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On qualitative analysis for a two competing fish species model with a combined non-selective harvesting effort in the presence of toxicity
1. | College of Mathematics and Information Science, Shaanxi Normal University, Xi'an, Shaanxi 710062, China |
2. | College of Mathematics and Information Science, Shaanxi Normal University, Xi’an, Shaanxi 710119 |
3. | Department of Applied Mathematics, National Sun Yat-sen University, Kaohsiung 80424, Taiwan |
References:
[1] |
K. J. Arrow and M. Kurz, "Public Investment, The Rate of Return and Optimal Fiscal Policy," John Hopkins Press, Baltimore, 1970. |
[2] |
C. Azar, J. Holmberg and K. Lindgren, Stability analysis of harvesting in a predator-prey model, J. Theor. Biol., 174 (1995), 13-19.
doi: 10.1006/jtbi.1995.0076. |
[3] |
H. Berguland, Simulation of growth of two marine algae by organic substances excreted by enteromorpha linza in unialgal and axenic cultures, Physiol. Plant., 22 (1969), 1069-1079. |
[4] |
J. Chattopadhyay, Effect of toxic substances on a two-species competitive system, Ecol. Model., 84 (1996), 287-289.
doi: 10.1016/0304-3800(94)00134-0. |
[5] |
J. Chattopadhyay, G. Ghosal and K. S. Chaudhuri, Nonselective harvesting of a prey-predator community with infected prey, Korean J. Compute. Appl. Math., 6 (1999), 601-616. |
[6] |
J. Chattopadhyay, R. R. Sarkar and S. Mandal, Toxinproducing plankton may act as a biological control for planktonic blooms field study and mathematical modelling, J. Theor. Biol., 215 (2002), 333-344.
doi: 10.1006/jtbi.2001.2510. |
[7] |
K. S. Chaudhuri, Dynamic optimization of combined harvesting of a two-species fishery, Ecol. Model., 41 (1988), 17-25.
doi: 10.1016/0304-3800(88)90041-5. |
[8] |
K. S. Chaudhuri and R. S. Saha, On the combined harvesting of a prey-predator system, J. Biol. Syst., 4 (1996), 373-389.
doi: 10.1142/S0218339096000259. |
[9] |
C. W. Clark, "Mathematical Bioeconomics: the Optimal Management of Renewable Resources," Wiley, New York, 1976. |
[10] |
E. Conway, R. Gardner and J. Smoller, Stability and bifurcation of steady-state solutions for predator-prey equations, Adv. Appl. Math., 3 (1982), 288-334.
doi: 10.1016/S0196-8858(82)80009-2. |
[11] |
M. G. Crandall and P. H. Rabinowitz, Bifurcation, perturbation of simple eigenvalues and linearized stability, Arch. Rational Mech. Anal., 52 (1973), 161-180.
doi: 10.1007/BF00282325. |
[12] |
G. Dai and M. Tang, Coexistence region and global dynamics of a harvested predator-prey system, SIAM J. Appl. Math., 58 (1998), 193-210.
doi: 10.1137/S0036139994275799. |
[13] |
J. T. De Luna and T. G. Hallam, Effect of toxicants on population: a qualitative approach iv. Resource-consumer-toxicant models, Ecol. Model., 35 (1987), 249-273.
doi: 10.1016/0304-3800(87)90115-3. |
[14] |
W. Feng and X. Lu, On diffusive population models with toxicants and time delays, J. Math. Anal. Appl., 233 (1999), 373-386.
doi: 10.1006/jmaa.1999.6332. |
[15] |
W. Feng and X. Lu, Global periodicity in a class of reaction-diffusion systems with time delays, Discrete Contin. Dyn. Syst. Ser. B, 3 (2003), 69-78. |
[16] |
H. I. Freedman and J. B. Shukla, Models for the effect of toxicant in a single-species and predator-prey systems, J. Math. Biol., 30 (1991), 15-30.
doi: 10.1007/BF00168004. |
[17] |
T. G. Hallam and C. E. Clark, Nonautonomous logistic equations as models of populations in a deteriorating environment, J. Theor. Biol., 93 (1981), 303-311.
