DENSITY-DEPENDENT SEARCHING TIME: IMPLICATIONS IN 

 SURPLUS-PRODUCTION MODELS 1 



Richard E. Condrey 2 



ABSTRACT 



An initial theoretical consideration is presented to show how density-dependent searching time can be 

 incorporated into surplus-production models of fisheries. A simple simulation is used to demonstrate 

 the management implications associated with failure to account for this parameter in fisheries where 

 handling time reduces the total time initially available for searching. 



The failure to measure density-dependent search- 

 ing time in assessing fishing effort can lead to 

 erroneous conclusions concerning the collapse of a 

 fishery. In this paper, I will develop my argument 

 using two simple models: Graham's equilibrium 

 yield model (Graham 1935; Ricker 1975) and Hol- 

 ling's (1959b) "disk" equation. I will conclude by 

 drawing parallels between this simple theoretical 

 treatment and patterns that have been observed in 

 searching fisheries. 



DEVELOPMENT OF THE MODEL 



I begin with the normal definition of the instan- 

 taneous rate of fishing mortality, F, as 



F = q  f=CIN 



(1) 



where q is the instantaneous catchability coeffi- 

 cient and is the proportion of the stock (N) that is 

 caught (C) by one unit of fishing effort (/); this 

 fishing effort (/*) is the total gear in use for a 

 specific time (Ricker 1975). Following the example 

 of Beddington (1979) and Fowler (1980), I depart 

 from the normal treatment of f, by considering 



If searching time is a constant, independent of 

 stock abundance N, then a linear relationship is 

 normally expected between Clf (or Clf) and N up 

 to some theoretical limit, as in a Holling (1959a) 

 Type I curve (Fig. 1A) or in the Palohemio and 

 Dickie (1964) model. 



f=f't' 



(2) 



Stock Density 



FIGURE 1.— Two types of functional responses of catch to stock 

 density, after Holling (1959a). 



If, however, fishermen must expend a substan- 

 tial amount of time in harvesting the catch once it 

 is sighted, this handling time (t' h ) will reduce the 

 time initially available for searching (e.g., Gul- 

 land 1956, 1964; Rothschild and Suda 1977). Hol- 

 ling (1959b) describes one such dependence of t' g 

 oniV as 



where/*' is a physical measure of the total fishing 

 gear in use and t ' is the proportion of the total 

 fishing time (t ') which is available for and used in 

 searching. 



Contribution No. LSU-CEFI-84-02 from the Coastal Ecology 

 and Fisheries Institute, Louisiana State University, Baton 

 Rouge, LA 70803-7503. 



2 Center for Wetland Resources, Coastal Ecology and Fisheries 

 Institute, Louisiana State University, Baton Rouge, LA 70803- 

 7503. 



Manuscript accepted February 1984. 

 FISHERY BULLETIN: VOL. 82, NO. 3, 1984. 



t' =t' -t'iCIf). 



(3) 



Equation (3) describes a curvilinear decline in Clf 

 with increasing stock abundance (Fig. IB). Sub- 

 stituting Equation (3) into Equation (2) into Equa- 

 tion (1), we obtain 



F = qf it' -f h - Clf), 

 which can be arranged to 



(4) 



449 



V4* 



