zbMATH — the first resource for mathematics

Differential movement and movement bias models for marine protected areas. (English) Zbl 1262.34049
In connection with marine protected areas for overfished stocks, the authors consider mathematical models constructed by means of boundary value problems for partial (and for describing the steady state – ordinary) differential equations. In the first of the considered models, the parameters are piecewise constant with the diffusion coefficient and mortality rate different inside and outside the protected area. Next, a movement bias model is studied. It is supposed here that at the boundary, the probability of moving to the protected area is higher than of moving outside. Finally, a model with constant diffusion coefficient and smooth in the space variable mortality rate is considered. Different models are analyzed and compared with empirical evidence.

34C60 Qualitative investigation and simulation of ordinary differential equation models
34B60 Applications of boundary value problems involving ordinary differential equations
35Q92 PDEs in connection with biology, chemistry and other natural sciences
Full Text: DOI
[1] Baskett ML, Levin SA, Gaines SD, Dushoff J (2005) Marine reserve design and the evolution of size at maturation in harvested fish. Ecol Appl 15: 882–901 · doi:10.1890/04-0723
[2] Cantrell RS, Cosner C (1999) Diffusion models for population dynamics incorporating individual behavior at boundaries: applications to refuge design. Theor Popul Biol 55: 189–207 · Zbl 0958.92028 · doi:10.1006/tpbi.1998.1397
[3] Cantrell RS, Cosner C, Lou Y (2007) Advection-mediated coexistence of competing species. Proc R Soc Edinburgh 137A: 487–518 · Zbl 1139.35048
[4] Cantrell RS, Cosner C (2003) Spatial ecology via reaction–diffusion equations. Wiley, New York · Zbl 1059.92051
[5] Chen X, Lou Y (2008) Principal Eigenvalue and Eigenfunctions of an elliptic operator with large advection and its application to a competition model. Indiana Univ Math J 57: 627–658 · Zbl 1153.35056 · doi:10.1512/iumj.2008.57.3204
[6] Claudet J, Osenberg CW, Benedetti-Cecchi L, Domenici P, Garcia-Charton JA, Perez-Ruzafa A, Badalamenti F, Bayle-Sempere J, Brito A, Bulleri F, Culioli JM, Dimech M, Falcón JM, Guala I, Milazzo M, Sánchez-Meca J, Somerfield PJ, Stobart B, Vandeperre F, Valle C, Planes S (2008) Marine reserves: size and age do matter. Ecol Lett 11: 481–489 · doi:10.1111/j.1461-0248.2008.01166.x
[7] Claudet J, Osenberg CW, Domenici P, Badalamenti F, Milazzo M, Falcón JM, Bertocci I, Benedetti-Cecchi L, Garcia-Charton J-A, Goñi R, Borg JA, Forcada A, De Lucia A, Pérez-Ruzafa A, Afonso P, Brito A, Guala I, Le Diréach L, Sanchez-Jerez P, Somerfield PJ, Planes S (2010) Marine reserves: fish life history and ecological traits matter. Ecol Appl 20: 830–839 · doi:10.1890/08-2131.1
[8] Eggleston DB, Parsons DM (2008) Disturbance-induced ’spill-in’ of Caribbean lobster to marine reserves. Mar Ecol Prog Ser 371: 213–220 · doi:10.3354/meps07699
[9] Gerber LR, Botsford LW, Hastings A, Possingham HP, Gaines SD, Palumbi SR, Andelman S (2003) Population models for marine reserve design: a retrospective and prospective synthesis. Ecol Appl 13: 47–64 · doi:10.1890/1051-0761(2003)013[0047:PMFMRD]2.0.CO;2
[10] Goñi R, Adlerstein S, Alvarez-Berastegui D, Forcada A, Reñones O, Criquet G, Polti S, Cadiou G, Valle C, Lenfant P, Bonhomme P, Pérez-Ruzafa A, Sánchez-Lizaso FL, Garcia-Charton JA, Bernard G, Stelzenmüller V, Planes S (2008) Spillover from six western Mediterranean marine protected areas: evidence from artisanal fisheries. Mar Ecol Prog Ser 366: 159–174 · doi:10.3354/meps07532
[11] Halpern BS (2003) The impact of marine reserves: do reserves work and does reserve size matter?. Ecol Appl 13: 117–137 · doi:10.1890/1051-0761(2003)013[0117:TIOMRD]2.0.CO;2
