A model of a fishery with fish stock involving delay equations

Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences

Volumn 367, Issue 1908, 2009, Pages 4907-4922

  Auger, P ^a   Ducrot, Arnaud ^b  

^a Sorbonne Universités   (France)

^b UNIVERSITÉ DE BORDEAUX   (France)

Author keywords

Delay differential equation; Fishery models; Hopf bifurcation

Indexed keywords

DIFFERENTIAL EQUATIONS; FISHERIES; MATHEMATICAL MODELS;

BIFURCATION ANALYSIS; DELAY DIFFERENTIAL EQUATION; DELAY DIFFERENTIAL EQUATIONS; DELAY EQUATIONS; FISH STOCK; INFINITE DELAY; NEW MATHEMATICAL MODEL; STEADY STATE;

HOPF BIFURCATION;

ANIMAL; ARTICLE; BIOLOGICAL MODEL; COMPUTER SIMULATION; ECONOMICS; FISH; FOOD INDUSTRY; GROWTH, DEVELOPMENT AND AGING; METHODOLOGY; STATISTICAL MODEL;

ANIMALS; COMPUTER SIMULATION; FISHERIES; FISHES; MODELS, BIOLOGICAL; MODELS, ECONOMIC;

EID: 73349120941     PISSN: 1364503X     EISSN: None     Source Type: Journal    
DOI: 10.1098/rsta.2009.0147     Document Type: Article

References (19)

1. Adimy, M., Bouzahir, H. & Ezzinbi, K. 2004a Local existence or a class of partial neutral functional differential equations with infinite delay. Diff. Equ. Dyn. Syst. 12, 353-370.
2. Adimy, M., Bouzahir, H. & Ezzinbi, K. 20046 Existence and stability for some partial neutral functional differential equations with infinite delay. J. Math. Anal. Appl. 294, 438-461. (doi:10.1016/j.jmaa.2004.02. 033)
3. Adimy, M., Alia, M. & Ezzinbi, K. Submitted. Partial neutral functional differential equations with infinite delay in extrapolation spaces.
4. Aubin, J.-P. 1991 Viability theory, p. 543. Boston, MA: Birkhäuser.
5. Barbier, E. B., Strand, I. & Sathirathai, S. 2002 Do open access conditions affect the valuation of an externality? Estimating the welfare effects of mangrove-fishery linkages. Environ. Resour. Econ. 21, 343-367. (doi:10.1023/A:1015129502284)
6. Clark, C. W. 1990 Mathematical bioeconomics. The optimal management of renewable resources, 2nd edn. New York, NY: John Wiley & Sons.
7. Cury, P. M., Mullon, C., Garcia, S. M. & Shannon, L. J. 2005 Viability theory for an ecosystem approach to fisheries. ICES J. Mar. Sci. 62, 577-584.
8. Diekmann, O., van Gils, S. A., Verduyn Lunel, S. M. & Walter, H.-O. 1995 Delay equations: functional-, complex-, and nonlinear analysis. New York, NY: Springer-Verlag.
9. Ducrot, A., Liu, Z. & Magal, P. 2008 Essential growth rate for bounded linear perturbation of non-densely defined Cauchy problems. J. Math. Anal. Appl. 341, 501-518. (doi:10.1016/j.jmaa.2207.09.074)
10. p spaces. Can. J. Math.
11. Ducrot, A., Magal, P. & Ruan, S. Submitted. Projectors on the generalized eigenspaces for partial differential equations with delay.
12. Hale, J. K. & Kato, J. 1978 Phase space for retarded equations with infinite delay. Funkcial Ekvac. 21, 11-41.
13. Hassard, B. D., Kazarinoff, N. D. & Wan, Y. H. 1981 Theory and applications of Hopfbifurcation. Cambridge, UK: Cambridge University Press.
14. Liu, Z., Magal, P. & Ruan, S. 2008 Projectors on the generalized eigenspaces for functional differential equations using integrated semigroups. J. Diff. Eq. 244, 1784-1809. (doi:10.1016/j.jde.2008.01.007)
15. Magal, P. & Ruan, S. 2009 Center manifold theorem for semilinear equations with non-dense domain and applications to Hopf bifurcation in age structured models. Mem. Am. Math. Soc. 202, 951. (doi:10.1090/S0065-9266-09-00568-7)
16. Mchich, R., Auger, P., Bravo de la Parra, R. & Raïssi N. 2002 Dynamics of a fishery on two fishing zones with fish stock dependent migrations: aggregation and control. Ecol. Model. 158, 51-62. (doi:10.1016/S0304-3800(02)00237-5)
17. Mchich, R., Charouki, N., Auger, P., Raïssi, N. & Ettahiri, O. 2006 Optimal spatial distribution of the fishing effort in a multi fishing zone model. Ecol. Model. 197, 274-280. (doi:10.1016/j.ecolmodel.2006. 03.026)
18. Smith, V. L. 1968 Economics of production from natural resources. Am. Econ. Rev. 58, 409-431.
19. Smith, V. L. 1969 On models of commercial fishing. J. Political Econ. 77, 181-198. (doi:10.1086/259507)