A New Approach on Exact Controllability of Semilinear Predator Prey Model

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Authors

  • Chandan Shukla Rajkiya Engineering College, Kannauj, Uttar Pradesh-209732, India Author
  • Anurag Shukla Rajkiya Engineering College, Kannauj, Uttar Pradesh-209732, India Author
  • Rajeev Kumar Rajkiya Engineering College, Kannauj, Uttar Pradesh-209732, India Author
  • Arun Kumar Singh Rajkiya Engineering College, Kannauj, Uttar Pradesh-209732, India Author
  • Arstu Gautam Rajkiya Engineering College, Kannauj, Uttar Pradesh-209732, India Author

DOI:

https://doi.org/10.5890/DNC.2024.03.002

Abstract

Using the fundamentals of functional differential equations, we provide explicit controllability results for a family of semilinear nonautonomous predator-prey systems in this study. The suggested method is simple in terms of hefty estimations as compared to standard ways because it avoids the well-known fixed point theory approach. It's also effective because it doesn't require any of the unnatural conditions that fixed point theory requires. Finally, we looked at a case study with simulation findings.

References

[1] Albrecht, F., Gatzke, H., Haddad, A., and Wax, N. (1976), On the control of certain interacting populations, Journal of Mathematical Analysis and Applications, 53(3), 578-603, DOI:doi.org/10.1016/0022-247X(76)90094-9.

[2] Freedman, H.I. (1980), Deterministic Mathematical Model in Population Ecology, Marcel Dekker, New York, New York, DOI:doi.org/10.4236/as.2014.53024.

[3] Scudo, F.M. (1971), Vito Volterra and theoretical ecology, Theoretical Population Biology, 2(1), 1-23, DOI:doi.org/10.1016/0040-5809(71)90002-5.

[4] Backwinkel-Schillings, M. (1976), Existence theorems for generalized Hammerstein equations, Journal of Functional Analysis, 23(3), 177-194, DOI:doi.org/10.1016/0022-1236(76)90046-X.

[5] Cheng, H. and Yuan, R. (2017), Existence and stability of traveling waves for Leslie Gower predator-prey system with nonlocal diffusion, Discrete and Continuous Dynamical Systems-A, 37(10), 5433, DOI:doi.org/10.3934/dcds.2017236.

[6] Fu, S. and Miao, L. (2020), Global existence and asymptotic stability in a predator prey chemotaxis model, Nonlinear Analysis: Real World Applications, 54, 103079, DOI:doi.org/10.1016/j.nonrwa.2019.103079.

[7] Ghanbari, B. (2020), On approximate solutions for a fractional prey predator model involving the Atangana Baleanu derivative, Advances in Difference Equations, (1), 1-24, DOI:doi.org/10.1016/j.rinp.2023.106489.

[8] Kumar, S., Kumar, R., Cattani, C., and Samet, B. (2020), Chaotic behaviour of fractional predator-prey dynamical system. Chaos, Solitons and Fractals, 135, 109811, DOI:doi.org/10.1016/j.chaos.2020.109811.

[9] Curtain, R.F. and Zwart, H. (2012), An introduction to infinite-dimensional linear systems theory, Springer Science & Business Media, 21, DOI:10.1137/1038096.

[10] Fattorini, H.O. (1966), Some remarks on complete controllability, SIAM Journal on Control, 4(4)m 686-694, DOI:doi.org/10.1137/0304048,

[11] Zabczyk, J. (1997), Mathematical control theory: an introductionmm Applications of Mathematics-New York, 42(1), 80.

[12] Bashirov, A.E. and Mahmudov, N.I. (1999), On concepts of controllability for deterministic and stochastic systems, SIAM Journal on Control and Optimization, 37(6), 1808-1821, DOI:doi/10.1137/S036301299732184X.

[13] Bashirov, A.E. and Jneid, M. (2013), On Partial Complete Controllability of Semilinear Systems, In Abstract and Applied Analysis, Hindawi, 2013, DOI:doi.org/10.1155/2013/521052,

[14] Bashirov, A.E. and Jneid, M. (2014), Partial complete controllability of deterministic semilinear systems, TWMS Journal of Applied and Engineering Mathematics, 4(2), 216-225.

[15] Balachandran, K. and Dauer, J.P. (2002), Controllability of nonlinear systems in Banach spaces: a survey, Journal of Optimization Theory and Applications, 115(1), 7-28, DOI:doi.org/10.1023/A:1019668728098.

[16] Klamka, J. (2013), Controllability of dynamical systems. A survey, Bulletin of the Polish Academy of Sciences. Technical Sciences, 61(2), DOI:10.2478/bpasts-2013-0031.

[17] Echarroudi, Y., Maniar, L., and Ainseba, B. (2020), Null controllability of a cascade model in population dynamics. DOI:doi.org/10.48550/arXiv.1701.04083.

[18] Gu, J.J. and Wang, X.M. (2012), Null exact controllability of predator prey population dynamics, Applied Mathematical Sciences, 6(2), 55-62.

[19] Zhang, H., Georgescu, P., and Chen, L. (2008), On the impulsive controllability and bifurcation of a predator pest model of IPM, BioSystems, 93(3), 151-171, DOI:10.1016/j.biosystems.2008.03.008.

[20] Silverman, L.M. and Meadows, H.E. (1967), Controllability and observability in time-variable linear systems, SIAM Journal on Control, 5(1), 64-73, DOI:doi.org/10.1137/0305005.

