X. Let these vessels be fishery X_{1} and the remainder of X be X_{2}. This is equivalent to a proportional reduction e of fishing intensity if all vessels in fishery X are identical. We wish to assess the effects on the catches of X and Y of fishery X_{1} ceasing to catch and retain fish.
In a hypothetical experiment, suppose that all fish of a selected yearclass caught by X_{1} are tagged and released, i.e., a number N = eC_{x} of fish with biomass W = eK_{x} are tagged. Also suppose that tagged fish recaptured by X_{1} are all rereleased. Consequently fishery X_{1} does not retain any fish for sale and so the population of live fish, both tagged and untagged, become subject to fishing by fisheries X_{2} and Y. Thus all the tagged fish are those that fishery X_{1} would have caught and retained under the existing pattern of fishing, while the untagged fish would be the corresponding live population under the existing fishing regime. Hence, the tagged fish caught by Y represent the increase in the yield of Y, i.e., DK_{y} = R_{y}. The increase in the yield of fishery X is equal to the biomass, R_{x} of the tagged fish caught by X_{2} less the biomass W = eK_{x} foregone by X_{1}. So Equations (1) and (2) are intuitively justified.
In an actual tagging experiment a representative sample of fish caught by X would be tagged, and eC_{x}/N (or eK_{x}/W) is the factor needed to scale up the experiment to the size of the commercial catch. These arguments also apply if fishery X is defined to be that part of the fishing gear which catches fish less than a certain size, with fishery Y defined as the remaining part of the gear, i.e that which catches fish at or above that size. Therefore, questions regarding size limits can also be addressed by the application of Equations (13).
If both fishing and population are assumed steadystate (i.e., recruitment, the pattern and intensity of fishing, catches, fish growth and behaviour are consistent from year to year) an alternative approach is to tag a representative sample of fish from the catch of X during one year over all yearclasses. Here K_{x} and K_{y} represent stabilized annual yields of fisheries X and Y, rather than lifetime yields of a cohort. However, once the fishing intensity of X is changed there will be a transition period (up to the lifespan of the species) before the age structure of the population stabilizes again. The change in yield, DK_{y}, as derived from Equations (1) and (3 a), is an estimate of the change in the annual yield of Y between one stable population and the next. An advantage of this method, together with the steadystate assumption, is that, although it is based on an underlying agedstructured model, it does not require the age of any fish to be known, i.e., it is a sizebased method.
If the effect of a marginal change to fishery X were to be estimated, then e would be close to zero, and so an approximate estimate of DK_{y}, could be calculated from Equation (3a), with R_{x} and R_{y} being obtained from an experiment with no rereleasing. On the other hand, it may be planned to close fishery X altogether (i.e., e = 1). In this case R_{x} = 0 because fishery X would retain no tagged fish. Consequently, Equation (3a) becomes

(4) 
Equations (1) and (2) can be combined to give the total change DK_{T} in the catch of both fisheries, viz.

(5) 
2.3 Population Model 1  HomogeneouslyMixed Population
Consider two separate fisheries, X and Y, harvesting from a homogeneouslymixed population. The dynamics of the population of a particular yearclass are assumed to be dependent only on age. The number of fish, P(t), at age t can be described by

(6) 

(7) 
2.4 Population Model 2: Two Fisheries Harvesting in Separate Grounds
For convenience let the separate grounds that fisheries X and Y operate in be called grounds X and Y, respectively. The dynamics of a particular yearclass of fish can be described by


