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

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.
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 & 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, (1970). Google Scholar

[2]

C. Azar, J. Holmberg and K. Lindgren, Stability analysis of harvesting in a predator-prey model,, J. Theor. Biol., 174 (1995), 13. doi: 10.1006/jtbi.1995.0076. Google Scholar

[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. Google Scholar

[4]

J. Chattopadhyay, Effect of toxic substances on a two-species competitive system,, Ecol. Model., 84 (1996), 287. doi: 10.1016/0304-3800(94)00134-0. Google Scholar

[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. Google Scholar

[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. doi: 10.1006/jtbi.2001.2510. Google Scholar

[7]

K. S. Chaudhuri, Dynamic optimization of combined harvesting of a two-species fishery,, Ecol. Model., 41 (1988), 17. doi: 10.1016/0304-3800(88)90041-5. Google Scholar

[8]

K. S. Chaudhuri and R. S. Saha, On the combined harvesting of a prey-predator system,, J. Biol. Syst., 4 (1996), 373. doi: 10.1142/S0218339096000259. Google Scholar

[9]

C. W. Clark, "Mathematical Bioeconomics: the Optimal Management of Renewable Resources,", Wiley, (1976). Google Scholar

[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. doi: 10.1016/S0196-8858(82)80009-2. Google Scholar

[11]

M. G. Crandall and P. H. Rabinowitz, Bifurcation, perturbation of simple eigenvalues and linearized stability,, Arch. Rational Mech. Anal., 52 (1973), 161. doi: 10.1007/BF00282325. Google Scholar

[12]

G. Dai and M. Tang, Coexistence region and global dynamics of a harvested predator-prey system,, SIAM J. Appl. Math., 58 (1998), 193. doi: 10.1137/S0036139994275799. Google Scholar

[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. doi: 10.1016/0304-3800(87)90115-3. Google Scholar

[14]

W. Feng and X. Lu, On diffusive population models with toxicants and time delays,, J. Math. Anal. Appl., 233 (1999), 373. doi: 10.1006/jmaa.1999.6332. Google Scholar

[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. Google Scholar

[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. doi: 10.1007/BF00168004. Google Scholar

[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. doi: 10.1016/0022-5193(81)90106-5. Google Scholar

[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. doi: 10.1007/BF00275908. Google Scholar

[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. doi: 10.1016/S0022-5193(84)80090-9. Google Scholar

[20]

V. Hutson, J. López-Gómez, K. Mischaikow and G. Vickers, Limit behavior for a competing species problem with diffusion,, in, (1995), 343. doi: 10.1142/9789812796417_0022. Google Scholar

[21]

M. Ito, Global aspect of steady-states for competitive-diffusion systems with homogeneous Dirichlet conditions,, Phys. D, 14 (1984), 1. doi: 10.1016/0167-2789(84)90002-2. Google Scholar

[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. doi: 10.1016/S0096-3003(01)00033-9. Google Scholar

[23]

S. J. Maynard, "Models in Ecology,", Cambridge Univ. Press, (1974). Google Scholar

[24]

M. Mesterton-Gibbons, On the optimal policy for the combined harvesting of independent species,, Nat. Resour. Model., 2 (1987), 109. Google Scholar

[25]

M. Mesterton-Gibbons, On the optimal policy for the combined harvesting of predator and prey,, Nat. Resour. Model., 3 (1988), 63. Google Scholar

[26]

T. J. Monahan and F. R. Trainor, Stimulatory properties of filtrated from green alga hormotila blennista. I. Description,, J. Phycol., 6 (1970), 263. doi: 10.1111/j.0022-3646.1970.00263.x. Google Scholar

[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. doi: 10.1007/BF00173294. Google Scholar

[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. doi: 10.1090/S0002-9947-05-04010-9. Google Scholar

[29]

C. V. Pao, On nonlinear reaction-diffusion systems,, J. Math. Anal. Appl., 87 (1982), 165. doi: 10.1016/0022-247X(82)90160-3. Google Scholar

[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. doi: 10.1007/s12190-009-0301-2. Google Scholar

[31]

R. R. Sarkar and J. Chattopadhayay, A technique for estimating maximum harvesting effort in a stochastic fishery model,, J. Biosci., 28 (2003), 497. doi: 10.1007/BF02705124. Google Scholar

[32]

J. B. Shukla and B. Dubey, Simultaneous effects of two toxicants on biological species: a mathematical model,, J. Biol. Syst., 4 (1996), 109. doi: 10.1142/S0218339096000090. Google Scholar

[33]

J. Smoller, "Shock Waves and Reaction-Diffusion Equations,", Springer-verlag, (1999). Google Scholar

[34]

D. Xiao and S. Ruan, Bogdanov-Takens bifurcations in predator-prey systems with constant rate harvesting,, in, 21 (1997), 493. Google Scholar

[35]

Q. Ye and Z. Li, "Introduction to Reaction-Diffusion Equations,", Science Press, (1990). Google Scholar

[36]

C. Zhong, X. Fan and W. Chen, "Introduction to Nonlinear Functional Analysis,", Lanzhou Univ. Press, (1998). Google Scholar

show all references

References:
[1]

