Project title: Mathematical Models of Fish Populations in Marine Reserves

Transcription

1 Applcaton for Fundng (Malaspna Research Fund) Date: November 0, 2005 Project ttle: Mathematcal Models of Fsh Populatons n Marne Reserves Dr. Lev V. Idels Unversty College Professor Mathematcs Department Malaspna Unversty-College Buldng 360/304 Phone: & Emal: delsl@mala.bc.ca Dr. Jon Schnute Fsheres and Oceans Canada Pacfc Bologcal Staton 390 Hammond Bay Road Nanamo, Brtsh Columba, Canada V9T 6N7 Phone: & Emal: schnutej@pac.dfo-mpo.gc.ca Research abstract Fshery management typcally nvolves many sources of uncertanty. Fsh populatons respond to varable events n nature, often unknown to fshery managers. The avalable data usually contan hgh levels of measurement error. Marne reserves offer a potental management tool for dealng wth these uncertantes. Protected populatons n areas closed to all fshng could act as buffers that mtgate excessve fshng elsewhere. Furthermore, reserves could help mantan healthy ecosystems wth natural levels of speces and habtat dversty. As the world populaton grows, pressure ncreases to harvest unsustanable quanttes of our marne resources. The defnton of a safe harvest level often becomes controversal. Marne reserves offer an alternatve strategy that could supplement or even replace the attempt to defne safe harvest lmts. In ths project we use mathematcal models to examne management strateges assocated wth fully protected marne reserves. We consder two lnked populatons n adjacent areas, where one s exposed to fshng and the other s not. Our models allow us to explore the nfluence of a reserve on the levels of stock bomass and catch. Assocated software would make t easy for students and researchers to nvestgate potental optons for effectve harvest management n the context of protected marne populatons. MUC Research Fund Proposal: Models of marne reserves Page

2 Potental mpacts of the research. Enhance the reputaton of MUC through nternatonal conferences and presentatons at Canadan unverstes. The prncpal nvestgators routnely work wth colleagues at the Unversty of BC, Smon Fraser Unversty, the Unversty of Alberta, and the Unversty of Calgary. 2. Encourage collaboraton wth other MUC Departments (ncludng Fsheres, Bology, and Computng Scence) and other outsde agences (ncludng the Pacfc Bologcal Staton n Nanamo and the Centre for Integratve Genetcs, Norwegan Unversty of Lfe Scences, As, Norway). 3. Produce tangble outcomes, ncludng artcles n scentfc journals, presentatons at nternatonal conferences, and reports on software and smulaton results. 4. Provde nsghts nto solutons for practcal problems n fshery management. 5. Involve undergraduate and postgraduate students n actve scentfc research. 6. Promote broader understandng of the ssues nvolved n fshery management, a subject hghly relevant to Brtsh Columba s coastal communtes. Research Proposal. Introducton Tradtonal methods of controllng fshng nclude: lmtng the effort (restrctng the fshng season); lmtng the catch (restrctng the number or bomass of fsh captured); lmtng the fsh sze (restrctng the sze of fsh captured); lmtng the locaton (restrctng the places avalable for fshng). The last opton ncludes the establshment of marne reserves, areas permanently closed to all fshng. As descrbed n the abstract, these can supplement or potentally replace other methods of regulaton. For ths reason, the reserve concept has recently attracted great nterest n the communty of fsheres scentsts and managers. 2. Outlne of proposed research Our research wll use relatvely smple mathematcal models to nvestgate the effects of a marne reserve on the fsh populatons nsde and outsde the reserve. For example, suppose that two regons have habtat areas A ( =,2), and let B ( t ) denote the bomass of fsh at tme t n regon. Assume that fshng takes place only n regon 2, wth regon establshed as a marne reserve. Then the coupled system of logstc dfferental equatons MUC Research Fund Proposal: Models of marne reserves Page 2

