FISHERY BULLETIN: VOL. 84, NO. 3 



redistributed populations. Consistent with the first- 

 order nature of the formulas, certain simplifications 

 are expected to be incorporated in the ecological 

 representation including implicit modeling of lower 

 trophic levels (e.g., phytoplankton and zooplankton) 

 and functional groupings of less important species 

 as competitors, predators, and prey. 



Growth and Mortalities 



First-order differential equations of the following 

 general form are typically used to describe the 

 growth and mortalities of a population P, under 

 competitive and predator-prey influences with itself 

 and other populations: 



d t 



(r t - u % P - v,P + WjP)Pi 



(1) 



where r, = survival/growth parameter 



P = population vector 



= \P\ > " 2 t • • • t "it •  • ) Pn) 



u { = competition coefficient vector 



— {tin , u, { 



2)  



U; 



u in ) 



Vi = 



W: = 



predation coefficient vector 

 prey coefficient vector. 



The coefficient vectors u t , v { , and w x contain ap- 

 propriate zeros such that only the active interactions 

 between populations are defined. (Note that vector 

 multiplication is implied by the forms such as u^P.) 

 The competition terms correspond to the standard 

 Gause model, while the predator-prey terms corres- 

 pond to the simple Lotka-Volterra model (Pielou 

 1977). The population variables P { can be expressed 

 in units of either numbers of individuals or total 

 biomass, with the coefficients defined accordingly. 



Assuming a small time step (At) relative to the 

 characteristic time of the system (1/r), a discrete- 

 time approximation is found directly by integrating 

 Equation (1) to give 



P t (M) = e r > M 



-u PM 



-v P&t 



w, PM 



■Pr(0) (2) 



These exponential terms form the basis of the dif- 

 ference model. However, some modification and in- 

 terpretation of terms is required in order to describe 

 a general form appropriate over a range of popula- 

 tion levels. 



The most obvious inadequacy of Equation (2) is 

 the positive exponential prey term, e w < PM , which 

 gets increasingly larger as prey increases without 

 ever reaching a saturation condition. A more ap- 



propriate form is the predator feeding model given 

 by Ivlev (1961): 



F = F max (1 - e-V) 



(3) 



where F is the predator feeding ration and i, is an 

 associated prey coefficient, assuming that this form 

 can also be used to describe the predator's growth/ 

 survival as a function of prey density. 



The proposed difference equation for expressing 

 population growth and mortalities during a At time 

 step is 



Pi(t + 1) = S t e~°. p e-W (1 - R t e-^)P t (t) (4) 



where S t = maximum survival/growth rate per 

 time period 



a t = discrete form of competition coeffi- 

 cient vector 



Pi = discrete form of predation coefficient 

 vector 



R t = starvation mortality factor 



Yi = discrete form of prey coefficient 

 vector. 



The terms in this generalized form need further 

 discussion and interpretation. 



The maximum survival/growth rate factor, S, ac- 

 counts for population births (if single age class), 

 growth (if biomass units), and certain mortalities 

 such as fishing, disease, and old age. It also accounts 

 for predatory deaths caused by populations not ex- 

 plicitly included in the ecosystem model. It does not 

 account for predation, competition, and prey avail- 

 ability effects associated with the modeled popula- 

 tions, which are explicitly stated by the other terms 

 of Equation (4). Maximum survival/growth is defined 

 under ideal conditions when competition and pre- 

 dation influences are negligible and there is an abun- 

 dant supply of prey. 



The a competition coefficient is the exponential 

 equivalent to the Gause term in Equation (1) and 

 represents a basic damping factor inhibiting popula- 

 tion expansion. Self-competition generally relates 

 to the essential environmental resources such as 

 food supply and habitat space. Additional intra- 

 population effects can come into play at the extreme 

 ranges of population densities to complicate this in- 

 terpretation, such as decreased fecundity caused by 

 crowding (Parrish 1975) and decreased birth rates 

 at very low densities (May 1973). Competition be- 

 tween population groups involves considerations of 

 niche overlap relative to the common resources for 

 which they compete (May 1973). Active competition 



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