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September  2013, 12(5): 1927-1941. doi: 10.3934/cpaa.2013.12.1927

On qualitative analysis for a two competing fish species model with a combined non-selective harvesting effort in the presence of toxicity

Yunfeng Jia 1, , Jianhua Wu 2, and  Hong-Kun Xu 3,

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

Received  October 2011 Revised  August 2012 Published  January 2013

In this paper, a two competing fish species model with combined harvesting is concerned, both the species obey the law of logistic growth and release a toxic substance to the other. Use spectrum analysis and bifurcation theory, the stability of semi-trivial solution, positive constant solution and the bifurcation solutions of model are investigated. We discuss bifurcation solutions which emanate from positive constant solution and trivial solution by taking the growth rate as bifurcation parameter. By the monotonic method, the existence result of positive steady-state of the model is discussed. The possibility of existence of a bionomic equilibrium is also obtained by taking the economical factor into consideration. Finally, some numerical examples are given to illustrate the results.
Keywords: Competing model, stability, bifurcation theory, monotonic method, bionomic equilibrium..
Mathematics Subject Classification: Primary: 92D25; 93C20; Secondary: 35K5.
Citation: Yunfeng Jia, Jianhua Wu, Hong-Kun Xu. On qualitative analysis for a two competing fish species model with a combined non-selective harvesting effort in the presence of toxicity. Communications on Pure and Applied Analysis, 2013, 12 (5) : 1927-1941. doi: 10.3934/cpaa.2013.12.1927
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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