[21] Joshi, M.C. and George, R.K. (1992), On the controllability of predator-prey systems, Journal of Optimization Theory and Applications, 74(2), 243-258, DOI:10.1007/BF00940893.

[22] Leiva, H. (2015), Controllability of semilinear impulsive nonautonomous systems, International Journal of Control, 88(3), 585-592, DOI:10.1080/00207179.2014.966759.

[23] Kar, T.K. and Matsuda, H. (2006), Controllability of a harvested prey predator system with time delay, Journal of Biological Systems, 14(02), 243-254, DOI:doi.org/10.1142/S0218339006001775.

[24] Sakthivel, K., Devipriya, G., Balachandran, K., and Kim, J.H. (2010), Controllability of a reaction-diffusion system describing predator prey model, Numerical Functional Analysis and Optimization, 31(7), 831-851, DOI:doi.org/10.1080/01630563.2010.493128.

[25] Dineshkumar, C., Udhayakumar, R., Vijayakumar, V., Shukla, A., and Nisar, K.S. (2021), A note on approximate controllability for nonlocal fractional evolution stochastic integrodifferential inclusions of order $rin(1,2)$ with delay, Chaos, Solitons and Fractals, 153, 111565, DOI: 10.1016/j.chaos.2021.111565.

[26] Dineshkumar, C., Udhayakumar, R., Vijayakumar, V., Nisar, K.S., and Shukla, A. (2021), A note on the approximate controllability of Sobolev type fractional stochastic integro-differential delay inclusions with order $1

[27] Haq, A. and Sukavanam, N. (2020), Existence and approximate controllability of Riemann-Liouville fractional integrodifferential systems with damping, Chaos, Solitons and Fractals, 139, 110043, DOI:doi.org/10.1016/j.chaos.2020.110043.

[28] Haq, A. and Sukavanam, N. (2021), Mild solution and approximate controllability of retarded semilinear systems with control delays and nonlocal conditions. Numerical Functional Analysis and Optimization, 42(6), 721-737, DOI:doi.org/10.1080/01630563.2021.1928697.

[29] Mohan Raja, M., Vijayakumar, V., Shukla, A., Nisar, K.S., and Rezapour, S. (2021), New discussion on nonlocal controllability for fractional evolution system of order $1

[30] Shukla, A., Sukavanam, N., and Pandey, D.N. (2015), Approximate controllability of semilinear fractional control systems of order $alphain(1,2]$. In 2015 Proceedings of the Conference on Control and its Applications, Society for Industrial and Applied Mathematics, 175-180, DOI:doi.org/10.1137/1.9781611974072.25.

[31] Shukla, A., Sukavanam, N., and Pandey, D.N. (2015), Complete controllability of semi-linear stochastic system with delay, Rendiconti del Circolo Matematico di Palermo (1952-), 64(2), 209-220.

[32] Shukla, A., Sukavanam, N., and Pandey, D.N. (2014), Controllability of semilinear stochastic system with multiple delays in control, IFAC Proceedings Volumes, 47(1), 306-312, DOI:doi.org/10.3182/20140313-3-IN-3024.00107.

[33] Shukla, A., Sukavanam, N., and Pandey, D.N. (2015), Approximate controllability of semilinear stochastic control system with nonlocal conditions, Nonlinear Dynamics and Systems Theory, 15(3), 321-333, DOI:doi.org/10.1002/mma.8444.

[34] Vijayakumar, V. and Murugesu, R. (2019), Controllability for a class of second-order evolution differential inclusions without compactness, Applicable Analysis, 98(7), 1367-1385, DOI:doi.org/10.1080/00036811.2017.1422727

[35] Vijayakumar, V., Udhayakumar, R., and Dineshkumar, C. (2021), Approximate controllability of second order nonlocal neutral differential evolution inclusions, IMA Journal of Mathematical Control and Information, 38(1), 192-210, DOI:doi.org/10.1093/imamci/dnaa001.

[36] Vijayakumar, V. (2018), Approximate controllability for a class of second-order stochastic evolution inclusions of Clarke subdifferential type, Results in Mathematics, 73(1), 1-23, DOI:doi.org/10.1007/s00025-018-0807-8.

[37] Vijayakumar, V., Panda, S.K., Nisar, K.S., and Baskonus, H.M. (2021), Results on approximate controllability results for second-order Sobolev-type impulsive neutral differential evolution inclusions with infinite delay, Numerical Methods for Partial Differential Equations, 37(2), 1200-1221, DOI:10.1002/num.22573.

[38] Zhang, X. (2000), Exact controllability of semilinear evolution systems and its application, Journal of Optimization Theory and Applications, 107(2), 415-432, DOI:10.1023/A:1026460831701.

[39] Bashirov, A.E. (2021), On exact controllability of semilinear systems, Mathematical Methods in the Applied Sciences, 44, 7455- 7462, DOI:doi.org/10.1002/mma.6265.

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PublishedMarch 2024

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How to Cite

Shukla, C., Shukla, A., Kumar, R., Singh, A. K., & Gautam, A. (2026). A New Approach on Exact Controllability of Semilinear Predator Prey Model. Discontinuity, Nonlinearity, and Complexity, 13(1), 17-26. https://doi.org/10.5890/DNC.2024.03.002