(8) 
and 




(9) 
Note that Equations (8) and (9) have been used previously to describe the dynamics of fishing acting on separate grounds, e.g., Beverton and Holt (1957) and Sibert (1984). The latter estimated natural and fishing mortality rates and transfer rates by analysing data from tagging in two fishing grounds and corresponding catch and effort information, but the rates of natural mortality and transfer were assumed constant. In the model represented by Equations (8) and (9), these rates are assumed to be dependent only on age, and for each ground the fish are assumed to be homogeneously mixed. Fish may be recruited to either or both fishing grounds, and r is the age that fish begin to be captured.
The solutions x(t) and y(t) of the differential Equations (8) and (9) have no general analytic expressions. However, if b(t) = 0 (i.e., fish migrate from X to Y, but not the reverse) analytic solutions can be obtained and the justification of Equations (1) and (2) follows a similar procedure to that given in Appendix A. If b(t) ¹ 0 a different approach is required, which is presented in Appendix B.
3. MODIFICATIONS
3.1 Increasing the Fishing Intensity of X
The experimental Condition 6 is designed for estimating the effect on catches of a specific reduction in the fishing intensity. However, modifying it to require all tagged fish that are recaptured by X be rereleased is more appropriate to general analyses. Importantly, the formulae obtained, when this condition is implemented, are applicable to assessing the effects of increasing the fishing intensity of fishery X.
If management plans a proportional reduction of e (which would be negative if an increase were planned) in the fishing intensity of X, the estimates of R_{x} and R_{y} would be


(10) 
and 




(11) 
The biomass of tagged fish recaptured for the first time by X is R_{x,}_{0}. In the experiment that assesses the effects of reducing the fishing intensity of X, the fraction, 1  e, of these fish would be retained by fishery X which is equivalent to adding (1e) R_{x,0} to R_{x} as in Equation (10). Also a fraction e of fish caught by X would be rereleased, which is equivalent to rereleasing all fish recaptured by X and giving each a value of e, which compounds each time the fish is rereleased.
Clearly, Equations (10) and (11) have direct intuitive meanings only for 0 £ e £ 1. However, they do give correct estimates of R_{x} and R_{y} for negative values of e, i.e., when it is proposed to increase the fishing intensity of X by a factor e. To illustrate this, consider a proposal to double the fishing intensity of fishery X. Here e = 1 and the interpretation of Equation (10) follows. A tagged fish recaptured by X for the first time has double its caught weight added to R_{x} (i.e., 1  e = 2), once for itself and once for an equivalent tagged fish that would have been caught if the fishing intensity of X had been doubled. However, this equivalent fish is still in the water so it is compensated for by rereleasing the tagged fish to the water with a value of minus one (i.e., e = 1). This explains why some terms in the expressions for R_{x} and R_{y} in Equations (10) and (11) become negative for negative values of e.
The implementation of the modified form of Condition 6 is preferred, as it allows a range of values of e (e £ 1) to be analysed, rather than one. Information is also obtained from more fish, which would tend to reduce the variance of the estimate of R_{x} and R_{y}. Additionally, it has the distinct advantage of simplifying rerelease instructions to fishers.
A significant and representative proportion b of fishers may reliably rerelease tagged fish without harm and record their weights (or lengths), e.g., if independent observers are on board. In such a case instruct other fishers not to rerelease fish and modify Equations (10) and (11) by replacing e^{i} with (e/b)^{i}, but leave unaltered the (1e) term in Equation (10).
The effect on the catch of fishery Y of starting a fishery in ground X, where none existed before, can also be assessed using this approach. Fish, with the same age composition as that intended for the catch of X, should be tagged during the period of its projected fishing season. Because no fish are presently being caught in ground X, then R_{x} = 0 and there can also be no rereleasing, so the effects of fishing in ground X cannot be correctly estimated from Equations (10) and (11). Nevertheless, if moderate yields from X are planned, a linear extrapolation of Equation (3a), with R_{x} = 0, would be approximately correct, i.e., Equation (4).
3.2 Changing the Fishing Intensity of Both Fisheries
Using simple interaction models, Beverton and Holt (1957) show that, for an overharvested fish stock, a unilateral reduction of the fishing intensity of X always increases the catch of Y, but increases the catch of X only if its stock is overharvested on a yieldperrecruit basis. To achieve an equitable share of the benefits of a reduction in fishing intensity, fisheries managers need to know the likely effects of changing the fishing intensity of both fisheries.
To achieve this objective, we suggest modifying Condition 5 by requiring that a random selection of fish caught by Y also be tagged. Like Condition 6, it is further required that a proportion g of all tagged fish recaptured by Y be rereleased, including those tagged in ground X. Similarly a proportion g of all tagged fish recaptured by X should be rereleased. The estimates of the changes in the catch of Y and X are:


