AGGREGATION OF FISH THROUGH VARIABLE DIFFUSIVITY 



Ashley J. Mullen^ 



ABSTRACT 



It is argued that the commonly used model for the dispersal of tagged fish may be inappropriate; for 

 yellowfin tuna at least, it is unable to reproduce the observed spatial variation of abundance. An alter- 

 native model, in which the local environment affects both the local population dynamics and the disper- 

 sal of fish, is presented. 



Fishing is introduced using a simple bioeconomic model; the effect on the distribution of the popula- 

 tion is surprising. Routine management questions such as maximizing production become difficult, if 

 not impossible, within this heterogeneous model. Of particular interest are the interactions between a 

 region of high production and its surroundings: at steady state with low rates of exploitation, there is 

 net emigration from areas that can sustain larger populations, but the direction of net migration reverses 

 as fishing pressure increases. Interaction between zones where different technologies are applied is 

 investigated. 



Skellam (1951) suggested adopting a diffusion model 

 for the dispersion of inert particles for describing 

 the motility of living organisms, based upon their 

 random motion. Beverton and Holt (1957) put it 

 within a fisheries context, and Jones (1959 and 1976) 

 explained its use in detail. The method is simple: a 

 velocity vector is determined for each recovery, the 

 mean of these is calculated, the differences between 

 each vector and their mean are obtained, and final- 

 ly the mean of the squares of these residuals is 

 calculated. This term, the "diffusion coefficient" or 

 "diffusivity", a~, and the mean velocity vector, v, 

 are used to characterize the movements of entire 

 populations. The first governs the dispersion of a 

 population while the second parameterizes any 

 directed, often seasonal, migration. 



If there is no directed migration then, for a fish 

 with constant range, the effective area searched per 

 unit time is determined solely by a'. Analysis of 

 tagging experiments in the eastern Pacific Ocean 

 have not yet shown clear seasonal direction in move- 

 ments of either yellowfin, Thunnus albacares, or 

 skipjack tuna, Katswonus pelamis, in this area 

 (Hunter et al. 1986). Directed migration, v, is not 

 explicitly incorporated into the model presented, but 

 could be. 



Previous mathematical models for the dispersion 

 of fish have assumed the coefficient of diffusion to 

 be constant, so that the rate of transport due to 



'Inter-American Tropical Tuna Commission, do Scripps Institu- 

 tion of Oceanography, La JoUa, CA 92093 (for correspondence); 

 and Renewable Resource Assessment Group, Centre for Environ- 

 mental Technology, Imperial College of Science and Technology, 

 48 Prince's Gardens, London SW7 INA, England. 



dispersion is proportional to the gradient of abun- 

 dance. Bayliff and Rothschild (1974) and Bayliff 

 (1979, 1984), however, reported that estimates of 

 a'~ varied by between one and two orders of mag- 

 nitude for both yellowfin and skipjack tuna in the 

 eastern Pacific. There appeared to be some pattern 

 to these results; for instance, close to islands and 

 shallow banks, where it has been suggested that 

 prey is more abundant (Sund et al. 1981), a'~ was 

 often less. However, there has been no attempt to 

 formulate the pattern formally, and variations in 

 measured coefficients of diffusion have been treated 

 simply as noise or errors of measurement. 



Taking a typical value for the coefficient of diffu- 

 sion leads to a problem in the case of yellowfin ttma; 

 any single value for a ~ estimated from tagging ex- 

 periments predicts an almost homogeneous distribu- 

 tion. This is not observed; catch rates tend to be high 

 where prey are believed to be abundant (Sund et al. 

 1981). Spatial variability of production is unlikely 

 to be sufficient to maintain the variability of abun- 

 dance that is demonstrated by variability in catch 

 between areas. 



This inconsistency does not arise with a variable 

 coefficient of diffusion. Kareiva and Odell (1987) con- 

 sidered a diffusion process for ladybugs preying 

 upon aphids in which the probability of course rever- 

 sal was increased when the aphid had recently eaten. 

 They showed that this foraging mechanism concen- 

 trated predators in areas of high prey density. A 

 similar mechanism is suggested for pelagic fish; if 

 the coefficient of diffusion is a function of local 

 habitat then distributions of tuna can be more 

 realistically simulated. The mechanism involves den- 



Manuscript accepted January 1989. 

 Fishery Bulletin, U.S. 87:363-362. 



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