[12] Hilborn R, Stokes K, Maguire JJ, Smith T, Botsford LW, Mangel M, Orensanz J, Parma A, Rice J, Bell J, Cochrane KL, Garcia S, Hall SJ, Kirkwood GP, Sainsbury K, Stefansson G, Walters C (2004) When can marine reserves improve fisheries management?. Ocean Coast Manage 47: 197–205 · doi:10.1016/j.ocecoaman.2004.04.001
[13] Keller EF, Segel LA (1970) Initiation of slime mold aggregation viewed as an instability. J Theor Biol 26: 399–415 · Zbl 1170.92306 · doi:10.1016/0022-5193(70)90092-5
[14] Kellner JB, Tetreault I, Gaines SD, Nisbet RM (2007) Fishing the line near marine reserves in single and multispecies fisheries. Ecol Appl 17: 1039–1054 · doi:10.1890/05-1845
[15] Le Quesne WJF, Codling EA (2009) Managing mobile species with MPAs: the effects of mobility, larval dispersal, and fishing mortality on closure size. ICES J Mar Sci 66: 122–131 · doi:10.1093/icesjms/fsn202
[16] Lou Y (2006) On the effects of migration and spatial heterogeneity on single and multiple species. J Differ Equ 223: 400–426 · Zbl 1097.35079 · doi:10.1016/j.jde.2005.05.010
[17] Malvadkar U, Hastings A (2008) Persistence of mobile species in marine protected areas. Fish Res 91: 69–78 · doi:10.1016/j.fishres.2007.11.023
[18] Moffitt EA, Botsford LW, Kaplan DM, O’Farrell MR (2009) Marine reserve networks for species that move within a home range. Ecol Appl 19: 1835–1847 · doi:10.1890/08-1101.1
[19] Nagai T, Senba T, Yoshida K (1997) Application of the Trudinger–Moser inequality to a parabolic system of chemotaxis. Funkcilaj Ekvacioj 40: 411–433 · Zbl 0901.35104
[20] Osenberg CW, Bolker BM, White JS, St. Mary C, Shima JS (2006) Statistical issues and study design in ecological restorations: lessons learned from marine reserves. In: Falk D, Palmer N, Zedler J (eds) Foundations of restoration ecology. Island Press, Washington, DC, pp 280–302
[21] Ovaskainen O, Cornell SJ (2003) Biased movement at a boundary and condditional occcupancy times for diffusion processes. J Appl Probability 40: 557–580 · Zbl 1078.60061 · doi:10.1239/jap/1059060888
[22] Pérez-Ruzafa A, Martín E, Marcos C, Zamarro JM, Stobart B, Harmelin-Vivien M, Polti S, Planes S, García-Charton JA, González-Wangüemert M (2008) Modeling spatial and temporal scales for spill-over and biomass exportation from MPAs and their potential for fisheries enhancement. J Nat Conserv 16: 234–255 · doi:10.1016/j.jnc.2008.09.003
[23] Roberts CM, Bohnsack JA, Gell F, Hawkins JP, Goodridge R (2001) Effects of marine reserves on adjacent fisheries. Science 294: 1920–1923 · doi:10.1126/science.294.5548.1920
[24] Rodwell LD, Barbier EB, Roberts CM, McClanahan TR (2003) The importance of habitat quality for marine reserve-fishery linkages. Can J Fisheries Aquat Sci 60: 171–181 · doi:10.1139/f03-009
[25] Sale PF, Cowen RK, Danilowicz BS, Jones GP, Kritzer JP, Lindeman KC, Planes S, Polunin NVC, Russ GR, Sadovy YJ, Steneck RS (2005) Critical science gaps impede use of no-take fishery reserves. Trends Ecol Evol 20: 74–80 · doi:10.1016/j.tree.2004.11.007
[26] Selig ER, Bruno JF (2010) A global analysis of the effectiveness of marine protected areas in preventing coral loss. PloS ONE 5
[27] Shigesada N, Kawasaki K, Teramoto E (1986) Traveling periodic waves in heterogeneous environments. Theor Popul Biol 30: 143–160 · Zbl 0591.92026 · doi:10.1016/0040-5809(86)90029-8
[28] Warner RR, Hughes TP (1988) The population dynamics of reef fishes. In: Proceedings of 6th international coral reef symposium, vol 1. Townsville, pp 146–155
[29] West CD, Dytham C, Righton D, Pitchford JW (2009) Preventing overexploitation of migratory fish stocks: the efficacy of marine protected areas in a stochastic environment. ICES J Mar Sci 66: 1919–1930 · doi:10.1093/icesjms/fsp159
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.