THE CATCH PROCESS. Deaths, both sources. M only F only Both sources. = N N_SMF 0 t. N_SM t. = N_SMF t. = N_SF t
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1 THE CATCH PROCESS Usually we canno harves all he fish from a populaion all a he same ime. Insead, we cach fish over some period of ime and gradually diminish he size of he populaion. Now we will explore a model for his cach process. Suppose we have a cohor of fish in isolaion, and in he absence of fishing 8% survive each ime period. N.8 N + Now suppose we sar harvesing fish wih fishing gear ha removes 5% of he fish each ime period. The percenage of fish ha survive boh forms of moraliy is given by he produc of he wo survival fracions, 8% x 5% 4%. This follows from a basic rule of probabiliy. If A and B are wo independen evens, hen he probabiliy ha boh evens occur is equal o he produc of he individual probabiliies, P( A and B ) P( A ) x P( B ). We canno apply his rule o he moraliy probabiliies (as opposed o he survival probabiliies) because an individual can only die once. Deah by naural causes and deah by fishing are no independen evens. Below is a hypoheical example showing how he abundance of a cohor is affeced by differen moraliy sources operaing individually and hen operaing ogeher. In he firs case he survival rae is from naural moraliy only, S M.8. In he second case he survival rae is from fishing moraliy only, S F.5. In he hird case he survival rae is from boh naural moraliy and fishing moraliy. The iniial abundance for he cohor is N. I is impossible for here o be no naural moraliy, bu his is a hypoheical example. M only F only Boh sources Deahs, boh sources N_SM N_SF N_SMF N N_SMF Consider he numbers of fish dying in he differen scenarios. Deahs if M only: Deahs if F only: Deahs if M and F: During s period During 2nd period The wo moraliy processes "compee" for fish. (Deahs if M only) + (Deahs if F only) (Deahs if M and F). FW43/53 Copyrigh 28 by David B. Sampson Cach - Page 33
2 We know how many fish die. I is simply N_Deahs N N_Survivors. Bu how do we deermine how many of hese deahs represen caches, and how many represen naural deahs? One way o answer his is by working wih he insananeous moraliy raes. Naural Survival during Inerval : Fishing Survival during Inerval : ( ) S M exp M > M ( ) S F exp F > F ln( S M ) ln( S F ) Insananeous Rae of Naural Moraliy: Insananeous Rae of Fishing Moraliy: M -ln(.8 ) /.223 / F -ln(.5 ) /.693 / Insananeous Rae of Toal Moraliy: ( F + M ) /.963 / Noe ha mos fisheries scieniss use he same symbols as above o denoe he insananeous raes of naural (M), fishing (F), and oal moraliy (), bu someimes you will find oher noaion. The insananeous moraliy raes M and F are added ogeher o ge he insananeous rae of oal moraliy. The fracion of hese deahs ha is due o fishing is he raio F /. Toal Deahs: Cumulaive Cach: D( ) N( ) N( ) N( ) ( exp( ) ) C( ) D( ) F N( ) ( exp( ) ) F The Cach Equaion This las equaion is widely used in fisheries models; i is generally known as he cach equaion. C() N() x ( F / ) x [ - exp( - ) ] Cach accruing during inerval (,). No. fish Fracion of alive a x deahs due x. o fishing. Fracion dying during inerval (,). Here is a numerical example based on N M.223 F.693 N( ) D( ) C( ) F.7565 FW43/53 Copyrigh 28 by David B. Sampson Cach - Page 34
3 Here is he graph of C() 8 N F As >, C() > N() F/. As ime passes all he fish evenually die and F/ of hem die because hey were caugh. C() Explore he effecs of F and M using he Excel demonsraion. The cach equaion can also be derived as he soluion o a differenial equaion (DE) for he insananeous cach rae. d FN dn d ( ) N Solve he second DE for N()... N( ) N( ) exp[ ( ) ] We solved for N() in a previous lecure bu he DE only involved M.... and subsiue he resul for N in he firs DE, and hen solve for C(). d FN( ) exp[ ( ) ] FN( ) exp[ ( ) ] d Separae he variables C,... FN( ) exp[ ( ) ] d... and inegrae. exp(a X) dx (/a) exp(a X) + consan C( ) FN( ) exp [ ( ) ] + K Here K is he arbirary consan because C is he focal variable. This is he general soluion. Unlike he differenial equaions for dn/d and dl/d, he equaion for /d is no a simple funcion of C. For iniial condiions, le us specify ha C(). Now solve for K. C( ) FN( ) exp [ ( ) ] + K FW43/53 Copyrigh 28 by David B. Sampson Cach - Page 35