(12) 
and 




(13) 
Note that Equations (12) and (13) can be extended to estimating interactions among many fisheries. To achieve this objective, a random selection of fish caught by each fishery, having biomass W_{m} (m = 1,2 ... n), is tagged. It is required that a proportion e_{m} of all tagged fish recaptured by fishery m be rereleased, including those tagged in other grounds. The estimate of the change in the biomass, DK_{j}, of the catch of fishery j is

(14) 
3.3 Substitutes for Fish that cannot be Rereleased
In very few fisheries would fishers be expected to reliably rerelease tagged fish without harm. Majkowski et al. (1988) proposed two methods for accounting for fish recaptured by fishery X that were not rereleased. One has subsequently been found to be invalid^{4}. However, their other proposal to tag and release equivalent fish for each one that is not rereleased is a viable solution. This would require a lowlevel tagging program during the period when tagged fish are liable to be caught by X.
^{4} The method discounted recaptures by fishery X from the number (or biomass) of fish released. While this method allowed for natural mortality, the spatial structure of the population was not taken into account.If one can not implement a substitute tagging program, the experiment can be simply conducted without requiring rereleases and then equivalent tagged fish can be randomly chosen to serve as the substitutes. If one of the "pseudo" substitute fish is recaptured it would be counted twice, once as a fish recaptured for the first time and again as a substitute for another fish. A variation of this technique is illustrated in Appendix C for a special case.
Or it can be assumed that the population is stable and a random sample of the catch of X over all ages is tagged during one season. An approximation to rereleasing may be to let the substitute fish be one that was tagged at about the recapture size (but not necessarily at the same time) as the fish that was not rereleased. This requires the assumption that temporal differences are not important.
For an homogeneouslymixed population it is readily shown from Equation (7) that for fish tagged at age t the proportion recaptured later by Y between ages t and t + dt (t £ t) is:
if there is no rereleasing. Comparing this equation with Equation (A8), which incorporates rereleasing, it can be seen that for every tagged fish recaptured by Y when there is no rereleasing there would berecaptured if the required fish were rereleased (note that a(t) = a(t)  eF_{x}(t), see Equation (A4)). Given a value of the natural mortality, one could approximately estimate F_{x}(t) by cohort analysis of the tagging data (see Bayliff, 1971). Alternatively, if the effort data of fishery X and an estimate of the catchability coefficient are available, one can estimate R_{y} without rereleasing tagged fish. A similar procedure would be applied if an estimate of R_{x} was required, but it would need to be multiplied by 1  e to allow for fish not retained by X. For fisheries in two grounds, these adjustments would be valid only while fish were in ground X (i.e., for some unknown period smaller than tT); therefore the values of R_{y} and R_{x} obtained in this way would be overestimated.3.4 Estimating Standard Errors
If DK_{y} is estimated from the first form of Equation (1) then, using the delta method (see Seber, 1973, equation 1.11 p9) with C_{x} and R_{y} independent and assuming the term involving Var(C_{x})Var(R_{y}) is negligible,

(15) 
However, if DK_{y} is estimated from the second part of Equation (1), then similarly

