THE CATCH PROCESS (coninued) In our previous derivaion of e relaionsip beween CPUE and fis abundance we assumed a all e fising unis and all e fis were spaially omogeneous. Now we explore wa appens wen is is no e case. Spaial Effecs on CPUE Fis and fising gear will almos never be uniformly disribued over e fising grounds. How does is affec e relaionsip beween CPUE and abundance? Suppose ere are wo fising areas, A and B, and no inercange of fis beween em. Te ime-averaged abundance for bo areas combined is avn A_B 1 ( N A ( u) N B ( u) ) Tis is a simple exension of e ime-averaged abundance a we explored in e Cac1 lecure. 1 N A ( u) N B ( u) Te inegral of a sum is e sum of e inegrals. avn A avn B CPUE A CPUE B from CPUE q avn q A q B If q A q B q, en q avn A_B CPUE A CPUE B and in general CPUE A CPUE B f A f B f A f B To combine cac and effor daa from differen areas we sould ake e sum of e indivial raios of cac over effor; we sould NOT ake e raio of e sum of e cac over e sum of e effor, aloug a is wa people commonly do. Here is a ypoeical numerical example wi N A () N B () 1 a illusraes e poin. M.1 /yr q.1 /yr f' A 1 unis f' B 3 unis Suppose a afer ree mons (1/4 yr) of fising we ave e following: Area A: 24.4 f A 2.5 CPUE A 9.75 Area B: 71.4 f B 7.5 CPUE B 9.52 Combined: _B 95.8 f A_B 1. CPUE A_B 9.58 Te sum of e CPUE values is CPUE A CPUE B 19.27. Te sum of e caces (oal cac) over e sum of e effor (oal effor) is CPUE A_B 9.58. Te raio of e oal cac over e oal effor is equivalen o e weiged average fis densiy per area, wi e "weig" for eac area equal o e fising effor. FW431/531 Copyrig 28 by David B. Sampson Cac3 - Page 44
ΣC Σf Σ f C 1 Wa is a weiged average? f Σf We wan our measure of CPUE o be proporional o e oal ime-averaged abundance, no e average (wi respec o area) ime-averaged abundance. Te quaniy (ΣC)/(Σf) may or may no be proporional o e ime-averaged abundance. I will depend on e disribuion of fising effor relaive o e disribuion of e fis. Anoer way o view e problem is in erms of e insananeous CPUE. Suppose e insananeous CPUE a any locaion is proporional o e fis abundance a a locaion (i.e. e fis densiy). d CPUE Loc d q N( Loc) Te oal number of fis over all locaions is given by e following inegral Loc N Toal q N Toal Loc N( u) CPUE( u) Fis Densiy Locaion Te area under e insananeous CPUE curve (e doed line) is q imes e area under e N curve (e solid line). You can ge some very srange relaionsips beween e raio (ΣC)/(Σf) and oal abundance. For example, e following wo fiseries ave idenical average fis densiies (and idenical cac/effor raios), bu one populaion as wice e abundance of e oer. Populaion A Populaion B Fis Densiy N A Fis Densiy N B CPUE A CPUE B Locaion Locaion Te fis densiies and e average insananeous CPUE values are e same for bo populaions. Te areas under e densiy and insananeous CPUE curves, owever, are proporional o oal abundance. FW431/531 Copyrig 28 by David B. Sampson Cac3 - Page 45
Effecive Fising Moraliy and Effecive Fising Effor An our of fising in an area were ere are few fis will ave muc less impac (will cause a smaller F) on e fis populaion an an our of fising in an area were e fis are very abundan. In general, if fising effor is no uniform over all spaial regions, en e relaionsip beween oal effor and e fising moraliy coefficien is no longer a simple one. Consider again e earlier numerical example. From e cac equaion we ave N A N B F [ M F 1 exp [ ( M F ) ] ] F 95.8 2 [ 1 exp[ (.1 F).25] ].1 F We canno solve direcly for F bu we can use numerical ecniques. (Tis is easy wi Excel using Solver.) F.1988 /yr, wic is NOT equal o q f' A f' B.4 /yr. Te sum of e fising effor values sould be proporional o F, wi q as e consan of proporionaliy, bu i is no. Te problem arises because e fis in eir spaial disribuions are no equally vulnerable o all e unis of fising effor. To solve is problem of F q f' we need o calculae e effecive overall fising moraliy coefficien, someimes denoed as F wi a ilde (~) over i. ~ F Σ ΣC i C F i Toal_Cac Te summaions are Σ ( Cac_per_Uni_Fising_Moraliy) over all locaions. Tis resul comes direcly from e noion a for eac area C F N( u). For our numerical example we ave q f' A q f' B 24.4.1 95.8 71.4.3.1988 < We've recovered e correc value for F. We can also calculae e effecive overall fising effor, f wi a ilde. ~ f Σ ΣC i C f' i Toal_Cac Noe a i is f' Σ ( Cac_per_Uni_Fising_Gear) ere, no f'. f' A f' B 24.4 1 95.8 71.4 3 19.88 FW431/531 Copyrig 28 by David B. Sampson Cac3 - Page 46