doi: 10.1016/0022-5193(81)90106-5. |
[18] |
T. G. Hallam, C. E. Clark and G. S. Jordan, Effects of toxicants on populations: a qualitative approach II. First order kinetics, J. Math. Biol., 18 (1983), 25-37.
doi: 10.1007/BF00275908. |
[19] |
T.G. Hallam and J.T. De Luna, Effects of toxicants on populations: a qualitative approach III. Environmental and food chain pathways, J. Theor. Biol., 109 (1984), 411-429.
doi: 10.1016/S0022-5193(84)80090-9. |
[20] |
V. Hutson, J. López-Gómez, K. Mischaikow and G. Vickers, Limit behavior for a competing species problem with diffusion, in "Dynamical Systems and Applications" (R. P. Agarwal ed.), World Scientific, Singapore, (1995), 343-358.
doi: 10.1142/9789812796417_0022. |
[21] |
M. Ito, Global aspect of steady-states for competitive-diffusion systems with homogeneous Dirichlet conditions, Phys. D, 14 (1984), 1-28.
doi: 10.1016/0167-2789(84)90002-2. |
[22] |
S. Kumar, S. K. Srivastava and P. Chingakham, Hopf bifurcation and stability analysis in a harvested one-predator-two-prey model, Appl. Math. Comput., 129 (2002), 107-108.
doi: 10.1016/S0096-3003(01)00033-9. |
[23] |
S. J. Maynard, "Models in Ecology," Cambridge Univ. Press, Cambridge, 1974. |
[24] |
M. Mesterton-Gibbons, On the optimal policy for the combined harvesting of independent species, Nat. Resour. Model., 2 (1987), 109-134. |
[25] |
M. Mesterton-Gibbons, On the optimal policy for the combined harvesting of predator and prey, Nat. Resour. Model., 3 (1988), 63-90. |
[26] |
T. J. Monahan and F. R. Trainor, Stimulatory properties of filtrated from green alga hormotila blennista. I. Description, J. Phycol., 6 (1970), 263-269.
doi: 10.1111/j.0022-3646.1970.00263.x. |
[27] |
M. R. Myerscough, B. F. Gray, W. L. Hogarth and J. Norbury, An analysis of an ordinary differential equation model for a two species predator-prey system with harvesting and stocking, J. Math. Biol., 30 (1992), 389-411.
doi: 10.1007/BF00173294. |
[28] |
W.-M. Ni and M. Tang, Turing patterns in the Lengyel-Epstein system for the CIMA reactions, Trans. Amer. Math. Soc., 357 (2005), 3953-3969.
doi: 10.1090/S0002-9947-05-04010-9. |
[29] |
C. V. Pao, On nonlinear reaction-diffusion systems, J. Math. Anal. Appl., 87 (1982), 165-198.
doi: 10.1016/0022-247X(82)90160-3. |
[30] |
D. Sadhukhan, L. N. Sahoo, B. Mondal and M. Maiti, Food chain model with optimal harvesting in fuzzy environment, J. Appl. Math. Comput., 34 (2010), 1-18.
doi: 10.1007/s12190-009-0301-2. |
[31] |
R. R. Sarkar and J. Chattopadhayay, A technique for estimating maximum harvesting effort in a stochastic fishery model, J. Biosci., 28 (2003), 497-506.
doi: 10.1007/BF02705124. |
[32] |
J. B. Shukla and B. Dubey, Simultaneous effects of two toxicants on biological species: a mathematical model, J. Biol. Syst., 4 (1996), 109-130.