(16) 
Adjustments would need to be made to the variance formulae in Majkowski et al. (1988) for them to be suitable for Equations (2), (3a), (5), (12), and (13). The procedures used to adjust results when assumptions are not complied with may be too complex for an analytic expression of variance. In such a case either of the jackknife or bootstrap methods could be used, as all the expressions are algebraic.
If fish form large schools, a tagging vessel would tag fish from some of the schools that would be subject to fishing by X. As the integrity of schools may persist for some time, the assumption of independence between fish may not hold. In such a case a bootstrap approach with schools as the basic unit may be preferable.
4. DISCUSSIONS AND CONCLUSIONS
We have demonstrated that Equations (1) and (2) generally hold for both increasing and decreasing effort. Therefore, tagging the catch from one ground (X) is sufficient if assessment of the effects of changing the fishing intensity of X is the only knowledge required. (Methods that derive, with good precision, a full solution of the dynamics of the population require that fish also be tagged in Y, e.g Bertignac, 1995.) Additionally, our formulae require estimates of the number (C_{x}) or biomass (K_{x}) of fish caught by fishery X and its age composition, but do not need detailed effort statistics. Tagging in ground X and not in Y would make more resources available to tag substitute fish or put independent observers aboard vessels in ground X to rerelease fish. However, if a number of management options were being considered the experimental design would need to allow for all cases, which could be more expensive (e.g., tagging in both grounds).
The estimate of in Equation (3b) requires only tagging data and the age distribution of the catch, (i.e., it does not need estimates of C_{x} or K_{x}), and is less sensitive to e than the Equation (1) estimate, which could be important if e is not proportional to the measured change in effort (i.e., Condition 2 not met). Moreover, it does not require management to be committed to forecasting the absolute value of the change in catch of Y, which may be critical if recruitment is variable. It also addresses the question of what is the catch weight gained by fishery Y per unit of weight foregone by fishery X, which is particularly relevant when there is no existing fishery in ground X (in which case Equations (1) and (2) cannot be applied).
If the fish caught by X are young the lengthfrequency modes may be distinct, thus allowing reliable yearclass identification. If so Equation (3b) essentially becomes a sizebased formula. Also tagging in ground X is likely to be completed in only two or three years, thus increasing the feasibility of this approach. Moreover, if the average size offish caught by Y is much larger than those caught by X there exists a potential for a strong fishery interaction, i.e may be substantial, which will be of concern to fishers operating in fishery Y and to managers. Additionally, the natural mortality and migration rates of young fish may vary markedly as they grow older: our approach deals with this aspect. We thus have a situation where a fishery interaction may be substantial, for which this method is likely to be particularly appropriate.
Experimental design and formulae, which are analogues of Equations (1) and (2), were developed for assessing interactions where the fishing intensity of two (or more) fisheries may be changed. This requires tagging in two grounds and catch information from each. The results generated from this procedure are expressed in terms of two (or more) control variables, which allows managers to allocate a more equitable share between fisheries of the benefit or loss that flows from the changes. Unfortunately, there is no analogue of Equations (3a) for this experimental design.
One difficulty with our approach is that a change in effort is not likely to be uniform over time and space (Condition 2). For example, an area may not be fished at all or the fishing season may be shortened. If one could predict how the reduction in effort was going to be distributed, then the fishery could be considered as a number of subfisheries. The experiment would be designed accordingly and Equation (14) used for analyses.
In Equation (11) R_{y,0} would be the dominant term in R_{y} unless the catch of X is very high. This dominant term is not affected by nonrereleasing and/or nonreporting by fishery X. If a reduction in the fishing intensity of X is proposed, then R_{y,}_{0} is a lower bound for R_{y}, and hence a lower bound for DK_{y} can be estimated from Equation (1). Similarly, R_{x,}_{0} is a lower bound for R_{x} and so a lower bound for the ratio can be obtained from Equation (3b).
The use of Equations (8) and (9) with general agedependent coefficients allows Equations (1) and(2) to be established for a simple spatial population model, a considerable advance over Majkowski et al. (1988), who established their formulae only for homogeneous populations. One may wish to construct more elaborate population models that incorporate spatiallyvarying advection and diffusion as, for example, those of Sibert and Fournier (1994). In such a case the proof of Equations (1) and (2) could be established in a similar way to that in Appendix B, by examining the differential properties of an analogue of Equations (B 11) and (B 12).
An important element of this experimental design  rereleasing a certain proportion of fish  is difficult to implement, but alternative strategies are to, rerelease all fish, tag substitute fish, make mathematical adjustments, or use effort data in some cases. The value of placing independent observers on fishing vessels is not only to ensure the rerelease of tagged fish but also to allow assessments of the catch and/or tag nonreporting, which are difficult problems to address by other means.
Equations (13) essentially represents a Petersen approach (see Ricker, 1975, p77). The essential concept is intuitively simple, even if difficult to establish for a complex population model. Even so, a number of assumptions need to be met, particularly those relating to the tagging experiment. Other tagging approaches must also meet the normal tagging assumptions (Condition 3), but tagging a representative sample of the catch of X (Condition 5) and rereleasing recaptured fish (Conditions 6) are requirements specific to this method. An advantage of the approach is that it avoids the need to derive the agevarying coefficients implicit in Equations (6), (8) and (9).
The sum of DK_{y} and DK_{x} (= DK_{T}) from Equations (12) and (13) gives the total change in yield in terms of the change of fishing intensities e and g in fisheries X and Y. This allows curves of equal value of DK_{T} to be plotted against e and g for the information of managers. The application of Equation (14) would extend this approach to evaluate the effects of changing the fishing intensities of several fisheries.
The estimates of fisheries interaction obtained from such experiments could make a valuable contribution to helping managers evaluate proposed changes in the fishing intensity of different fishery components.
5. ACKNOWLEDGMENTS
We thank Drs K. Allen, W. Bayliff, C. Fandry, V. Lyne, V. Mawson, T. Polacheck and Mr G. Leigh for helpful comments on an earlier draft of the paper. We are also grateful for Dr J. Majkowski's suggestion that some recaptured fish be rereleased, and Dr D. Jackett for advice about linear differential equations.
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APPENDIX A HOMOGENEOUSLY MIXED POPULATION: PROOF OF EQUATIONS (1) AND (2)
The rate that fishery X catches fish is F_{x}(t)P(t), so from Equation (7) the number (C_{x}) and biomass (K_{x}) of fish caught by X from the selected yearclass are, respectively,


