ALTRNATV MODLS FOR CPU AND ABUNDANC Fishing is funamenally a localize process. Tha is, fishing gear operaing in a paricular geographic area canno cach fish ha are no in ha area. Here we will evelop wo simple moels for a cach process harvesing fish from a local populaion ha is replenishe from an exernal source. A issue is wheher he CPU for his local cach process is a reasonable inex of abunance for he oal fish populaion. Moel 1: Ranom Movemen beween Fishe an Unfishe Areas Fishing operaions occur only in harves area H an wihin his area here are f fishing operaions operaing inepenenly an wihou inerference. ach fishing operaion caches fish a an insananeous rae of q per fish so ha he insananeous rae of fishing moraliy in he area is F f q. Fish migrae inepenenly ino area H a an insananeous rae of an emigrae inepenenly from he area a an insananeous rae of. Naural moraliy occurs everywhere a an insananeous rae of M. N O N H F C Le N H be he number of living fish insie area H, le N O be he number of living fish ousie area H, an le C be he cumulaive cach of fish. The behavior of N H, N O, an C are governe by he following sysem of ifferenial equaions: N O( ) N O ( ) ( + M + F) N H ( ) ( + M) N O ( ) + N H ( ) The ifferenial equaions accoun for all he movemens an losses of fish. C( ) F N H ( ) This sysem of equaions can be solve analyically bu he soluion process is a bi complicae. Some of he eails are provie in Sampson (1991) on he Supplemenal Reaing lis. nsea of rying o ge exac soluions o he ifferenial equaions, insea le's evelop numerical soluions an see wha we can learn from heir behavior. For he numerical soluions we can use he same approach we use for solving he ifferenial equaion for survival, bu we nee o simulaneously solve wo of he hree equaions. The equaion for N H involves N O an vice versa. FW431/531 Copyrigh 2008 by Davi B. Sampson Cach4 - Page 49
Here are he uler approximaion formulas for he soluions o he hree ifferenial equaions. ( ) N H N H + N O + ( ) ( ) N O ( ) + Number of fish in he harvese region + N O( ) Number of fish in he ousie (unharvese) region ( ) C C + ( ) + C( ) Number of fish caugh We nee o specify values for parameers M, F,, an, an iniial values for he hree variables. f iniially here is no ne migraion of fish beween he wo areas, hen... N O ( 0) N H ( 0) an N O ( 0) N H ( 0) Flow in flow ou... an he oal number of fish in boh areas iniially is N H ( 0) + N O ( 0) N H ( 0) 1 +. Here is a hypoheical example wih 0.1 0.2 F 0.4 M 0.2 NH 0 800 600 400 200 The soli line is N H (), he ashe line is N O (), an he oe line is C(). The cach-per-uni-effor (CPU) realize over he inerval (0,) is CPU( ) The ime-average abunance for he whole fish sock is av( N( ) ) C( ). f 1 ( N H ( τ) + N O ( τ) ) τ. 0 FW431/531 Copyrigh 2008 by Davi B. Sampson Cach4 - Page 50
Below are he graphs of CPU/q (he soli line) an av(n) (he oe line) for our example. 1500 500 Noice ha he CPU is NOT proporional o he oal average sock abunance. A ime 0 he CPU/q is abou 2/3 of he oal N, whereas a ime 20 i is abou half. n he usual moel for CPU an sock abunance, he raio of CPU over abunance is equal o a consan q, he cachabiliy coefficien. Here ha raio varies wih ime. Sampson (1991) examines he behavior of he raio CPU( ) av( N( ) ) an shows ha lim 0 q + an lim q + + a F where a +, which is equivalen o he fracion of he oal populaion ha is iniially inaccessible o fishing. f is very small (which means ha a will be small) or if F is very small, hen he wo limis will be nearly equal, an CPU will be he usual inex of sock abunance. f we ha a fish sock ha was enirely seenary an i no move (e.g., clams in a mu fla) hen he cach rae in he fishe area woul ell us nohing abou he abunance of clams ousie he fishe area. The cach rae in he fishe area provies us informaion abou he sock as a whole only o he exen ha he fishe area encompasses he whole sock (1-a). Moel 2: Migraion hrough a Fishe Area Again fishing operaions occur only in harves area H an wihin his area here are f fishing operaions operaing inepenenly an wihou inerference, an each fishing operaion caches fish a an insananeous rae of q per fish so ha F f q. Here he fish migrae ino area H from a ownsream populaion a an insananeous rae of A an emigrae o an upsream populaion a an insananeous rae of B. Naural moraliy occurs everywhere a an insananeous rae of M. FW431/531 Copyrigh 2008 by Davi B. Sampson Cach4 - Page 51
C A F B N D N H N U Downsream area Harvese area Upsream area Le N D be he number of living fish in he ownsream area, le N H be he number insie area H, le N U be he number of living fish in he upsream area H, an le C be he cumulaive cach of fish. The behavior of N D, N H, N U, an C are governe by he following sysem of ifferenial equaions: N D( ) ( A + M) N D ( ) N U( ) B N H ( ) M N U ( ) A N D ( ) ( B + M + F) N H ( ) C( ) F N H ( ) This sysem of equaions can be solve sequenially saring wih he one for N D bu we will use he numerical approach as we i for Moel 1. For iniial coniions assume ha he enire populaion is ownsream an ha no fish have been caugh ye. Here is an example wih A 2 B 0.5 F 0.5 M 0.2 ND 0 800 600 400 200 The soli line is N D (), he ashe line is N H (), he oe line is N U (),an he o-ash line is C(). FW431/531 Copyrigh 2008 by Davi B. Sampson Cach4 - Page 52
An here are he graphs of CPU/q (he soli line) an av(n) (he oe line). 500 Again noice ha he CPU is no proporional o he oal average sock abunance. The relaionship beween cach-per-uni-effor an fish abunance epens in a funamenal manner on he mechanisms of fish movemen an reisribuion. FW431/531 Copyrigh 2008 by Davi B. Sampson Cach4 - Page 53