doi: 10.1142/S0218339096000090. |
[33] |
J. Smoller, "Shock Waves and Reaction-Diffusion Equations," Springer-verlag, New York, 1999. |
[34] |
D. Xiao and S. Ruan, Bogdanov-Takens bifurcations in predator-prey systems with constant rate harvesting, in "Differential Equations with Applications to Biology," Halifax, NS, (1997), 493-506, Fields Inst. Commun., 21, Amer. Math. Soc., Providence, RI, 1999. |
[35] |
Q. Ye and Z. Li, "Introduction to Reaction-Diffusion Equations," Science Press, Beijing, 1990. |
[36] |
C. Zhong, X. Fan and W. Chen, "Introduction to Nonlinear Functional Analysis," Lanzhou Univ. Press, Lanzhou, 1998. |
show all references
References:
[1] |
K. J. Arrow and M. Kurz, "Public Investment, The Rate of Return and Optimal Fiscal Policy," John Hopkins Press, Baltimore, 1970. |
[2] |
C. Azar, J. Holmberg and K. Lindgren, Stability analysis of harvesting in a predator-prey model, J. Theor. Biol., 174 (1995), 13-19.
doi: 10.1006/jtbi.1995.0076. |
[3] |
H. Berguland, Simulation of growth of two marine algae by organic substances excreted by enteromorpha linza in unialgal and axenic cultures, Physiol. Plant., 22 (1969), 1069-1079. |
[4] |
J. Chattopadhyay, Effect of toxic substances on a two-species competitive system, Ecol. Model., 84 (1996), 287-289.
doi: 10.1016/0304-3800(94)00134-0. |
[5] |
J. Chattopadhyay, G. Ghosal and K. S. Chaudhuri, Nonselective harvesting of a prey-predator community with infected prey, Korean J. Compute. Appl. Math., 6 (1999), 601-616. |
[6] |
J. Chattopadhyay, R. R. Sarkar and S. Mandal, Toxinproducing plankton may act as a biological control for planktonic blooms field study and mathematical modelling, J. Theor. Biol., 215 (2002), 333-344.
doi: 10.1006/jtbi.2001.2510. |
[7] |
K. S. Chaudhuri, Dynamic optimization of combined harvesting of a two-species fishery, Ecol. Model., 41 (1988), 17-25.
doi: 10.1016/0304-3800(88)90041-5. |
[8] |
K. S. Chaudhuri and R. S. Saha, On the combined harvesting of a prey-predator system, J. Biol. Syst., 4 (1996), 373-389.
doi: 10.1142/S0218339096000259. |
[9] |
C. W. Clark, "Mathematical Bioeconomics: the Optimal Management of Renewable Resources," Wiley, New York, 1976. |
[10] |
E. Conway, R. Gardner and J. Smoller, Stability and bifurcation of steady-state solutions for predator-prey equations, Adv. Appl. Math., 3 (1982), 288-334.
doi: 10.1016/S0196-8858(82)80009-2. |
[11] |
M. G. Crandall and P. H. Rabinowitz, Bifurcation, perturbation of simple eigenvalues and linearized stability, Arch. Rational Mech. Anal., 52 (1973), 161-180.
doi: 10.1007/BF00282325. |
[12] |
G. Dai and M. Tang, Coexistence region and global dynamics of a harvested predator-prey system, SIAM J. Appl. Math., 58 (1998), 193-210.
doi: 10.1137/S0036139994275799. |
[13] |
J. T. De Luna and T. G. Hallam, Effect of toxicants on population: a qualitative approach iv. Resource-consumer-toxicant models, Ecol. Model., 35 (1987), 249-273.
doi: 10.1016/0304-3800(87)90115-3. |
[14] |
W. Feng and X. Lu, On diffusive population models with toxicants and time delays, J. Math. Anal. Appl., 233 (1999), 373-386.
doi: 10.1006/jmaa.1999.6332. |
[15] |
W. Feng and X. Lu, Global periodicity in a class of reaction-diffusion systems with time delays, Discrete Contin. Dyn. Syst. Ser. B, 3 (2003), 69-78. |
[16] |
H. I. Freedman and J. B. Shukla, Models for the effect of toxicant in a single-species and predator-prey systems, J. Math. Biol., 30 (1991), 15-30.
doi: 10.1007/BF00168004. |
[17] |
T. G. Hallam and C. E. Clark, Nonautonomous logistic equations as models of populations in a deteriorating environment, J. Theor. Biol., 93 (1981), 303-311.
doi: 10.1016/0022-5193(81)90106-5. |
[18] |
T. G. Hallam, C. E. Clark and G. S. Jordan, Effects of toxicants on populations: a qualitative approach II. First order kinetics, J. Math. Biol., 18 (1983), 25-37.