(A1) 
and 




(A2) 

(A3) 

(A4) 
If the changed regime, with its associated fish population represented by Equation (A4), operated for the entire lifetime of the selected yearclass, the biomasses, K^{*}_{x} and K^{*}_{y}, of fish caught by fisheries X and Y would be


and 





(A5) 
and 




(A6) 

(A7) 
Derivation of Equation (1)
By replacing t for r in Equation (A4) it can be seen that, of fish tagged from the catch of X at age t, the proportion recaptured later by Y between ages t and t + dt (t £ t) is expected to be

(A8) 

(A9) 

(A10) 

(A11) 
Derivation of Equation (2)
Of fish tagged from X at age t the proportion recaptured later by X between ages t and t + dt (t £ t) and retained is expected to be

(A12) 

(A13) 

(A14) 

which from Equations (A2) and (A5)
= DK_{x} + DK_{y} thus establishing Equation
(2).
APPENDIX B TWO FISHING GROUNDS: PROOF OF EQUATIONS (1) AND (2)
At age t the rate that X catches fish is F_{x}(t)x(t)) and hence the total number, C_{x} and biomass, K_{x} of fish caught by X from the selected yearclass are


(B1) 
and 




(B2) 

(B3) 


(B4) 
and 




(B5) 
Under the proposed new fishing regime, the corresponding biomasses caught by fisheries X and Yon their respective grounds are, similarly to Equations (B2) and (B3), expected to be


(B6) 
and 




(B7) 


(B8) 
and 




(B9) 

(B10) 
Simplification of Proofs
If one fish is tagged in X at age t and subject to the proposed fishing regime, then the probability of its being in X at age t is defined as and in Y as (i.e., Green's functions). We aim to show that it is sufficient to prove


(B11) 
and 




(B12) 
Of fish tagged in X at age t, the proportion recaptured later by Y between ages t and t + dt (t £ t) is expected to be

(B13) 

(B14) 

