International Journal of Nonlinear Analysis and Applications
Qualitative study of an Eco-Toxicant model with migration
Document Type : Research Paper
Authors
Department of mathematics, College of Science, University of Baghdad, Baghdad, Iraq
Abstract
In this article, an ecology toxicant food chain system with Lotka-Volterra functional response for predator population includes prey protection zone has been suggested and studied. Toxins are excreted by all organisms as a form of defence. The prey follows the logistic growth law. The equilibrium points have been established. The analytic approach has been used to investigate the local stability for each acceptable equilibrium point. The global dynamics of this model was studied using the Lyapunov function. Lastly, numerical simulations and graphical illustrations were used to back up our analytic results.
Keywords
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[9] D. R. French and J. M. Travis, Density-dependent dispersal in host-parasitoid assemblages, Oikos 95 (2001)
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[11] T. Hallam and C. Clark, Non-autonomous logistic equations as models of populations in a deteriorating environment, J. Theo. Bio. 93 (1981) 303–311.
[12] T. Hallam, C. Clark and R. Lassiter, Effects of toxicants on populations: a qualitative approach I. Equilibrium
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[13] T. Hallam, C. Clark and G. Jordan, Effects of toxicants on populations: a qualitative approach II. First order
kinetics, J. Math. Bio. 18 (1983) 25–37.
[14] T. Hallam and J. De Luna, Effects of toxicants on populations: a qualitative: approach III. Environmental and
food chain pathways, J. Theo. Bio. 109 (1984) 411–429.
[15] J.K. Hale, Ordinary Differential Equation, Wiley-Interscience, New York, 1969.
[16] M.H. Ismaeel and A.A. Majeed, The Dynamical behavior of stage structured prey-predator model in the harvesting
and toxin presence, J. Southwest Jiaotong Univ. 54 (2019).
[17] V. A. A. Jansen, Theoretical Aspects of Metapopulation Dynamics, PhD thesis, Institute for Biodiversity and
Ecosystem Dynamics, 1994.
[18] T. Kar and A. Batabyal, Persistence and stability of a two prey one predator system, Int. J. Engin. Sci. Tech. 2
(2010) 174–190.[19] A.T. Keong, H.M. Safuan and K. Jacob, Dynamical behaviours of prey-predator fishery model with harvesting
affected by toxic substances, MATEMATIKA: Malays. J. Indust. Appl. Math. 34 (2018) 143–151.
[20] T.K. Kar and K. Chaudhuri, On non-selective harvesting of two competing fish species in the presence of toxicity,
Eco. Model. 161 (2003) 125–137.
[21] A.J. Kadhim and A.A. Majeed, The impact of toxicant on the food chain ecological model, AIP Conf. Proc. (2020)
040001.
[22] S. Kumar and H. Kharbanda, Stability analysis of prey-predator model with infection, migration and vaccination
in prey, arXiv preprint arXiv:1709.10319, (2017).
[23] M.A. Lafta and A.A. Majeed, The food web prey-predator model with toxin, AIP Conf. Proc. (2020) 030015.
[24] A.A. Majeed and R.R. Saadi, The persistence of prey-predator model with competition hosts, Int. J. Appl. Math.
Res. 4 (2015) 351.
[25] O. Misra and A.R. Babu, A model for the effect of toxicant on a three species food-chain system with food-limited
growth of prey population, Global J. Math. Anal. 2 (2014) 120–145.
[26] R. Mchich, P. Auger and J.-C. Poggiale, Effect of predator density dependent dispersal of prey on stability of a
predator–prey system, Math. Biosci. 206 (2007) 343–356.
[27] K. Olli, Diel vertical migration of phytoplankton and heterotrophic flagellates in the Gulf of Riga, J. Marine Syst.
23 (1999) 145–163.
[28] P. Panja and S.K. Mondal, Stability analysis of coexistence of three species prey–predator model, Nonlinear Dyn.
81 (2015) 373–382.
[29] P. Pillai, A. Gonzalez and M. Loreau, Evolution of dispersal in a predator-prey metacommunity, Amer. Natur.
179 (2012) 204–216.
[30] R. Rani and S. Gakkhar, The impact of provision of additional food to predator in predator–prey model with
combined harvesting in the presence of toxicity, J. Appl. Math. Comput. 60 (2019) 673–701.
[31] S. K. Roy and B. Roy, Analysis of prey-predator three species fishery model with harvesting including prey refuge
and migration, Int. J. Bifurc. Chaos 26, (2016) 1650022.
[32] S. Sharma, Modeling and analysis of a biological population: effects of toxicants (pollutants) emitted from external
sources as well as formed by precursors, Int. J. Appl. Res. 1(10) (2015) 494–500.
[33] P. Srinivasu and I. Gayatri, Influence of prey reserve capacity on predator–prey dynamics, Eco. Model. 181 (2005)
191–202.
[34] D. Tilman and P. Kareiva, Spatial ecology Princeton University Press, Princeton, NJ, USA, 1997.
bloom to oligotrophy in the open NW Mediterranean and effects of wind events. 2. Vertical distributions and
migrations, J. Plankton Rese. 23 (2001) 243–261.
[2] I. Al-Darabsah, X. Tang, and Y. Yuan, A prey-predator model with migrations and delays, Disc. Contin. Dyn.
Syst.-B, 21 (2016) 737.
[3] A.A. Berryman, The orgins and evolution of predator-prey theory, Eco. 73 (1992) 1530–1535.
[4] Y. Chen and F. Zhang, Dynamics of a delayed predator–prey model with predator migration, Appl. Math. Model.