doi: 10.1007/BF00275908. |
[19] |
T.G. Hallam and J.T. De Luna, Effects of toxicants on populations: a qualitative approach III. Environmental and food chain pathways, J. Theor. Biol., 109 (1984), 411-429.
doi: 10.1016/S0022-5193(84)80090-9. |
[20] |
V. Hutson, J. López-Gómez, K. Mischaikow and G. Vickers, Limit behavior for a competing species problem with diffusion, in "Dynamical Systems and Applications" (R. P. Agarwal ed.), World Scientific, Singapore, (1995), 343-358.
doi: 10.1142/9789812796417_0022. |
[21] |
M. Ito, Global aspect of steady-states for competitive-diffusion systems with homogeneous Dirichlet conditions, Phys. D, 14 (1984), 1-28.
doi: 10.1016/0167-2789(84)90002-2. |
[22] |
S. Kumar, S. K. Srivastava and P. Chingakham, Hopf bifurcation and stability analysis in a harvested one-predator-two-prey model, Appl. Math. Comput., 129 (2002), 107-108.
doi: 10.1016/S0096-3003(01)00033-9. |
[23] |
S. J. Maynard, "Models in Ecology," Cambridge Univ. Press, Cambridge, 1974. |
[24] |
M. Mesterton-Gibbons, On the optimal policy for the combined harvesting of independent species, Nat. Resour. Model., 2 (1987), 109-134. |
[25] |
M. Mesterton-Gibbons, On the optimal policy for the combined harvesting of predator and prey, Nat. Resour. Model., 3 (1988), 63-90. |
[26] |
T. J. Monahan and F. R. Trainor, Stimulatory properties of filtrated from green alga hormotila blennista. I. Description, J. Phycol., 6 (1970), 263-269.
doi: 10.1111/j.0022-3646.1970.00263.x. |
[27] |
M. R. Myerscough, B. F. Gray, W. L. Hogarth and J. Norbury, An analysis of an ordinary differential equation model for a two species predator-prey system with harvesting and stocking, J. Math. Biol., 30 (1992), 389-411.
doi: 10.1007/BF00173294. |
[28] |
W.-M. Ni and M. Tang, Turing patterns in the Lengyel-Epstein system for the CIMA reactions, Trans. Amer. Math. Soc., 357 (2005), 3953-3969.
doi: 10.1090/S0002-9947-05-04010-9. |
[29] |
C. V. Pao, On nonlinear reaction-diffusion systems, J. Math. Anal. Appl., 87 (1982), 165-198.
doi: 10.1016/0022-247X(82)90160-3. |
[30] |
D. Sadhukhan, L. N. Sahoo, B. Mondal and M. Maiti, Food chain model with optimal harvesting in fuzzy environment, J. Appl. Math. Comput., 34 (2010), 1-18.
doi: 10.1007/s12190-009-0301-2. |
[31] |
R. R. Sarkar and J. Chattopadhayay, A technique for estimating maximum harvesting effort in a stochastic fishery model, J. Biosci., 28 (2003), 497-506.
doi: 10.1007/BF02705124. |
[32] |
J. B. Shukla and B. Dubey, Simultaneous effects of two toxicants on biological species: a mathematical model, J. Biol. Syst., 4 (1996), 109-130.
doi: 10.1142/S0218339096000090. |
[33] |
J. Smoller, "Shock Waves and Reaction-Diffusion Equations," Springer-verlag, New York, 1999. |
[34] |
D. Xiao and S. Ruan, Bogdanov-Takens bifurcations in predator-prey systems with constant rate harvesting, in "Differential Equations with Applications to Biology," Halifax, NS, (1997), 493-506, Fields Inst. Commun., 21, Amer. Math. Soc., Providence, RI, 1999. |
[35] |
Q. Ye and Z. Li, "Introduction to Reaction-Diffusion Equations," Science Press, Beijing, 1990. |
[36] |
C. Zhong, X. Fan and W. Chen, "Introduction to Nonlinear Functional Analysis," Lanzhou Univ. Press, Lanzhou, 1998. |
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