(B15) 
Of the fish tagged in ground X at age t, the proportion expected to be caught and retained by fishery X between ages t and t+dt (t t) is

(B16) 

(B17) 

(B18) 

Proof of Equations (B11) and (B12)
Let the LHSs of Equations (B11) and (B 12) be

F(t) =
x^{*}(t)x(t) 
and 


Y(t) =
y^{*}(t)y(t) 


and 





(B19) 
and 




(B20) 
Furthermore,

(B21) 




(B22) 



The linear differential Equations (B19) and (B20) for F(t) and Y(t) are identical to Equations (B21) and (B22) for F(t) and Y(t) and the initial conditions are also equal. Therefore F(t) = F(t) and Y(t) = Y(t), thus proving Equations (B 11) and (B 12) and so justifying Equations (1) and (2).
APPENDIX C ASSESSING THE EFFECT OF ONE PULSE FISHERY UPON ANOTHER: AN EXACT SOLUTION WHEN NO FISH ARE RERELEASED
Consider two separate fishing grounds X and Y. Assume fishing takes place during a short period at the same time each year for both fisheries. For fish of a particular cohort, let P_{x1} fish of age 1 year and P_{x2} fish of age 2 years be recruited to ground X just before the corresponding fishing season. The numbers, P_{y2} and P_{y3}, of fish recruited to ground Y are similarly defined.
Proportions f_{x1} and f_{x2} of fish aged 1 and 2 years respectively, are caught by fishers in ground X and proportions f_{y2} and f_{y3} of fish aged 2 and 3 years are caught by fishers in ground Y. The survival parameter S_{xx1} is the fraction of fish initially in X and remaining in X which survives from the beginning of season 1 to the beginning of season 2, i.e it includes the effect of fishing, migration and natural mortality. Parameter S_{xy1} is the proportion of fish of age 1 in X that moves to Y and survives to age 2. Parameters S_{xy2} and S_{yy2} are similarly defined. It is assumed that natural mortality and migration can be ignored during the short fishing season.
The numbers of fish in the commercial catches are C_{x1}, C_{x2}, C_{y2} and C_{y2}. Symbols in bold represent numbers of fish that need to be observed, or estimated by some means, which are the input information into the interaction estimation procedure.
The purpose of this analysis is illustrate how to estimate the effect on the catch numbers of Y of closing down fishery X when no tagged fish are rereleased. It is assumed that the mortality parameters are independent of population size and that 1 and 2 year old fish in X can be identified with sufficient accuracy. We construct a table of the numbers of fish caught under the existing fishing regime.
Table C1. The commercial catch numbers of the present fishing regime, by age and fishing ground, in terms of recruitment, survival and fishing intensity.
Ground caught 
Catch numbers at age 

1 
2 
3 

X 
C_{x1} 
C_{x2} 

P_{x1}f_{x1} 
{P_{x1}S_{xx1}+P_{x2}}f_{x2} 


Y 

C_{y2} 
Cy3 

{P_{x1}S_{xy1}+P_{y2}}f_{y2} 
{P_{x1}S_{xx1}+P_{x2}} S_{xy2}f_{y3
}+ {P_{x1}S_{xy1}+P_{y2}}S_{yy2}f_{y3}+P_{y3}f_{y3} 

S_{xx1} =
S'_{xx1}(1f_{x1}) 
or 
S'_{xx1}S_{xx1} =
S_{xx1}f_{x1}/(1f_{x1}), 
(C1) 

S_{xy1} ^{=}
S'_{xy1}(1f_{x1}) 
or 
S'_{xy1}S_{xy1} =
S_{xy1}f_{x1}/(1f_{x1}), 
(C2) 
and 
S_{xy2} =
S'_{xy2}(1f_{x2}) 
or 
S'_{xy2}S_{xy2} =
S_{xy2}f_{x2}/(1f_{x2}). 
(C3) 
Table C2. After closing down fishery X: the commercial catch numbers by age, in terms of assumed recruitment, survival and fishing intensity.
Ground caught 
Numbers in the catch at age 

