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Finite Time Blow-up in a Delayed Diffusive Population Model with Competitive Interference

  • Rana D. Parshad , Suman Bhowmick , Emmanuel Quansah , Rashmi Agrawal and Ranjit Kumar Upadhyay EMAIL logo
Published/Copyright: May 23, 2017

Abstract

In the current manuscript, an attempt has been made to understand the dynamics of a time-delayed predator-prey system with modified Leslie-Gower and Beddington-DeAngelis type functional responses for large initial data. In Ref. Upadhyay and Agrawal, 83(2016) 821–837, it was shown that the model possesses globally bounded solutions, for small initial conditions, under certain parametric restrictions. Here, we show that actually solutions to this model system can blow-up in finite time, for large initial condition, even under the parametric restrictions derived in Ref. Upadhyay and Agrawal, 83(2016) 821–837. We prove blow-up in the delayed model, as well as the non-delayed model, providing sufficient conditions on the largeness of data, required for finite time blow-up. Numerical simulations show that actually the initial data does not have to be very large, to induce blow-up. The spatially explicit system is seen to possess non-Turing instability. We have also studied Hopf-bifurcation direction in the spatial system, as well as stability of the spatial Hopf-bifurcation using the central manifold theorem and normal form theory.

References

[1] R.M. May (ed.), Theoretical ecology: principles and applications, Ch.5, Blackwell, Oxford, 1976.Search in Google Scholar

[2] May R.M., Mutualistic interactions among species, Nature 296 (1982), 803–804.10.1038/296803a0Search in Google Scholar

[3] Lou Y., Nagylaki T. and Ni W., On diffusion induced blowups in a mutualistic model, Nonlinear Anal. 45 (2001), 329–342.10.1016/S0362-546X(99)00346-6Search in Google Scholar

[4] Lin Z., Blowup estimates for a mutualistic model in ecology, E. J. Qualitative Theory of Diff. Equ. 8 (1) (2002), 1–14.10.14232/ejqtde.2002.1.8Search in Google Scholar

[5] Zhou P. and Lin Z., Global existene and blowup of a nonlocal problem in space with free boundary. J. Funct. Anal. 262 (2012), 3409–3429.10.1016/j.jfa.2012.01.018Search in Google Scholar

[6] Ninomiya H. and Weinberger H.F., Pest control may make the pest population explode. Zeitschrift f”ur angewandte Mathematik und Physik ZAMP 54 (5) (2003), 869–873.10.1007/s00033-003-3210-5Search in Google Scholar

[7] Stuart A.M. and Floater M.S., On the computation of blow up, Eur. J. Appl. Math. 1 (1990), 47–71.10.1017/S095679250000005XSearch in Google Scholar

[8] Hirota C. and Ozawa K., Numerical method of estimating the blow up time and rate of solution of ODE- an approach to the blow up problems of PDEs, J. Comput. Appl. Math. 193 (2006), 614–637.10.1016/j.cam.2005.04.069Search in Google Scholar

[9] Fila M., Ninomiya H. and Vazauez J., Dirichlet boundary conditions can predict blow up in reaction-diffusion equations and systems, Dis. Cont. Dyn. Syst. Ser. A 14 (2006), 63–74.10.3934/dcds.2006.14.63Search in Google Scholar

[10] Parshad R.D., Abderrahmane H.A., Upadhyay R.K. and Kumari N., Finite time blow up in a realistic food chain model, ISRN Biomathematics (2013).10.1155/2013/424062Search in Google Scholar

[11] Parshad R.D., Kumari N. and Kouachi S., A remark on “Study of a Leslie-Gower-type tritrophic population model [Chaos, Solitons and Fractals 14 (2002), 1275–1293], Chaos, Solitons Fractals 71 (2) (2015), 22–28.10.1016/j.chaos.2014.11.014Search in Google Scholar

[12] Upadhyay R.K. and Iyengar S.R.K., Introduction to mathematical modeling and chaotic dynamis. CRC Press, Boca Raton, 2013.10.1201/b15317Search in Google Scholar

[13] Upadhyay R.K. and Agrawal R., Dynamics and responses of a predator-prey system with competitive interference and time delay, Nonlinear Dyn 83(2016), 821–837.10.1007/s11071-015-2370-0Search in Google Scholar

[14] Xu C. and Yuan S., Spatial periodic solutions in a delayed diffusive predator-prey model with herd behavior, Int. J. Bifurcation Chaos 25 (11) (2015), 1–14.10.1142/S0218127415501552Search in Google Scholar

[15] Li Y. and Wang M., Hopf bifurcation and global stability of a delayed predator-prey model with prey harvesting, Comput. Math. Appl. 69 (2015), 398–410.10.1016/j.camwa.2015.01.003Search in Google Scholar

[16] Shigesada N. and Kawasaki K., Biological invasions: theory and practice, Oxford University Press, Oxford, 1997.10.1093/oso/9780198548522.001.0001Search in Google Scholar

[17] Murray J.D., Mathematical biology, Springer-Verlag, Berlin, 1993.10.1007/978-3-662-08542-4Search in Google Scholar

[18] Core Team R, R: A language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria, 2015. http://www.R-project.org/Search in Google Scholar

[19] Hassard B.D., Kazarinoff N.D. and Wan Y., Theory and applications of Hopf-bifurcation. Cambridge University Press, Cambridge, 1981.Search in Google Scholar

[20] Wu J., Theory and applications of partial functional differential equations. Springer-Verlag, NY, 1996.10.1007/978-1-4612-4050-1Search in Google Scholar

[21] Joglekar M., E.Ott and Yorke J.A., Scaling of chaos versus periodicity: How certain is in that an attractor is chaotic? Phys. Rev. Lett. 113 (2014), 084101–084104.10.1103/PhysRevLett.113.084101Search in Google Scholar PubMed

[22] Joglekar M., Ott E. and Yorke J.A., Uncertainty as to whether or not a system has a chaotic attractor. Nonlinearity 28 (2015), 3803–3820.10.1088/0951-7715/28/11/3803Search in Google Scholar

[23] Gomulkiewicz R., Thompson J.N., Holt R.D., Nuismer S.L. and Hochberg M.E., Hot spots, cold spots, and the geographic mosaic theory of coevolution, The American Naturalist 156 (2) (2000), 156–174.10.1086/303382Search in Google Scholar PubMed

[24] Parshad R.D., Quansah E., Beauregard M. and Kouachi S., On “Small” data blow-up in a three species food chain model, Computers and Mathematics with Applications 73(4) (2017), 576–587.10.1016/j.camwa.2016.12.018Search in Google Scholar

[25] Parshad R.D., Quansah E., Black K. and Beauregard M., Biological control via ecological damping: an approach that attenuates non-target effects, Math. Biosci. 273 (2016), 23–44.10.1016/j.mbs.2015.12.010Search in Google Scholar PubMed

Received: 2015-12-2
Accepted: 2017-5-4
Published Online: 2017-5-23
Published in Print: 2017-7-26

© 2017 Walter de Gruyter GmbH, Berlin/Boston

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