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Lakshmi Narayan, K.
- Optimal Harvesting of Prey in Three Species Ecological Model with a Time Delay on Prey and Predator
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Authors
Affiliations
1 Department of Mathematics, JNTUK, University College of Engineering,Vizianagarm-535003, IN
2 Department of Mathematics, VITS, Deshmukhi, Hyderabad - 508284, IN
1 Department of Mathematics, JNTUK, University College of Engineering,Vizianagarm-535003, IN
2 Department of Mathematics, VITS, Deshmukhi, Hyderabad - 508284, IN
Source
Research Journal of Science and Technology, Vol 9, No 3 (2017), Pagination: 368-376Abstract
This paper deals with the stability analysis of three species Ecological model consists of a Prey (N1), a predator (N2) and a competitor (N3) .The competitor (N3) is competing with the Predator Species (N2) and neutral with the prey (N1). In this model the third species is competing with the predator for food other than the prey (N1).In addition to that, the death rates, carrying capacities of all three species are also considered , a delay in the interaction between the prey and the predator (gestation period of the predator) and harvesting effort of prey population is also considered. The model is characterized by a couple of integro- differential equations. All the eight equilibrium points of the model are identified and their local stability is discussed for interior equilibrium point. The global stability is studied by constructing a suitable Lyapunov’s function. Suitable parameter are identified for Numerical simulation which shows that this continuous time delay model exhibits rich dynamics and time delay can further stabilize or destabilize the system.Keywords
Prey, Predator, Competitor, Equilibrium Points, Local Stability, Global Stability, Harvesting Numerical Simulation.
References
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- Kapur J. N., 1988, Mathematical Modeling, Wiley-Eatern.
- Kapur, J. N., 1985 ,Mathematical Models in Biology and Medicine, Affiliated East-west,.
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- Paul Colinvaux., 1986: Ecology, John Wiley and Sons Inc., New York.
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- Yang kuang. Delay differential equations with applicatios in population dynamics. Academic Press, Inc.1993.
- paparao A.V.,K. Lakshmi Narayan., Shahnaz. Bathul, 2012, A Three Species Ecological Model with a Prey, Predator and a Competitor to both the Prey and Predator, International journal of mathematics and scientific computing vol. 2, no. 1. Pp 70-75
- paparao A.V.,K. Lakshmi Narayan.,Shahnaz. Bathul, A Three Species Ecological Model with a Prey, Predator and a Competitor to the Predator. Mathematics Science International Research journal, Vol 1 No 1 2012.
- paparao A.V.,K. Lakshmi Narayan.,Shahnaz. Bathul, A Three Species Ecological Model with a Prey, Predator and a Competitor to the prey and optimal harvesting of the prey.Journal of Advanced Research in Dynamical and Control Systems, Vol. 5, Issue. 1, 2013, pp. 37-49
- Vidyanath T, Laxmi Narayan K and Shahnaz Bathul, A three species ecological model with a predator and two preying species, International Frontier Sciences Letters, 9; 2016: 26-32.
- Paparao AV and Lakshmi Narayan K, Dynamics of a prey predator and competitor model with time delay. International Journal of Ecology and Development. 32(1); 2017: 75-86.
- Stability Analysis of a Viral Model with Intercellular Delay
Abstract Views :117 |
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Authors
Affiliations
1 Department of Basic Science and Humanities, Vignan Institute of Technology and Science, Deshmukhi, Hyderabad-508284. TS., IN
2 Department of Mathematics, JNTU College of Engineering, Kukatpally, Hyderabad-5000080, TS, IN
1 Department of Basic Science and Humanities, Vignan Institute of Technology and Science, Deshmukhi, Hyderabad-508284. TS., IN
2 Department of Mathematics, JNTU College of Engineering, Kukatpally, Hyderabad-5000080, TS, IN
Source
Research Journal of Science and Technology, Vol 9, No 3 (2017), Pagination: 435-440Abstract
A three compartment model with healthy cells, infected cells, and free virus has been considered incorporating time delays. We derived the conditions for global asymptotic stability and showed that the chronic infected equilibrium is asymptotically stable for all delay. Numerical simulations are presented to illustrate the results.Keywords
Compartment Model, Asymptotic Stability, Delay, Chronic, Viral Infection.References
- MacDonald N. Time Delays in Biological Models. Springer, Heidelberg, 1978.
- Hale JK, VerduynLunel S. Introduction to Functional Differential Equations. Springer, NewYork 1993.
- Perelson, AS, Kirschner DE, de Boer R. Dynamics of HIV infection of CD4 T cells. Mathematical Biosciences. 1993; 114: 81-125.