Tese wo derived quaniies are relaed by ~ ~ F f q See Beveron and Hol (1957) pages 148-151 for e formal derivaions of e equaions for F ilde and f ilde. Te ideas of effecive fising moraliy and effecive fising effor ofen do no receive muc aenion, even oug e spaial aspecs of fising are likely o dominae e relaionsip beween cac raes and fis abundance for many socks. Beveron and Hol (1957, secion 8.3) and Gulland (1983), on e Recommended Reading lis, consider e issues of non-uniform fising in space and ime, and Roscild (1977) describes various oer complicaions a can disor e relaionsip beween fising effor and e insananeous fising moraliy a i can generae. Cac Process wi Handling We ave o be careful ow we define fising effor. Te examples we've explored so far ave no made is clear. Here is a model a illusraes one kind of problem. To simplify e derivaion assume a e number of fis removed by e cac process ring a fising rip is small relaive o e overall size of e populaion, and a e number of naural deas ring a fising rip is also small. In is case we can use e following approximaion for e cac equaion, C q N F were F is e ime spen fising e gear. Suppose also a for eac fis caug a cerain amoun of ime is required o andle and process e fis, and a ring eac andling period i is no possible o deploy e fising gear. Here are some examples of is penomenon: wen rawling i akes ime o sor ou e cac before e rawl can be reurned o e waer; wen purse seining i akes ime o pump ou e ne; wen fising wi ook and line i akes ime o reel in any fis a are caug. Te oal ime spen andling e cac is H C were H is e andling ime. Finally, suppose eac rip requires ime O for ravel and oer aciviies. Te oal raion of e rip is T F H O Noe a is pariioning of e fising rip is no appropriae for fis raps or oer forms of gear a fis wile unaended by e fisermen. Now look a e cac rae (cac over ime). C q N F q N T H O from F H O and e definiion of T. C q N T C O from e definiion H C. C q N T q N C q N O Solve for C. C q N C q N T q N O Collec on e lef e erms wi C. C ( 1 q N ) q N T q N O > C q N T O 1 q N FW431/531 Copyrig 28 by David B. Sampson Cac3 - Page 47
Le denoe e fracion of a rip spen fising e gear and andling e cac. is e Greek leer au. T O > C T > T 1 q N C T 1 q N Le T approac zero and we ge e insananeous cac rae. d 1 q N Te slope a e origin is q. /d Te asympoe is d To deermine e i of /d as goes o infiniy, we ave o use L'Hôpial's Rule. N d 1 q N Take e i of e raio of e derivaives. > q q Now go back o e grap of /d versus N. Wen N is near zero, we do no cac fis very quickly so we do no lose muc ime o andling and e cac rae is almos sricly proporional o N, wic is our usual assumpion. Wen N is large, owever, we spend los of ime andling e fis and a is ow muc we can cac. For example, in e waling insry i ook abou wo ours o sube a fin wale afer i was arpooned; consequenly, i was impossible o cac more an abou 12 fin wales per day per cacer boa. Explore e effecs of q,, and using e Excel demonsraion.. If andling ime is significan, en cac-per-rip will no be a good index of fis abundance. However, e cac per our of fising will be proporional o fis abundance. Te lesson is: Be Careful How You Define Fising Effor!!! Paloeimo and Dickie (1964) on e Supplemenal Reading lis derive a similar form of curved relaionsip for cac rae and sock abundance a can arise wen fis occur in scools. Socasic versions of e cac model wi andling ime are given in Deriso and Parma (1987) and Sampson (1988) on e Supplemenal Reading lis. FW431/531 Copyrig 28 by David B. Sampson Cac3 - Page 48