MULLEN: FISH THROUGH VARIABLE DIFFUSIVITY 



in particular it predicts that heavy fishing will 

 enhance immigration. In a practical model, one 

 might use a'"s directly when possible and find a 

 convenient empirical function for indirect estima- 

 tion of a- at other times. 



The dimensions of a^ are (distanced/time); a^ can 

 be thought of as the average distance moved before 

 taking another direction, multiplied by the mean 

 speed over that interval (Beverton and Holt 1957). 

 Even at constant speed, an individual fish can 

 reduce its a - simply by changing course more fre- 

 quently, so the fish could maintain an almost con- 

 stant position if it were to change direction frequent- 

 ly enough. The upper limit is determined by the 

 fish's ability to hold a course. Walker et al. (1985) 

 showed that yellowfin tuna can detect a geomag- 

 netic field and suggested that they might use it for 

 navigation. This appeared incongruous in a fish 

 whose direction at any time is said to be random. 

 But, if it allows each fish to hold any random course 

 longer, then it allows the fish to get away from an 

 area it has found to be unsatisfactory. Fish in an 

 isolated undesirable area will all be, in a sense, 

 navigating away from that area, but in different 

 directions; that is why the population in that area 

 does not exhibit any directed migrations; they all 

 cancel each other. 



Given that there is heterogeneity in the distribu- 

 tion of prey, it is not surprising that predators have 

 evolved towards matching that distribution. A fish 

 cannot know where the greatest concentrations of 

 prey are and then navigate to them, but it can 

 reduce the chances of leaving a favorable region and 

 increase its search area when hunting is poor. A 

 tuna varying a- inversely with habitat quality has 

 advantage over any with constant a'. In poor 

 habitat the fish has high a~, and its net movement 

 over any period is greater. Upon entering a more 

 favorable area, a- drops, and the fish weaves a 

 more intertwined track over a smaller area. Thus 

 the individual spends more time in the more favor- 

 able areas; a population of such individuals accum- 

 ulates in the better habitat without any directed 

 migration. 



The most patently unrealistic aspect of this model 

 is its topology, that of a torus. This is a convenience 

 chosen to avoid boundary conditions at the spatial 

 limits of the model and to avoid speculating about 

 an additional mechanism that maintains the fish 

 within those limits. Specifying more realistic bound- 

 ary conditions might include seasonal changes in the 

 positions of those boundaries. 



A species constrained within such plastic bound- 

 aries would demonstrate seasonal changes in dis- 



tribution, and tagging would suggest directed move- 

 ment. But, no long-range directed navigation would 

 be necessary. At one end of their distribution, as 

 the boundary of intolerable conditions encroaches, 

 fish might retreat, or simply die. Elsewhere the 

 population might simultaneously be expanding by 

 the chance movement of individuals into freshly 

 habitable waters. 



For those species whose directed movements are 

 real, one can simply add a term, v, to the model. 

 Determining the (time varying) values for this might 

 require more tagging effort than that required sim- 

 ply for evaluating a^. 



There is probably some autocorrelation in the 

 direction of movement of the fish, but this would 

 not affect the conclusions drawn from this model. 

 Individuals emanating from a point source at first 

 show a clear orientation away from this source; 

 if each individual's course exhibits autocorrela- 

 tion, then this orientation persists but diminishes 

 through time. Eventually the individuals lose their 

 orientation to the source, and direction is inde- 

 pendent of position. The void initially created at 

 the source is filled, and the simpler diffusion equa- 

 tion may be considered an acceptable model for 

 describing the distribution of individuals (Skellam 

 1973). 



In a more realistic model, the topography of the 

 environment would be very complicated, with varia- 

 tion at many scales. This model is a very inexact 

 description of the population it purports to describe 

 (yellowfin tuna in the eastern Pacific), but habitat 

 dependent diffusion may be applicable to other 

 regions and other species. Kleiber' has shown that 

 the simple diffusion model is also inadequate for 

 skipjack tuna in the western Pacific. Beamish and 

 McFarlane (1988) suggested that the dispersal of 

 adult sablefish may be affected by the local density. 

 Sablefish are much more sedentary than tunas; 

 Beamish and McFarlane estimated that local fluc- 

 tuations of abundance are determined by recruit- 

 ment of juveniles and by fishing. 



Scientists who have examined data for tunas 

 tagged close to islands refer to two populations: one 

 which remains associated with the islands, which 

 must have a low a-; and one that breaks away, 

 which necessarily has a higher value for a-. Indivi- 

 duals that leave one island are, of course, still sus- 

 ceptible to "capture" by another island, or perhaps 

 a shallow bank. The model described here suggests 



^Pierre M. Kleiber, Southwest Fisheries Center La Jolla Labor- 

 atory. National Marine Fisheries Service, NOAA, P.O. Box 271, 

 La Jolla, CA 92038. pers. commun. June 1988. 



359 