37 (2013) 1400–1412.
[5] J. Chattopadhyay, Effect of toxic substances on a two-species competitive system, Eco. Model. 84 (1996) 287–289.
[6] J. De Luna and T. Hallam, Effects of toxicants on populations: a qualitative approach IV. Resource-consumertoxicant models, Eco. Model. 35 (1987) 249–273.
[7] T. Das, R. Mukherjee and K. Chaudhuri, Harvesting of a prey–predator fishery in the presence of toxicity, Appl.
Math. Model. 33 (2009) 2282–2292.
[8] A. El Abdllaoui, P. Auger, B. W. Kooi, R.B. De la Parra and R. Mchich, Effects of density-dependent migrations
on stability of a two-patch predator–prey model, Math. Biosci. 210 (2007) 335–354.
[9] D. R. French and J. M. Travis, Density-dependent dispersal in host-parasitoid assemblages, Oikos 95 (2001)
125–135.
[10] W. Feng, B. Rock and J. Hinson, On a new model of two-patch predator-prey system with migration of both
species, J. Appl. Anal. Comput. 1 (2011) 193203.
[11] T. Hallam and C. Clark, Non-autonomous logistic equations as models of populations in a deteriorating environment, J. Theo. Bio. 93 (1981) 303–311.
[12] T. Hallam, C. Clark and R. Lassiter, Effects of toxicants on populations: a qualitative approach I. Equilibrium
environmental exposure, Eco. Model. 18 (1983) 291–304.
[13] T. Hallam, C. Clark and G. Jordan, Effects of toxicants on populations: a qualitative approach II. First order
kinetics, J. Math. Bio. 18 (1983) 25–37.
[14] T. Hallam and J. De Luna, Effects of toxicants on populations: a qualitative: approach III. Environmental and
food chain pathways, J. Theo. Bio. 109 (1984) 411–429.
[15] J.K. Hale, Ordinary Differential Equation, Wiley-Interscience, New York, 1969.
[16] M.H. Ismaeel and A.A. Majeed, The Dynamical behavior of stage structured prey-predator model in the harvesting
and toxin presence, J. Southwest Jiaotong Univ. 54 (2019).
[17] V. A. A. Jansen, Theoretical Aspects of Metapopulation Dynamics, PhD thesis, Institute for Biodiversity and
Ecosystem Dynamics, 1994.
[18] T. Kar and A. Batabyal, Persistence and stability of a two prey one predator system, Int. J. Engin. Sci. Tech. 2
(2010) 174–190.[19] A.T. Keong, H.M. Safuan and K. Jacob, Dynamical behaviours of prey-predator fishery model with harvesting
affected by toxic substances, MATEMATIKA: Malays. J. Indust. Appl. Math. 34 (2018) 143–151.
[20] T.K. Kar and K. Chaudhuri, On non-selective harvesting of two competing fish species in the presence of toxicity,
Eco. Model. 161 (2003) 125–137.
[21] A.J. Kadhim and A.A. Majeed, The impact of toxicant on the food chain ecological model, AIP Conf. Proc. (2020)
040001.
[22] S. Kumar and H. Kharbanda, Stability analysis of prey-predator model with infection, migration and vaccination
in prey, arXiv preprint arXiv:1709.10319, (2017).
[23] M.A. Lafta and A.A. Majeed, The food web prey-predator model with toxin, AIP Conf. Proc. (2020) 030015.
[24] A.A. Majeed and R.R. Saadi, The persistence of prey-predator model with competition hosts, Int. J. Appl. Math.
Res. 4 (2015) 351.
[25] O. Misra and A.R. Babu, A model for the effect of toxicant on a three species food-chain system with food-limited
growth of prey population, Global J. Math. Anal. 2 (2014) 120–145.
[26] R. Mchich, P. Auger and J.-C. Poggiale, Effect of predator density dependent dispersal of prey on stability of a
predator–prey system, Math. Biosci. 206 (2007) 343–356.
[27] K. Olli, Diel vertical migration of phytoplankton and heterotrophic flagellates in the Gulf of Riga, J. Marine Syst.
23 (1999) 145–163.
[28] P. Panja and S.K. Mondal, Stability analysis of coexistence of three species prey–predator model, Nonlinear Dyn.
81 (2015) 373–382.
[29] P. Pillai, A. Gonzalez and M. Loreau, Evolution of dispersal in a predator-prey metacommunity, Amer. Natur.
179 (2012) 204–216.
[30] R. Rani and S. Gakkhar, The impact of provision of additional food to predator in predator–prey model with
combined harvesting in the presence of toxicity, J. Appl. Math. Comput. 60 (2019) 673–701.
[31] S. K. Roy and B. Roy, Analysis of prey-predator three species fishery model with harvesting including prey refuge
and migration, Int. J. Bifurc. Chaos 26, (2016) 1650022.
[32] S. Sharma, Modeling and analysis of a biological population: effects of toxicants (pollutants) emitted from external
sources as well as formed by precursors, Int. J. Appl. Res. 1(10) (2015) 494–500.
[33] P. Srinivasu and I. Gayatri, Influence of prey reserve capacity on predator–prey dynamics, Eco. Model. 181 (2005)
191–202.
[34] D. Tilman and P. Kareiva, Spatial ecology Princeton University Press, Princeton, NJ, USA, 1997.
)
Volume 12, Issue 2
November 2021Pages 1883-1902
- Receive Date: 12 April 2021
- Revise Date: 25 June 2021
- Accept Date: 11 July 2021