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- Smith HL, De Leenheer P. Virus dynamics: a global analysis. SIAM Journal of Applied Mathematics. 2003; 63: 1313-1327.
- Korobeinikov A. Global properties of basic virus dynamic models. Bulletin of Mathematical Biology. 2004; 66: 879-883.
- Wang L, Li MY, Mathematical analysis of the global dynamics of a model for HIV infection of CD4+ T cells. Mathematical Biosciences. 2006; 200: 44-57.
- Bonhoeffer S, May RM, Shaw GM, Nowak MA. Virus dynamics and drug therapy. Proceedings of the National Academy of Sciences USA. 1997; 94: 6971-6976.
- Herz AVM, Bonhoeffer S, Anderson RM, May RM, Nowak MA, Viral dynamics in vivo: limitations on estimates of intracellular delay and virus decay. Proceedings of the National Academy of Sciences USA. 1996; 93: 7247-7251.
- Culshaw RV and Ruan SG. A delay-differential equation model of HIV infection of CD4+T- cells. Mathematical Biosciences. 2000; 165: 27-39.
- Wang Y, Zhou Y, Wu J, Heffernan J, Oscillatory viral dynamics in a delayed HIV pathogenesis model. Mathematical Biosciences. 2009; 219: 104-112.
- Perelson AS and Castillo-Chavez C. Editors. Modelling the interaction of the immune system with HIV. Mathematical and Statistical Approaches to AIDS Epidemiology, Springer, Berlin. 1989; pp. 350-370.
- Mittler JE, Sulzer B, Neumann AU, Perelson AS. Influence of delayed viral production onviral dynamics in HIV-1 infected patients, Mathematical Biosciences. 1998; 152: 143-163.
- Perelson AS, Nelson PW. Mathematical analysis of HIV-I dynamics in vivo. SIAM Review. 1999; 41: 3-44.
- Mittler JE, Markowitz M, Ho DD, Perelson AS. Improved estimates for HIV-1 clearance rate and intracellular delay. AIDS. 1999; 13: 1415-1417.
- Tam J. Delay effect in a model for virus replication, IMA Journal of Mathematics Applied in Medicine and Biology. 1999; 16: 29-37.
- Dynamics of Three Species Food Chain Model with Neutralism and Ammensalism
Abstract Views :119 |
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Authors
Affiliations
1 Vidya Jyothi Institute of Technology, Hyderabad, IN
2 VITS, Hyderabad, IN
1 Vidya Jyothi Institute of Technology, Hyderabad, IN
2 VITS, Hyderabad, IN
Source
Research Journal of Science and Technology, Vol 9, No 3 (2017), Pagination: 447-452Abstract
The present investigation is an analytical study of three species system comprising two neutral species (First species N1 and Second species N2), which are ammensal on the third species (N3). The model is represented by a system of three first order non-linear ordinary differential equations. Interior equilibrium point is identified and the stability of interior equilibrium point was discussed using Routh-Hurwitz criterion. Further solutions of quasi-linearized equations are identified and the global stability is discussed by Lyapunov's function and the results are simulated by numerical examples using MatLab.Keywords
Neutralism, Ammensalism, Interior Equilibrium Point, Global Stability, Lyapunov Function.References
- Lotka A. J. 1925. Elements of Physical Biology, Williams and Wilking, Baltimore.
- Voltera V, Leconseen La Theori Mathematique De La Leitte Pou Lavie, Gauthier-Villars, Paris, 1931.
- Meyer. W. J 1925, Concepts of Mathematical Modeling, McGraw-Hill.
- Cushing. J. N. 1977 Integro-Differential Equations and Delay Models in Population Dynamics. Lecture Notes in Bio-Mathematics. Vol. 20. Springer Verlag.
- Kapur. J. N. 1985. Mathematical Modeling in Biology Affiliated East West.
- Kapur. J. N. 1985. Mathematical Modeling, Wiley Easter.
- Freedman. H. I, Deterministic Mathematical Models in Population Ecology, Decker, New York, 1980.
- Lakshmi Narayan K, 2005. A Mathematical Steady of a Prey-Predator Ecological Model with a Partial Cover and Alternate food for the Predator, Ph. D. Thesis. J. N. T. University, India.
- Lakshmi Narayan K and Pattabhiramacharyulu N. Ch, 2007. A Prey-Predator Model with Cover for Prey and Alternate food for the Predator and Time Delay, International Journal of Scientific Computing 1:7-14.
- Papa Rao. A. V, Lakshmi Narayan. K, Bathul. S. 2012, A Three Species Ecological Model with a Prey, Predator and a Competitor to both the Prey and Predator, International Journal of Mathematics and Scientific Computing ( ISSN: 2231-5330), Vol-2, No. 1, 2012.
- Kondala Rao K, Lakshmi Narayan K, 2015. Stability Analysis of Ammensal model Comprising Humans, Plants and Birds with Harvesting, Global Journal of Pure and Applied Mathematics (GJPAM), volume 11, Issue 2 (2015 Special issue) pp 115-120, ISSN: 0973-1768.
- Kondala Rao K and Lakshmi Narayan K, 2016. Dynamics of three species food chain model with harvesting and the paper was published in proceedings of “The 10th International Conference of IMBIC on Mathematical Sciences for Advancement of Science and Technology” (MSAST-2016) with ISBN No: 978-81-925832-4-2.
- Papa Rao. A. V, Lakshmi Narayan K, 2016. A Prey, Predator and a Competitor to the Predator Model with Gestation Period and the paper was published in proceedings of “The 10th International Conference of IMBIC on Mathematical Sciences for Advancement of Science and Technology” (MSAST-2016) with ISBN No: 978-81-925832-4-2.
- A Two Species Amensalism Model with a Linearly Varying Cover on the First Species
Abstract Views :420 |
PDF Views:0
Authors
Affiliations
1 Department of Mathematics, VJIT, Hyderabad-500085, IN
2 Department of Mathematics, VITS, Hyderabad - 508284, IN
1 Department of Mathematics, VJIT, Hyderabad-500085, IN
2 Department of Mathematics, VITS, Hyderabad - 508284, IN
Source
Research Journal of Science and Technology, Vol 9, No 4 (2017), Pagination: 511-520Abstract
The present paper is devoted to an analytical investigation of a two species ammensalism model with a cover linearly varying with the population on the first species (x) to protect from the attacks of the second species (y).All the equilibrium points are identified and the local stability is discussed and global stability is also discussed by constructing suitable Lyapunov’s function supported by the numerical simulation by using Matlab.Keywords
Ammensal, Cover, Equilibrium Point, Global Stability, Lyapunov Function.References
- Lakshmi Narayan K. A Mathematical study of Prey-Predator Ecological Models with a partial covers for the prey and alternative food for the predator. Ph.D thesis. 2004. JNTUH.
- Lakshmi Naryan K, and Pattabhi Ramacharyulu NCh. Some threshold theorems for prey-predator model with harvesting. Int. J. of Math.Sci. and Engg. Appls 2008; 2: 23-3.
- Leonardo of Pisa (Fibonacci). Liberabaci (Book of counting board), 1202.
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- Lucas WF Roberts SF and Thrall RM. Discreate and system models (vol (III) of modules in Applied Mathematics). Springer-verlag, Heidel berg, 1983.
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- Matsuda H and Abrams PA. Effects of predators-prey interaction and adaptive change on sustainable yield. Can. J. Fish. Aquat. Sci./J. Can. Sci. Halieut. Aquat. 2004; 61: 175-184.
- May RM. Stability and complexity in model Eco-Systems. Princeton University press. Princeton, 1973.
- Mesterton-Gibbons. M. A technique for finding optimal two species harvesting policies. Ecol.model. 1996; 92: 235-244.
- Meyer WJ. Concepts of Mathematical Modeling. McGraw –Hill, 1985.
- Paul Colinvaux. Ecology. John Wiley and Sons Inc., New York, 1986.
- Phanikumar N, Pattabhiramacharyulu NCh. A three species eco-system consisting of a prey predator and host commensal to the prey. International Journal of Open Problems Compt. Math, 2010; 3(1): 92-113.
- Rish S and Boucher DH. 1976 what ecologist looks for. Bulletin of the Ecological Society of America 57: 8-9.
- Varma VS. A note on Exact solutions for a Special prey-predator or competing Species System. Bull.Math.Biol. 1977; 39: 619-622.
- Volterra V. Leconssen la Theorie Mathematique de la Leitte Pou Lavie. Gauthier-Villars, Paris,1931.
- Liao SJ. beyond perturbation: introduction to the homotopy analysis method.CRC Press. Boca Raton: Chapman & Hall. 2003.
- Sita Rambabu B, Lakshmi Narayan K and Shahanaz Bathul. A Mathematical study of Two Species Amensalism Model With a Cover for the first Species by Homotopy Analysis Method. Advances in Applied Science Research. Pelagia Research Library. 2012; 3 (3): 1821-1826.