4 FN( ) + K > + K > K F C( ) N( ) [ exp[ ( ) ] ] The paricular soluion. Sochasic Version of he Cach Equaion I is possible o se up a sochasic version of he cach equaion ha describes he probabiliy ha a paricular number of fish are caugh during some ime inerval (,). The mahemaics are messy bu relaively sraighforward. Deerminisic & Coninuous Model. Sochasic & Discree Model. Cach Cach ime ime In he sochasic model changes in C() occur abruply, and he ime beween caches is a random variable. I urns ou ha for he sochasic model he equaion for he average cach as a funcion of ime is idenical o he deerminisic cach equaion. There are wo references on he Supplemenal Reading lis ha discuss sochasic models for he cach process, Mangel and Beder (985) and Sampson (988). Cach-per-Uni-Effor (CPUE) Suppose here are f' pieces of fishing gear and all are removing fish from he same fish populaion a he same rae q. d ( q + q q) N f' idenical unis of fishing gear. In his differenial equaion for he cach he parameer q is usually described as he cachabiliy coefficien. I is he insananeous rae of fishing moraliy caused by one uni of gear. We can also wrie he equaion as d ( qf' ) N FN where F qf' If we divide boh sides by f', we ge an equaion for he insananeous cach-per-uni-effor. Ofen cach-per-uni-effor is denoed as CPUE. FW43/53 Copyrigh 28 by David B. Sampson Cach - Page 36
5 f' d qn d d CPUE( ) The insananeous CPUE is proporional o abundance N(). In pracice we canno observe an insananeous cach rae. Insead, we observe caches as hey accumulae over ime. C( ) N( ) qf' ( exp( ) ) Recall ha F + M and F q f'. C( ) f' C( ) f' exp( ) qn( ) Divide boh sides by. exp( ) qn( ) The quaniy f' is he fishing effor, wih unis such as [rawl-hours] or [rap-days]. The quaniy N( ) exp( ) is he ime-averaged abundance, avn(). avn( ) N( u) du Noe ha u is a dummy variable ha represens he variable of inegraion. Time is he upper limi of he inegral avn( ) N( ) exp( u) du e ax dx (/a) e ax + C avn( ) N( ) exp( ) exp( ) avn( ) N( ) exp( ) We can wrie he relaionship for C()/(f' ) as CPUE qavn( ) The cach-per-uni-effor, which we measure using he raio of cach over effor, can be used as an index of fish abundance, meaning ha CPUE is proporional o fish abundance. Noe ha when auhors refer o cach-per-uni-effor, hey are no always consisen in heir definiion of fishing effor. You need o check he conex. If hey are using insananeous cach raes, hen he effor is f', he number of fishing gear unis. If hey are alking abou cumulaive cach, hen he effor is ( f' ), he applicaion of fishing gear over ime. In he lieraure ( f' ) is ofen denoed by jus f, wihou any prime. Remember he assumpions behind his model: () here are f' pieces of gear operaing independenly and wihou inerference, (which allows us o wrie F q f'); (2) all fish in he populaion are equally vulnerable o he fishing gear; and FW43/53 Copyrigh 28 by David B. Sampson Cach - Page 37
6 (3) he populaion is closed, meaning ha here is no immigraion, emigraion, or new recruimen. See Richards and Schnue (986) on he Supplemenal Reading lis for an empirical es of using CPUE as an index of abundance. Here are some graphs ha illusrae he relaionships beween N(), C(), ime-averaged N(), and CPUE(). N( ) C( ) ( f' ) 5 N( ) avn( ) cpue( ) Noe ha CPUE is no proporional o abundance, N(). I is proporional o ime-averaged abundance, shown as avn() above. Furhermore, ime-averaged abundance is always greaer han N(), because he average abundance always includes larger values of N from all he earlier poins in ime. Explore he effecs of q, M and F using he Excel demonsraion. Wha is he value for CPUE a? We canno direcly evaluae CPUE() because he denominaor is zero, bu we can use l'hôpial's Rule o evaluae he limi of CPUE() as goes o zero. L'Hôpial's Rule: If g(a), hen x lim a h( x) g( x) x lim a h' ( x) g' ( x) Differeniae he numeraor and denominaor, and evaluae he limi of he raio of he derivaives. For CPUE() we have exp( ) CPUE( ) qn( ) Derivaive of he numeraor d d ( exp( ) ) exp( ) Derivaive of he denominaor d d ( ) lim CPUE( ) exp( ) qn( ) qn( ) FW43/53 Copyrigh 28 by David B. Sampson Cach - Page 38
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