3 db 2 B rb B B = + γ dt K A2 A () db B B B = rb + γ ht ( ) B2 (2) dt K2 A A2 could represent the populaton dynamcs n each regon, where r and K denote the growth rate and carryng capacty n regon, γ represents a dsperson parameter for fsh movement toward a regon of lower bomass densty, and ht () s the harvest rate at tme t n regon 2. In partcular, the two equatons are lnked by the movement terms, whch depend on γ and a dfference of bomass denstes per unt of habtat area. The depleton of fsh n regon 2 wll generally decrease the fsh densty there and cause fsh to move away from a hgher densty n the protected regon. The addtonal assumpton K = κ A (3) would assocate the carryng capacty n regon wth the correspondng habtat area A, where the parameter κ specfes the capacty per unt area. Smlarly, we could assume a common growth rate r = r n both regons. Further model varatons would mmc regme changes n the envronment by allowng tme-dependent bologcal parameters r ( t ) and K ( t). The model () (2) would allow us to examne a varety of nterestng questons. For example,. How does the choce of parameters and harvest strategy ht ( ) nfluence the qualtatve behavour of the populaton dynamcs? Under what crcumstances does the reserve enhance the growth of the overall system? 2. Gven a total habtat area A, how should t be dvded nto a reserve area and fshery area 2 wth reserve? A A= A + A 2? What are the advantages and dsadvantages of ncreasng the sze of the 3. How do tme-varyng parameters nfluence ratonal harvestng strateges ht ( )? Under what crcumstances mght seasonal harvestng prove effectve? 4. What data should be collected, and how mght these data be used to set the harvest polcy? A MUC Research Fund Proposal: Models of marne reserves Page 3

4 3. Methods of nvestgaton Qualtatve and theoretcal methods: dfferental and ntegral nequaltes, stablty and oscllaton theorems for systems of ordnary dfferental equatons, control and optmzaton. Numercal methods: Computer smulatons usng Matlab and other modern scentfc software packages. Where possble, we wll encourage students to use free software packages, such as Sclab ( and R ( 4. References Ganes, S.D., B. Gaylord, and J.L. Larger Avodng current oversghts n marne reserve desgn. Ecologcal Applcatons 3: S32-S46. Gerber, L.R., L.W. Botsford, A. Hastngs, H.P. Possngham, S.D. Ganes, S.R. Palumb, and S. Andelman Populaton models for marne reserve desgn: a retrospectve and prospectve synthess. Ecologcal Applcatons 3: S47-S64. Halpern, B.S The mpact of marne reserve: do reserves work and does reserve sze matter? Ecologcal Applcatons 3: S7-S37. Halpern, B.S. and R.R. Warner Marne reserves have rapd and lastng effects. Ecology Letters 5: Shea, K., et al Management of populatons n conservaton, harvestng and control. Trends n Ecology and Evoluton, 3(9): Budget 6. Tmelnes Dec 2005 Feb 2006: Analyse and develop models of marne populaton dynamcs. Develop and justfy mathematcally a methodology for studyng marne reserves wth customsed models. March 05 Sep 06: Perform numercal smulatons Conduct qualtatve analyses of dynamc models for marne reserves systems. Oct 06 December 06: Prepare results for publcaton. MUC Research Fund Proposal: Models of marne reserves Page 4

5 7. Referees Prof. Fred Brauer Department of Mathematcs Unversty of Brtsh Columba, Vancouver BC V6T Z2 Canada, fax , phone brauer@math.ubc.ca Prof. Thomas Hllen, Department of Mathematcal and Statstcal Scences Unversty of Alberta, Edmonton, AB, T6G 2G, Canada phone: , fax: thllen@ualberta.ca Prof. Istvan Gyor Head of Department, Department of Mathematcs and Computng Unversty of Veszprem P.O. Box 58, 820 Veszprem, Hungary Phone/fax: (36) MUC Research Fund Proposal: Models of marne reserves Page 5