Fisheries Science 61(5), 752-754 (1995)

A Theoretical Study of Equations Used in Virtual Population Analysis

Kazuhiko Hiramatsu

National Research Institute of Far Seas Fisheries, Orido, Shimizu 424, Japan (Received December 16, 1994)

A generalization of the equations used in virtual population analysis is derived from the differen tial equations for survival and catch. This new equation relates exactly catch to population size and con tains the equations of virtual population analysis and cohort analysis as special cases. Examples are given to illustrate the flexibility and usefulness of this general equation. Key words: VPA, cohort analysis, stock assessment, generalization

Virtual population analysis (VPA) or cohort analysis is the population size at time t. Substitution of equation (4) extensively used in the stock assessment of exploited ma into equation (3) yields rine resources. The basic equations of this method are

Rearranging the solution of the above equation (see Ap where pendix), we have Ci= the catch in time period i M=the natural mortality coefficient Fi=the fishing mortality coefficient in time period i Ni=the population size at the beginning of time period i. Although many methods, including VPA, are based on these equations, unfortunately, one cannot solve them ana lytically for the unknown quantities in terms of the known This is an exact equation relating N, N;+j and the catch or assumed quantities such as C and M. Furthermore the during period i. assumption of constant M and constant F during each With a simple assumption of constant natural mortality, time period does not always hold. Pope,1) Ishioka and we have Kishida,2) MacCall,3)I Allen and Hearn,4) and Evans5) devel oped approximations to equations (1) and (2) that can be solved directly. The effects of variation in Fand M were in vestigated by Ulltang6) and Sims.7,8) We call equation (6) model I and equation (7) model 2. We present a more generalized form of the equations For simplicity, we use model 2 in the following analysis. which is exact and flexible for practical use. Almost all previous methods are shown to be special cases of a more Applications general approach that we develop here. This equation allows us to derive a simple formula that In this section, we consider four catching patterns and should be viewed not as an approximation to equations (1) derive special versions of model 2 for them. and (2) but as an approximation to real fishing processes. The equation allows one to easily investigate the effects of Pulse Fishing temporal variation in F and M. Assume that the entire catch during the period i is taken instantaneously at time i + u (0 ? u < 1). The change in The Model catch can be represented using Dirac's delta function:

A starting point with this model is the following differen tial equations for survival and catch: where C, is the entire catch during period i and

where F(t) and M(t) are the time dependent fishing mor tality coefficient and natural mortality coefficient, respec Substituting equation (8) into equation (7) gives tively. C(t) is the cumulative catch in number and N(t) is Theoretical Study of Equations Used in VPA 753

Discussion If the catch is taken at midyear (u=1/2), equation (11) is equivalentto the Cohort Analysis equation.1) If there are Although many methods have been based on equations pulse catches C; at time u; (where E;Ci;=C), then equa (1) and (2), all of them make the unrealistic assumption of tion (7) can be written constant fishing mortality. Moreover, this assumption ap parently leads to the disadvantage that N can not be represented as an explicit function of C and M. Hence, the solution must be obtained numerically and statistical analy Thisequation corresponds to equation (15) of MacCall.3) sis of VPA equations9) is usually complicated. The general model developed here has following two ad Constant Catch vantages. The appropriate model for a constant level of catch per (i) It is easy to accommodate the model to real fishing pro time is given by cesses in which mortality coefficients vary with time.

(úA) In most cases, analytical solutions to this equation can be obtained. Therefore, there is no good reason to use equa tions (1) and (2). Substituting equation (13) into (7) yields Pope's Cohort Analysis and other similar equations are usually considered to be approximations to the VPA equa tions. However, they (including VPA) all should be treat ed as special cases for model 2 as shown above. Constant Fishing Mortality Although the seasonal pattern of catches is usually This corresponds to the equations for Virtual Popula unknown, the fishing season may be known. Thus, in prac tion Analysis. The catch is given by tical applications, equation (21) should be a good approxi mation for many real fisheries. If the seasonal pattern of catches is known, much closer approximations are possi ble. Model 1 is usable in fewer practical applications, be cause we usually do not know how M varies through time. so that equation (7) may be rewritten as We can, however, investigate the effects of seasonal vari ation in M on estimates of N by applying this model. In conclusion, we can derive more realistic models from a general equation that exactly relates catch to population size. Although the model with constant fishing mortality whichis identical to the equations (1) and (2). has been traditionally used in fisheries sciences, it should Nagai's equation2) or MacCall's approximation3) can be be considered as a special case of this general model. derived by assuming that a pattern for catching process is Acknowledgments I am grateful to T. Koido, T. Akamine, and Y. Takeuchi for their valuable comments. I also thank D. B. Sampson and anonymous reviewers for their critical readings of the manuscript. instead of equation (15). When F=-M, letting F=M in References equation (15) yields 1) J. G. Pope: An investigation of the accuracy of virtual population analysis using cohort analysis. Int. Commn. NWAtlant. Fish. Res. Bull., 9, 65-74 (1972). Thisequation should be a slightly better approximation to 2) K. Ishioka and T. Kishida: Studies on the algorithm for solving the catch equation and its accuracy. Bull. Nansei reg. Fish. Res. Lab., VPAcompared to equation (17) in this case. Substitution 19, 111-120 (1985) (in Japanese). of equation (18) into (7) yields 3) A. D. MacCall: Virtual population analysis (VPA) equations for non homogeneous populations, and a family of approximations includ ing improvements on Pope's cohort analysis. Can. J. Fish. aquat. Sci., 43, 2406-2409 (1986). 4) K. R. Allen and W. S. Hearn: Some procedures for use in cohort Constant Catch with Seasonality analysis and other population simulations. Can. J. Fish. aquat. Sci., We consider the following seasonal fishery: 46, 483-488 (1989). 5) G. T. Evans: Rational approximations to solutions of the VPA equa tions. Can. J. Fish. aquat. Sci., 46, 1274-1276 (1989). 6) ƒÓ. Ulltang: Sources of errors in and limitations of Virtual Popula tion Analysis (Cohort Analysis). J. Cons. int. Explor. Mer, 37, 249 260(1977). - 7) S. E. Sims: The effect of unevenly distributed catches on stock-size es In this case , N, is given by timates using Virtual Population Analysis (Cohort Analysis). J. Cons. int. Explor. Mer, 40, 47-52 (1982). 8) S. E. Sims: An analysis of the effect of errors in the natural mortality 754 Hiramatsu

rate on stock-size estimates using Virtual Population Analysis (Co Hence, the relation between populations at time i and time hort Analysis). J. Cons. int. Explor. Mer, 41, 149-153 (1984). i + 1 are given by 9) D. B. Sampson: Variance estimators for Virtual Population Analy sis. J. Cons. int. Explor. Mer, 43, 149-158 (1987).

Note added in proof Readers should also refer the following paper; M. Aksland: A general cohort analysis method. Biometrics, 50, 917-932 (1994).

Appendix: The Derivation of General Equations or

Equation (5) can be solved analytically: