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i-manager's Journal on Mathematics (JMAT)

Current Issue Vol. 12 Issue 2

Statistical Analysis on Job Satisfaction of Academic Staff at College of Natural Science in Jimma University, Ethiopia: A Cross-Sectional Study
Tolerance Based Rough Algebras Induced by Blocks
Significance of Thermal Boundary Layer Analysis of MHD Chemically Radiative Dissipative Casson Nanofluid Flow over a Stretching Sheet with Heat Source
Transient Analysis of M/M/1 Fractional Queue
A Novel Multi-Viewpoints Based Cosine Similarity Visual Technique for an Effective Assessment of Clustering Tendency

Most Cited

Volume 8 Issue 3 July - September 2019

Research Paper

2-HH Norm and Birkhoff-James Orthogonality in Normed Spaces

Bhuwan Prasad Ojha* , Prakash Muni Bajracharya**

*-** Department of Mathematics at Tribhuvan University, Kathmandu, Nepal.

Ojha, B. P., and Bajracharya, P. M. (2019). 2-HH Norm and Birkhoff-James Orthogonality in Normed Spaces. i-manager's Journal on Mathematics, 8(3), 1-9. https://doi.org/10.26634/jmat.8.3.16746

Abstract

For any normed space X, the p-HH norms X were introduced by Kikianty and Dragomir on X² = X x X of normed spaces. p- norms and p-HH norms induce the same topology, so they are equivalent, but are geometrically different. Besides that, E. Kikianty and S. S. Dragimor introduced HH-P orthogonality and HH-I orthogonality by using 2-HH norm and discussed main properties of these orthogonalities. The main purpose of this paper is to focus on the concept of 2-HH norm to Birkhoff and a new orthogonality in normed spaces, and we discuss some properties of these orthogonalities. It is proved that Robert orthogonality via 2-HH norm implies Birkhoff-James orthogonality via 2-HH norm; however, it is not necessary for the converse part.

References

[1]. Alonso, J., & Benítez, C. (1988). Orthogonality in normed linear spaces. A survey I: Main properties. Extracta Mathematicae, 3(1), 1-15.

[2]. Alonso, J., Martini, H., & Wu, S. (2012). On Birkhoff orthogonality and isosceles orthogonality in normed linear spaces. Aequationes Mathematicae, 83(1-2), 153-189. https://doi.org/10.1007/s00010-011-0092-z

[3]. Dragomir, S. S., & Kikianty, E. (2010). Orthogonality connected with integral means and characterizations of inner product spaces. Journal of Geometry, 98(1-2), 33-49. https://doi.org/10.1007/s00022-010-0048-9

[4]. James, R. C. (1945). Orthogonality in normed linear spaces. Duke Mathematical Journal, 12(2), 291-302. https://doi.org/10.1215/S0012-7094-45-01223-3

[5]. Kikianty, E., & Dragomir, S. S. (2008). Hermite-Hadamard's inequality and the p-HH-norm on the cartesian product of two copies of a normed space. Research Report Collection, 11(1).

[6]. Kikianty, E., & Dragomir, S. S. (2009). On new notions of orthogonality in normed spaces via the 2-HH-norms. Research Report Collection, 12(1).

[7]. Kikianty, E., & Dragomir, S. S. (2012). On Carlsson type orthogonality and characterization of inner product spaces. Filomat, 26(4), 859-870.

[8]. Kikianty, E., & Sinnamon, G. (2009). The p− HH norms on Cartesian powers and sequence spaces. Journal of mathematical analysis and applications, 359(2), 765-779. https://doi.org/10.1016/j.jmaa.2009.06.020

[9]. Martini, H., & Spirova, M. (2010). A new type of orthogonality for normed planes. Czechoslovak Mathematical Journal, 60(2), 339-349. https://doi.org/10.1007/s10587-010-0039-x

[10]. Ojha, B. P. (2016). Some new types of orthogonalities in normed spaces and application in best approximation. Journal of Advanced College of Engineering and Management, 2, 1-4. https://doi.org/10.3126/jacem.v2i0.16092

[11]. Sikoraska. J (2015). Orthogonalities and functional equations. Aequationes Mathematicae , 89, 215-277. https://doi. org/10.1007/s00010-014-0288-0

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Research Paper

Modified Intuitionistic M-Fuzzy Metric Space and Some Common Fixed-Point Theorems

Dr Pranjali* , Shailesh Dhar Diwan**

* Department of Applied Mathematics, Shri Shankaracharya Institute of Professional Management and Technology, Raipur, Chhattisgarh, India.

** Department of Applied Mathematics, Government Engineering College, Raipur, Chhattisgarh, India.

Sharma, P., and Diwan, S. D. (2019). Modified Intuitionistic M-Fuzzy Metric Space and Some Common Fixed-Point Theorems. i-manager's Journal on Mathematics, 8(3), 10-16. https://doi.org/10.26634/jmat.8.3.16954

Abstract

In this paper, we modified the idea of M-fuzzy metric space defined in the work “A common fixed point theorem in two M- fuzzy metric spaces” by Sedghi and Shobein, published in the year 2007, with intuitionistic fuzzy metric space and we present the notion of modified intuitionistic M-fuzzy metric space with the help of continuous t-representable. We also prove some common fixed-point theorems for modified intuitionistic M-fuzzy metric space.

References

[1]. Atanassov, K. (1986). Intutionistic fuzzy metric spaces. Fuzzy Sets and System, 20, 87-96.

[2]. Beloul, S., & Tomar, A. (2019). Integral type common fixed point theorems in modified intuitionistic fuzzy metric spaces. Afrika Matematika, 30(3-4), 581-596. https://doi.org/10.1007%2Fs13370-019-00668-1

[3]. Chauhan, S., Imdad, M., & Samet, B. (2013). Coincidence and common fixed point theorems in modified intuitionistic fuzzy metric spaces. Mathematical and Computer Modelling, 58(3-4), 898-906. https://doi.org/10.1016/j.mcm.2013. 03.010

[4]. Deschrijver, G., & Kerre, E. E. (2003). On the relationship between some extensions of fuzzy set theory. Fuzzy Sets and Systems, 133(2), 227-235. https://doi.org/10.1016/S0165-0114(02)00127-6

[5]. Deschrijver, G., Cornelis, C., & Kerre, E. E. (2004). On the representation of intuitionistic fuzzy t-norms and t-conorms. IEEE Transactions on Fuzzy Systems, 12(1), 45-61. https://doi.org/10.1109/TFUZZ.2003.822678

[6]. Deshpande, B., & Handa, A. (2013). Fixed point theorems on modified intuitionistic fuzzy quasi-metric spaces with applications to the domain of words. Annals of Fuzzy Mathematics and Informatics, 6(1), 17-31.

[7]. Dhage, B. C. (1992). Generalised metric space and mappings with fixed point. Bulletin of the Calcutta Mathematical Society, 84(4), 329- 336.

[8]. George, A., & Veeramani, P. (1994). On some results in fuzzy metric spaces. Fuzzy Sets and Systems, 64(3), 395-399. https://doi.org/10.1016/0165-0114(94)90162-7

[9]. Gregori, V., Romaguera, S., &Veereamani, P. (2006). A note on intuitionistic fuzzy metric spaces. Chaos, Solitons & Fractals, 28, 902-905. https://doi.org/10.1016/j.chaos.2005.08.113

[10]. Kramosil, I., & Michálek, J. (1975). Fuzzy metrics and statistical metric spaces. Kybernetika, 11(5), 336-344.

[11]. Park, J. H. (2004). Intuitionistic fuzzy metric spaces. Chaos, Solitons & Fractals, 22(5), 1039-1046. https://doi.org/10.10 16/j.chaos.2004.02.051

[12]. Saadati, R., & Park, J. H. (2006). On the intuitionistic fuzzy topological spaces. Chaos, Solitons & Fractals, 27(2), 331- 344. https://doi.org/10.1016/j.chaos.2005.03.019

[13]. Saadati, R., Sedghi, S., & Shobe, N. (2008). Modified intuitionistic fuzzy metric spaces and some fixed point theorems. Chaos, Solitons & Fractals, 38(1), 36-47. https://doi.org/10.1016/j.chaos.2006.11.008

[14]. Sedghi, S., & Shobe, N. (2007). A common fixed point theorem in two M-fuzzy metric spaces. Communications-Korean Mathematical Society, 22(4), 513.

[15]. Tanveer, M., Imdad, M., Gopal, D., & Patel, D. K. (2012). Common fixed point theorems in modified intuitionistic fuzzy metric spaces with common property (EA). Fixed Point Theory and Applications, 2012(1), 36.https://doi.org/10.1186/1687- 1812-2012-36

[16]. Turkoglu, D., Alaca, C., & Yildiz, C. (2006). Compatible maps and compatible maps of types (α) and (β) in intuitionistic fuzzy metric spaces. Demonstratio Mathematica, 39(3), 671-684. https://doi.org/10.1515/dema-2006-0323

[17]. Vasuki, R., & Veeramani, P. (2003). Fixed point theorems and Cauchy sequences in fuzzy metric spaces. Fuzzy Sets and Systems, 135(3), 415-417. https://doi.org/10.1016/S0165-0114(02)00132-X

[18]. Zadeh, L. A. (1965). Fuzzy Sets, Information and Control. 8(3), pp. 338-353. https://doi.org/10.1016/S0019-9958(65) 90241-X

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Research Paper

Dynamics of Delayed Sirs Epidemic Model with a Non-Linear Incidence Rate

Appa Rao Dokala* , Shaik Kalesha Vali S. , Papa Rao A. V.*

* Department of Mathematics, IIIT Srikakulam, RGUKT, Andhra Pradesh, India.

-* Department of Mathematics, JNTUK, University College of Engineering, Vizianagaram, Andhra Pradesh, India.

Dokala, A. R., Vali, S. K., and Rao, P. A. V. (2019). Dynamics of Delayed Sirs Epidemic Model with a Non-Linear Incidence Rate. i-manager's Journal on Mathematics, 8(3), 17-27. https://doi.org/10.26634/jmat.8.3.16707

Abstract

In this work, we consider an SIRS epidemic model with a non-linear incidence rate with time delay. The time delay is incorporated in susceptible population with the interaction of susceptible (S) and removable (R) population. We also induce the saturated incidence rate in S, R population. By analyzing the model, the local stability of an endemic equilibrium point is discussed. The system undergoes Hopf bifurcation. The analytical results are supported with numerical simulation using MATLAB and it is shown that the system is locally asymptotically stable and exhibit Hopf bifurcation.

References

[1]. Anderson, R. M., & May, R. M. (1979). Population biology of infectious diseases: Part I. Nature, 280(5721), 361-367. https://doi.org/10.1038/280361a0

[2]. Beretta, E., & Takeuchi, Y. (1995). Global stability of an SIR epidemic model with time delays. Journal of Mathematical Biology, 33(3), 250-260.

[3]. Beretta, E., Hara, T., Ma, W., & Takeuchi, Y. (2001). Global asymptotic stability of an SIR epidemic model with distributed time delay. Nonlinear Analysis, Theory, Methods & Applications, 47(6), 4107-4115.

[4]. Capasso, V., & Serio, G. (1978). A generalization of the Kermack-McKendrick deterministic epidemic model. Mathematical Biosciences, 42(1-2), 43-61.

[5]. Chen, L., & Chen, J. (1993). Nonlinear Biological Dynamical System. Science, Beijing, China.

[6]. Kermack, W. O., & McKendrick, A. G. (1927). A contribution to the mathematical theory of epidemics. Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character, 115(772), 700-721. https://doi.org/10.1098/rspa.1927.0118

[7]. Kumar, G. R., Narayan, K. L., & Reddy, B. R. (n. d.). Stability analysis of epidemic model with nonlinear incidence rates and immigration. In Proceedings of National Conference on Pure and Applied Mathematics (pp. 978-93).

[8]. Li, G., & Wang, W. (2009). Bifurcation analysis of an epidemic model with nonlinear incidence. Applied Mathematics and Computation, 214(2), 411-423. https://doi.org/10.1016/J.amc.2009.04.012

[9]. Liu, W. M., Hethcote, H. W., & Levin, S. A. (1987). Dynamical behavior of epidemiological models with nonlinear incidence rates. Journal of Mathematical Biology, 25(4), 359-380.

[10]. Mena-Lorcat, J., & Hethcote, H. W. (1992). Dynamic models of infectious diseases as regulators of population sizes. Journal of Mathematical Biology, 30(7), 693-716.

[11]. Pang, G., & Chen, L. (2007). A delayed SIRS epidemic model with pulse vaccination. Chaos, Solitons & Fractals, 34(5), 1629-1635.

[12]. Ruan, S., & Wang, W. (2003). Dynamical behavior of an epidemic model with a nonlinear incidence rate. Journal of Differential Equations, 188(1), 135-163. https://doi.org/10.1016/s0022-0396(02)00089-x

[13]. Wen, L., & Yang, X. (2008). Global stability of a delayed SIRS model with temporary immunity. Chaos, Solitons & Fractals, 38(1), 221-226.

[14]. Xu, R., & Ma, Z. (2009). Stability of a delayed SIRS epidemic model with a nonlinear incidence rate. Chaos, Solitons & Fractals, 41(5), 2319-2325.

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Research Paper

Flexural Vibration of Piezo Electric Solid Cylinder of Class 6-Human Bone

E. S. Nehru *

Department of Mathematics, Colleges of Engineering and Universities in India, Oman, Saudi Arabia, and Bahrain.

Nehru, E. S. (2019). Flexural Vibration of Piezo Electric Solid Cylinder of Class 6-Human Bone. i-manager's Journal on Mathematics, 8(3), 28-34. https://doi.org/10.26634/jmat.8.3.17061

Abstract

This paper presents a Flexural vibration of Piezo Electric Solid Cylinder of Class 6-Human Bone. The frequency equations are obtained for the traction free surfaces with continuity condition at the interfaces. The boundary conditions are solved by using Fourier Collocation Method. The frequency equation is solved by using Muller's Method. In this paper, we have studied about the attenuation effect and Vibration characteristic for different wave numbers. Numerical results are carried out for the Human Bone material constants and the dispersion curves are compared with that of a solid piezoelectric cylinder and a similar model embedding a Carban Fiber Reinforced Plastic (CFRP) and Linear Elastics Material with Voids (LEMV).

References

[1]. Alizad, A., Walch, M., Greenleaf, J. F., & Fatemi, M. (2006). Vibrational characteristics of bone fracture and fracture repair: Application to excised rat femur. Journal of Biomechanical Engineering, 128(3), 300-308. https://doi.org/10.1115/ 1.2187037

[2]. Almudhaffar, M., Nassar, A. A., & Kareem, H. A. A. (2014). Vibration of Bones: A case study on Human Femur. Basrah Journal for Engineering Science, 14(2), 229-239.

[3]. Gupta, A., & Tse, K. M. (2014, December). Vibration analysis of femur bone using Elmer. Journal of Engineering Science and Technology. Special Issue on ICMTEA Conference (pp. 12-19).

[4]. Paul, H. S., & Nelson, V. K. (1994). Axisymmetric vibration of piezoelectric composite cylinders. In Proceedings of Third international Congress on Air and Structure- Borne Sound and Vibration, (Vol. 1,pp.137-144).

[5]. Paul, H. S., & Nelson, V. K. (1995). Wave propagation in piezocomposite plates. In Proceedings of Indian Natural Sciences Academy, 61, Part- A, (pp. 221-228).

[6]. Paul, H. S., & Nelson, V. K. (1996a). Axisymmetric vibration of piezocomposite hollow circular cylinder. Acta Mechanica, 116, 213-222. https://doi.org/10.1007/BF01171431

[7]. Paul, H. S., & Nelson, V. K. (1996b). Flexural vibration of piezocomposite hollow cylinder. Journal of the Acoustical Society of America, 99, 309-313.

[8]. Paul, H. S., & Venkatesh, M. (1989). Wave propagation in piezoelectric solid cylinder of arbitrary cross section of class 6 (Bone). International Journal Engineering Science, 27, 847-853. https://doi.org/10.1016/0020-7225(89)90050-5

[9]. Plona, T. J., Sinha, B. K., Kostek, S., & Chang, S. K. (1992). Axisymmetric wave propagation in fluid‐loaded cylindrical shells. II: Theory versus experiment. Journal of the Acoustical Society of America, 92(2), 1144-1155. https://doi.org/10.1121/ 1.404041

[10]. Sinha, B. K., Plona, T. J., Kostek, S., & Chang, S. K. (1992). Axisymmetric wave propagation in fluid–loaded cylindrical shells: Theory, Journal of the Acoustical Society of America, 92, 1132-1143. https://doi.org/10.1121/1.404040

[11]. Thomas, A. M. C., Luo, D. Z., & Dunn, J. W. (1991). Response of human femur to mechanical vibration. Journal of Biomedical Engineering, 13(1), 58-60. https://doi.org/10.1016/0141-5425(91)90045-9

[12]. Yang, G., Kabel, J., Van Rietbergen, B., Odgaard, A., Huiskes, R. I. K., & Cown, S. C. (1998). The anisotropic Hooke's law for cancellous bone and wood. Journal of Elasticity, 53(2), 125-146.

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Research Paper

Dynamics of a Prey and Two Predators with Time Delay in Prey Species

G. A. L. Satyavathi* , Paparao A. V., Sobhan Babu K.*

* Research Scholar, JNTUK, Kakinada, Andhra Pradesh, India.

Department of Mathematics, JNTUK, University College of Engineering (UCE), Vizianagaram, Andhra Pradesh, India.

* Department of Mathematics, JNTUK, University College of Engineering Narasaraopet (UCEN), Andhra Pradesh, India.

Satyavathi, G. A. L., Paparao, A. V., and Babu, S. K. (2019). Dynamics of a Prey and Two Predators with Time Delay in Prey Species. i-manager's Journal on Mathematics, 8(3), 35-45. https://doi.org/10.26634/jmat.8.3.17123

Abstract

We describe a mathematical model for a three species ecological model which consists a prey (x₁) and two neutral predators (x₂, x₃), surviving on the common prey (x₁). Here all the three species have limited own natural resources. A Distributed type of delay is included in the prey (x₁) species. The system is described by a couple of integro differential equations. The co-existing state is identified and the local stability analysis is studied at this state. Global stability is assured by choosing suitable Lyapunov's function. The effect of time delay is studied using Numerical simulation with different kernel strengths and it is proved that delays destabilizes the system.

References

[1]. Alfred, J. (1925). Elements of Physical Biology. Williams and Wilkins.

[2]. Braun, M. (1978). Differential Equations and their Applications- Applied Mathematical Science. Springer.

[3]. Colinvaux, P. (1986). Ecology. John Wiley and Sons Inc.

[4]. Cushing, J. M. (1977). Integro-Differential equations and delay models in population dynamics. Lecture Notes in Biomathematics, (Vol. 20). Springer-Verlag.

[5]. Freedman, H. I. (1980). Deterministic Mathematical Models in Population Ecology (Vol. 57). Marcel Dekker Incorporated.

[6]. Gopalaswamy, K. (1992). Mathematics and Its Applications Stability and Oscillations in Delay Differential Equations of Population Dynamics. Kluwer Academic Publishers (pp. 5-74).

[7]. Kapur, J. N. (1985). Mathematical Models in Biology and Medicine. Affiliated East-West Press.

[8]. Kapur, J. N. (1988). Mathematical Modeling. Wiley-Eatern.

[9]. Karuna, B. N. R., Narayan, K. L., & Reddy, B. R. (2015). A mathematical study of an infectious disease model with time delay in CTL response. Global Journal of Pure and Applied Mathematics, 11(2), 110-114.

[10]. Kuang, Y. (Ed.). (1993). Delay Differential Equations: With Applications in Population Dynamics. Academic Press.

[11]. MacDonald, N. (1978). Time Lags in Biological Models. Springer-Verlag.

[12]. May, R. M. (1973). Stability and Complexity in Model Eco-Systems. Princeton University Press.

[13]. Murray, J. D. (2007). Mathematical Biology: I. An Introduction (Vol. 17). Springer Science & Business Media.

[14]. Paparao A. V., &. Gamini, N. (2018). Dynamical behaviour of prey predators model with time delay. International Journal of Mathematics And its Applications, 6(3), 27-37.

[15]. Paparao, A. V., & Narayan, K. L. (2015). Dynamics of three species ecological model with time-delay in prey and predator. Journal of Calcutta Mathematical society, 11(2), 111-136.

[16]. Paparao, A. V., & Narayan, K. L. (2017a). A prey, predator and a competitor to the predator model with time delay. International Journal of Research in Science & Engineering, 27-38.

[17]. Paparao, A. V., & Narayan, K. L. (2017b). Optimal harvesting of prey in three species ecological model with a time delay on prey and predator. Research Journal of Science and Technology, 9(3), 368-376. https://doi.org/10.5958/2349- 2988.2017.00064.X

[18]. Paparao, A V., & Narayan, K. L. (2017c). Dynamics of a prey predator and competitor model with time delay. International Journal of Ecology & Development, 32(1), 75-86.

[19]. Paparao, A. V., Narayan. K. L., & Rao, K. K. (2019). Amensalism model: A mathematical study. International Journal of Ecological Economics & Statistics (IJEES), 40(3), 75-87.

[20]. Paparao, A.V., & Gamini, N. (2019). Stability analysis of a time delay three species ecological model. International Journal of Recent Technology and Engineering (IJRTE), 7(6), 839-845.

[21]. Ranjith Kumar, G., Lakshmi Narayan, K., & Ravindra Reddy, B. (2006). Stability and Hop bifurcation analysis of SIR epidemic model with time delay. Equilibrium, 1, 2.

[22]. Rao, V. S. H., & Rao, P. R. S. (2009). Dynamic Models and Control of Biological Systems. Springer Science & Business Media.

[23]. Reddy, K. S. (2013). Some mathematical aspects of ecological multiple prey-predator systems. (Doctorate Dissertation), Jawaharlal Nehru Technological University Hyderabad, Kukatpally, Hyderabad, India.

[24]. Simmons, G. F. (1974). Differential Equations with Applications and Historical notes. Tata McGraw-Hill.

[25]. Volterra, V. (1931). Le conssen La Theirie Mathematique De La Leitte Pou Lavie. Gauthier-Villars, Paris.

i-manager's Journal on Mathematics (JMAT)

Current Issue Vol. 12 Issue 2

Most Read

Most Cited

Volume 8 Issue 3 July - September 2019

2-HH Norm and Birkhoff-James Orthogonality in Normed Spaces

Bhuwan Prasad Ojha* , Prakash Muni Bajracharya**

*-** Department of Mathematics at Tribhuvan University, Kathmandu, Nepal.

Ojha, B. P., and Bajracharya, P. M. (2019). 2-HH Norm and Birkhoff-James Orthogonality in Normed Spaces. i-manager's Journal on Mathematics, 8(3), 1-9. https://doi.org/10.26634/jmat.8.3.16746

Abstract

References

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Modified Intuitionistic M-Fuzzy Metric Space and Some Common Fixed-Point Theorems

Dr Pranjali* , Shailesh Dhar Diwan**

* Department of Applied Mathematics, Shri Shankaracharya Institute of Professional Management and Technology, Raipur, Chhattisgarh, India. ** Department of Applied Mathematics, Government Engineering College, Raipur, Chhattisgarh, India.

Sharma, P., and Diwan, S. D. (2019). Modified Intuitionistic M-Fuzzy Metric Space and Some Common Fixed-Point Theorems. i-manager's Journal on Mathematics, 8(3), 10-16. https://doi.org/10.26634/jmat.8.3.16954

Abstract

References

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Dynamics of Delayed Sirs Epidemic Model with a Non-Linear Incidence Rate

Appa Rao Dokala* , Shaik Kalesha Vali S. **, Papa Rao A. V.***

* Department of Mathematics, IIIT Srikakulam, RGUKT, Andhra Pradesh, India. **-*** Department of Mathematics, JNTUK, University College of Engineering, Vizianagaram, Andhra Pradesh, India.

Dokala, A. R., Vali, S. K., and Rao, P. A. V. (2019). Dynamics of Delayed Sirs Epidemic Model with a Non-Linear Incidence Rate. i-manager's Journal on Mathematics, 8(3), 17-27. https://doi.org/10.26634/jmat.8.3.16707

Abstract

References

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Flexural Vibration of Piezo Electric Solid Cylinder of Class 6-Human Bone

E. S. Nehru *

Department of Mathematics, Colleges of Engineering and Universities in India, Oman, Saudi Arabia, and Bahrain.

Nehru, E. S. (2019). Flexural Vibration of Piezo Electric Solid Cylinder of Class 6-Human Bone. i-manager's Journal on Mathematics, 8(3), 28-34. https://doi.org/10.26634/jmat.8.3.17061

Abstract

References

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Dynamics of a Prey and Two Predators with Time Delay in Prey Species

G. A. L. Satyavathi* , Paparao A. V.**, Sobhan Babu K.***

* Research Scholar, JNTUK, Kakinada, Andhra Pradesh, India. ** Department of Mathematics, JNTUK, University College of Engineering (UCE), Vizianagaram, Andhra Pradesh, India. *** Department of Mathematics, JNTUK, University College of Engineering Narasaraopet (UCEN), Andhra Pradesh, India.

Satyavathi, G. A. L., Paparao, A. V., and Babu, S. K. (2019). Dynamics of a Prey and Two Predators with Time Delay in Prey Species. i-manager's Journal on Mathematics, 8(3), 35-45. https://doi.org/10.26634/jmat.8.3.17123

Abstract

References

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* Department of Applied Mathematics, Shri Shankaracharya Institute of Professional Management and Technology, Raipur, Chhattisgarh, India.

** Department of Applied Mathematics, Government Engineering College, Raipur, Chhattisgarh, India.

Appa Rao Dokala* , Shaik Kalesha Vali S. , Papa Rao A. V.*

* Department of Mathematics, IIIT Srikakulam, RGUKT, Andhra Pradesh, India.

-* Department of Mathematics, JNTUK, University College of Engineering, Vizianagaram, Andhra Pradesh, India.

G. A. L. Satyavathi* , Paparao A. V., Sobhan Babu K.*

* Research Scholar, JNTUK, Kakinada, Andhra Pradesh, India.

Department of Mathematics, JNTUK, University College of Engineering (UCE), Vizianagaram, Andhra Pradesh, India.

* Department of Mathematics, JNTUK, University College of Engineering Narasaraopet (UCEN), Andhra Pradesh, India.