K. J. Arrow and M. Kurz, "Public Investment, The Rate of Return and Optimal Fiscal Policy,", John Hopkins Press, (1970). Google Scholar

[2]

C. Azar, J. Holmberg and K. Lindgren, Stability analysis of harvesting in a predator-prey model,, J. Theor. Biol., 174 (1995), 13. doi: 10.1006/jtbi.1995.0076. Google Scholar

[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. Google Scholar

[4]

J. Chattopadhyay, Effect of toxic substances on a two-species competitive system,, Ecol. Model., 84 (1996), 287. doi: 10.1016/0304-3800(94)00134-0. Google Scholar

[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. Google Scholar

[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. doi: 10.1006/jtbi.2001.2510. Google Scholar

[7]

K. S. Chaudhuri, Dynamic optimization of combined harvesting of a two-species fishery,, Ecol. Model., 41 (1988), 17. doi: 10.1016/0304-3800(88)90041-5. Google Scholar

[8]

K. S. Chaudhuri and R. S. Saha, On the combined harvesting of a prey-predator system,, J. Biol. Syst., 4 (1996), 373. doi: 10.1142/S0218339096000259. Google Scholar

[9]

C. W. Clark, "Mathematical Bioeconomics: the Optimal Management of Renewable Resources,", Wiley, (1976). Google Scholar

[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. doi: 10.1016/S0196-8858(82)80009-2. Google Scholar

[11]

M. G. Crandall and P. H. Rabinowitz, Bifurcation, perturbation of simple eigenvalues and linearized stability,, Arch. Rational Mech. Anal., 52 (1973), 161. doi: 10.1007/BF00282325. Google Scholar

[12]

G. Dai and M. Tang, Coexistence region and global dynamics of a harvested predator-prey system,, SIAM J. Appl. Math., 58 (1998), 193. doi: 10.1137/S0036139994275799. Google Scholar

[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. doi: 10.1016/0304-3800(87)90115-3. Google Scholar

[14]

W. Feng and X. Lu, On diffusive population models with toxicants and time delays,, J. Math. Anal. Appl., 233 (1999), 373. doi: 10.1006/jmaa.1999.6332. Google Scholar

[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. Google Scholar

[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. doi: 10.1007/BF00168004. Google Scholar

[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. doi: 10.1016/0022-5193(81)90106-5. Google Scholar

[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. doi: 10.1007/BF00275908. Google Scholar

[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. doi: 10.1016/S0022-5193(84)80090-9. Google Scholar

[20]

V. Hutson, J. López-Gómez, K. Mischaikow and G. Vickers, Limit behavior for a competing species problem with diffusion,, in, (1995), 343. doi: 10.1142/9789812796417_0022. Google Scholar

[21]

M. Ito, Global aspect of steady-states for competitive-diffusion systems with homogeneous Dirichlet conditions,, Phys. D, 14 (1984), 1. doi: 10.1016/0167-2789(84)90002-2. Google Scholar

[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. doi: 10.1016/S0096-3003(01)00033-9. Google Scholar

[23]

S. J. Maynard, "Models in Ecology,", Cambridge Univ. Press, (1974). Google Scholar

[24]

M. Mesterton-Gibbons, On the optimal policy for the combined harvesting of independent species,, Nat. Resour. Model., 2 (1987), 109. Google Scholar

[25]

M. Mesterton-Gibbons, On the optimal policy for the combined harvesting of predator and prey,, Nat. Resour. Model., 3 (1988), 63. Google Scholar

[26]

T. J. Monahan and F. R. Trainor, Stimulatory properties of filtrated from green alga hormotila blennista. I. Description,, J. Phycol., 6 (1970), 263. doi: 10.1111/j.0022-3646.1970.00263.x. Google Scholar

[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. doi: 10.1007/BF00173294. Google Scholar

[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. doi: 10.1090/S0002-9947-05-04010-9. Google Scholar

[29]

C. V. Pao, On nonlinear reaction-diffusion systems,, J. Math. Anal. Appl., 87 (1982), 165. doi: 10.1016/0022-247X(82)90160-3. Google Scholar

[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. doi: 10.1007/s12190-009-0301-2. Google Scholar

[31]

R. R. Sarkar and J. Chattopadhayay, A technique for estimating maximum harvesting effort in a stochastic fishery model,, J. Biosci., 28 (2003), 497. doi: 10.1007/BF02705124. Google Scholar

[32]

J. B. Shukla and B. Dubey, Simultaneous effects of two toxicants on biological species: a mathematical model,, J. Biol. Syst., 4 (1996), 109. doi: 10.1142/S0218339096000090. Google Scholar

[33]

J. Smoller, "Shock Waves and Reaction-Diffusion Equations,", Springer-verlag, (1999). Google Scholar

[34]

D. Xiao and S. Ruan, Bogdanov-Takens bifurcations in predator-prey systems with constant rate harvesting,, in, 21 (1997), 493. Google Scholar

[35]

Q. Ye and Z. Li, "Introduction to Reaction-Diffusion Equations,", Science Press, (1990). Google Scholar

[36]

C. Zhong, X. Fan and W. Chen, "Introduction to Nonlinear Functional Analysis,", Lanzhou Univ. Press, (1998). Google Scholar

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