1 
2 
3 

X 
C'_{x1} 
C'_{x2} 

0 
0 


Y 

C'_{y2} 
C'_{y3} 

{P_{x1}S'_{xy1} + P_{y2}}f_{y2} 
{P_{x1}S'_{xx1} + P_{x2}}S'_{xy2}f_{y3
}+ {P_{x1}S'_{xy1} + P_{y2}}S_{yy2}f_{y3}
+ P_{y3}f_{y3} 



(C4) 
Tagging experiment
Just before the fishing season begins tag N_{x1} one year olds in X and a year later tag N_{x2} two year olds in the same ground. The number recaptured from ground X at age j from fish tagged at age i is r_{xij}. A similar definition applies to r_{yij}. It is assumed that the age of a fish is known at release with sufficient accuracy, if fish are young this could be determined from lengthfrequency modes. The numbers of recaptures expressed in terms of population parameters are:
Table C3. The numbers of recaptures of tagged fish of the present fishing regime, by age of release and by age and fishing ground of recapture, in terms of numbers tagged, survival and fishing intensity.
Number tagged 
Ground recaptured 
Numbers of recaptures at age 

1 
2 
3 

N_{x1}

X 
r_{x11} 
r_{x12} 

N_{x1}f_{x1} 
N_{x1}S_{xx1}f_{x2} 


Y 

r_{y12} 
r_{y13} 


N_{x1}S_{xy1}f_{y2} 
N_{x1}S_{xx1}S_{xy2}f_{y3
}+ N_{x1}S_{xy1}S_{yy2}f_{y3} 

N_{x2}

X 

r_{x22} 


N_{x2}f_{x2} 


Y 


r_{y23} 



N_{x2}S_{xy2}f_{y3} 

(C5) 
We now deal with noncompliance with the rereleasing requirement. Firstly, at the end of season i some N_{xi}r_{xii} fish remain of the N_{xi} initially tagged at the beginning of the season. If all fish had been rereleased then N_{xi} would have remained. To allow for this discrepancy, subsequent recaptures of these fish should be multiplied by the factor N_{xi}(N_{xi}r_{xii}) or 1/(1 f_{xi}). Accordingly (r_{y12} + r_{y13})N_{x1}/(N_{x1}r_{x11}) + r_{y23}N_{x1}N_{x2}C_{x2}/[C_{x1}N_{x2}(N_{x2}r_{x22})] will contribute to R_{y}.
Next, from the N_{x1} fish tagged in year 1 some r_{x12} N_{x1}/(N_{x1}r_{x11}) should have been caught in X at age 2 and rereleased, but were not. These need to be represented by the N_{x2}r_{x22} tagged fish that remain after season 2 from the N_{x2} tagged just before it began. So the additional contribution to R_{y} is r_{x12}r_{x23}N_{x1}/[(N_{x1}r_{x11})(N_{x2}r_{x22})].
Consequently, from Equation (C5) the estimate of the change in the catch of Y in terms of observed fish numbers is
DC_{y} =
C_{x1} [r_{y12} +
r_{y13} +
r_{x12}r_{y23}/(N_{x2} 
r_{x22})]/(N_{x1} 
r_{x11}) +
C_{x2}r_{y23}/(N_{x2} 
r_{x22}). 
C6 
It is noted one cannot solve for all model parameters because there are 6 equations represented in Equation (C1) and Table (C3) with 8 unknowns: f_{x1}, f_{x2}, f_{y2}, f_{y3}, S_{xx1}, S_{xy1}, S_{xy2}, and S_{yy2}. Other information contributes equal numbers of equations and unknowns.
Equations can be adjusted to estimate the change in the weight of fish caught by fishery Y. Additionally, this analysis could also be extended to allow f_{x1}, and f_{x2} to be control variables, e.g. f_{x1} = 0 is the conventional size limit case. It could also be extended to cases where more age classes are harvested in X and/or Y or where fish are tagged in both grounds.