The Fisheries Dissipative Effect Modelling. Through Dynamical Systems and Chaos Theory

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1 Applied Mahemaical Sciences, Vol. 8, 0, no., HIKARI Ld, hp://dx.doi.org/0.988/ams The Fisheries Dissipaive Effec Modelling Through Dynamical Sysems and Chaos Theory M. A. M. Ferreira, J. A. Filipe Insiuo Universiário de Lisboa (ISCTE IUL), BRU - IUL, Lisboa, Porugal M. Coelho SOCIUS, ISEG/Universidade de Lisboa Copyrigh 0 M. A. M. Ferreira, J. A. Filipe and M. Coelho. This is an open access aricle disribued under he Creaive Commons Aribuion License, which permis unresriced use, disribuion, and reproducion in any medium, provided he original work is properly cied. Absrac I is presened a very simple mahemaical model o sudy he dissipaive effec in fisheries, based in dynamical sysems and in is approach hrough chaos heory. Acually, due o consan misconducs, forgeing heir consequences and only prosecuing easy profis, he risk of exincion is very real for fisheries resources. From i resuls he imporance of his kind of models. Keywords: Chaos, dynamical sysems, fisheries. Inroducion Many sudies have been performed on fisheries in many environmens worldwide, using various frameworks. In his work, chaos heory is used o analyse how fisheries can be organized, rying o avoid is dissipaive effec. I is inended o consider chaos heory and complexiy o build a model o explain he caches dissipaive effec in fisheries sysems. Anoher work on his subjec is[ ].

2 57 M. A. M. Ferreira, J. A. Filipe and M. Coelho Equaions and Chaos Chaos is presen in many mahemaical compuer problems and in laboraory research. As soon as he idea of nonlineariy is inroduced ino heoreical models, chaos becomes obvious. The chaos heory and he complexiy heory reflec ha phenomena in many aciviies, such as fisheries. Williams (997), see [ ], refers ha phenomena happen over ime eiher a discree inervals or as coninuously. Discree inervals can be spaced evenly in ime or irregularly in ime. Coninuous phenomena in principle migh be measured coninuously. Bu generally hey are measured a discree inerval, being possible o obain good approximaions. Differenial equaions are ofen he mos adequae mahemaical way o describe a smooh coninuous evoluion. Bu, frequenly, o solve hem is no an easy ask. Difference equaions can be solved righ away if hey are linear. And, in his form, hey are ofen differenial equaions accepable approximaions. Olsen and Degn (985), see [ ], even risk o say ha difference equaions are he mos powerful vehicle o he undersanding of chaos. Ohers difference equaions, han he linear, presen difficulies even greaer han he differenial equaions. 3 The Model The fisheries mahemaical model o be presened follows he model presened in Berliner (99), see [ ], relaed o dissipaive sysems in he presence of chaos. Tha auhor shows ha non-inveribiliy is required o observe chaos for one-dimensional dynamic sysems. He refers also ha everywhere inverible maps in wo or more dimensions can exhibi chaoic behavior. The srange aracors

3 Fisheries dissipaive effec modelling 575 sudy shows ha in he long erm, as ime proceeds, he sysems rajecories may become rapped in cerain sysem sae space bounded regions. The model presened in Berliner (99), see again [ ], is an example, in wo dimensions, of he Hénon map, exhibiing he propery of having a srange aracor. y The Hénon map appears represened by he equaions: bx, for fixed values of a and b, 0,, x y ax and This inverible map has srange aracors and simulaneously srong sensiiviy o iniial condiions. The Hénon map, represening a ransformaion from equal o M o M, has Jacobian b. If 0 b, he Hénon map conracs he domains o which i is applied. These maps are said o be dissipaive. I is evidenly possible o sugges a model on his basis for fisheries. If a general siuaion is considered, he following equaions may represen a sysem in which fish socks, a ime, are given by x and caches by y. The model is as follows: x F( x ) y and y bx. I is a Hénon model generalizaion. The Jacobian is equal o b. As y is a porion of x, 0 b. So, i is a dissipaive model and he values of x are resriced o a bounded domain. Considering he paricular case below: so, x x y x x y, and y bx, and x x bx 0.Now, afer solving he characerisic equaion associaed o he difference equaion, see [ ], i is obained: b k b or k ; calling b and being 0 b,

4 576 M. A. M. Ferreira, J. A. Filipe and M. Coelho 3.So, 0 if 0when b. 0 b and 3 0 if b, being Consequenly, for 0 b, b b x A A. And for b, x A A. Finally, for b x b Acos arccos Asen arccos. b b In hese soluions, A and A are real consans. Noe ha he bases of powers are always beween 0 and. So, lim x 0 and whaever he value ofb, he dissipaive effec is real, even leading o he exincion of he specie. Of course, his is eviden according o he premises of his paricular siuaion of he model. Concluding his approach, he general model does no allow obaining in general such explici soluions. Bu, of course, wih simple compuaional ools i is possible o obain recursively concree ime series soluions afer esablishing he iniial value x 0 and o check he dissipaive effec.

5 Fisheries dissipaive effec modelling 577 Conclusions Chaos heory has become imporan in many conexs of dynamic sysems explanaion. Fish socks modeling may be considered on he basis of an approach associaed wih chaos heory, insead considering he usual prospec based on classical models. In fisheries analysis i is quie imporan o undersand ha overfishing may cause a problem of irreversibiliy in he recovering of several species. Anyway, o analyze his case specific siuaion, i is mandaory o obain enough daa o esimae he funcion which fis for ha paricular case. In his work, a model evidencing he dissipaive effec of caches on fisheries was presened. Addiionally a paricular model showed how socks are dissipaed and may end o he exincion. References [] G. P. Williams, Chaos Theory Tamed, Washingon, D. C., Joseph Henry Press, 997. [] J. A. Filipe, M. A. M. Ferreira, M. Coelho, The drama of he commons. An applicaion of Courno-Nash model o he sardine in Poruguese waers. The effecs of collusion, Journal of Agriculural, Food, and Environmenal Sciences, (008),. [3] L. F. Olsen, H. Degn, Chaos in biological sysems, Quarerly Revue of Biophysics, 8(985),, [] L. M. Berliner, Saisics, probabiliy and chaos, Saisical Science, 7(99),, 69-.

6 578 M. A. M. Ferreira, J. A. Filipe and M. Coelho [5] M. A. M. Ferreira, R. Menezes, Equações com Diferenças-Aplicações em Problemas de Finanças, Economia, Sociologia e Anropologia, Lisboa, Edições Sílabo, 99. [6] M. A. M. Ferreira, J. A. Filipe, M. Coelho, M. I. C. Pedro, Modelling he dissipaive effec of fisheries, China-USA Business Review, 0 (0),, 0-. [7] P. Bergé, Y. Pomeau, C. Vidal, Order wihin Chaos, New York, John Wiley, 98. Received: December 8, 03