Hostname: page-component-8448b6f56d-m8qmq Total loading time: 0 Render date: 2024-04-25T04:16:28.924Z Has data issue: false hasContentIssue false
  • English
  • Français

DYNAMIC RELATIONSHIP BETWEEN THE MUTUAL INTERFERENCE AND GESTATION DELAYS OF A HYBRID TRITROPHIC FOOD CHAIN MODEL

Part of: Dynamical problems Smooth dynamical systems: general theory Difference and functional equations, recurrence relations Nonlinear dynamics

Published online by Cambridge University Press:  26 February 2018

RASHMI AGRAWAL ,
DEBALDEV JANA ,
RANJIT KUMAR UPADHYAY Open the ORCID record for RANJIT KUMAR UPADHYAY [Opens in a new window]  and
V. SREE HARI RAO
RASHMI AGRAWAL*
Affiliation:
Department of Humanities and Science, S R Engineering College, Warangal, Telangana 506371, India email agrawal.rashmi2@gmail.com
DEBALDEV JANA
Affiliation:
Department of Mathematics and SRM Research Institute, SRM Institute of Science and Technology, Kattankulathur 603203, Tamil Nadu, India email debaldevjana.jana@gmail.com
RANJIT KUMAR UPADHYAY
Affiliation:
Department of Applied Mathematics, Indian Institute of Technology (ISM), Dhanbad, Jharkhand 826004, India email ranjit.chaos@gmail.com
V. SREE HARI RAO
Affiliation:
CEO, Turnin Innovation Technologies (Pvt) Ltd, Hyderabad, India email vshrao@turnin.in
*
✉agrawal.rashmi2@gmail.com
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We have proposed a three-species hybrid food chain model with multiple time delays. The interaction between the prey and the middle predator follows Holling type (HT) II functional response, while the interaction between the top predator and its only food, the middle predator, is taken as a general functional response with the mutual interference schemes, such as Crowley–Martin (CM), Beddington–DeAngelis (BD) and Hassell–Varley (HV) functional responses. We analyse the model system which employs HT II and CM functional responses, and discuss the local and global stability analyses of the coexisting equilibrium solution. The effect of gestation delay on both the middle and top predator has been studied. The dynamics of model systems are affected by both factors: gestation delay and the form of functional responses considered. The theoretical results are supported by appropriate numerical simulations, and bifurcation diagrams are obtained for biologically feasible parameter values. It is interesting from the application point of view to show how an individual delay changes the dynamics of the model system depending on the form of functional response.

Keywords

CM functional responseBD functional responseHV functional responsegestation delaymultiple time delaysHopf-bifurcationchaos

MSC classification

Primary: 70K50: Bifurcations and instability
Secondary: 37C75: Stability theory 65Q10: Difference equations 74H65: Chaotic behavior
Type
Research Article
Information
The ANZIAM Journal , Volume 59 , Issue 3 , January 2018 , pp. 370 - 401
Copyright
© 2018 Australian Mathematical Society 

References

Agrawal, R., Jana, D., Upadhyay, R. K. and Sree Hari Rao, V., “Complex dynamics of sexually reproductive generalist predator and gestation delay in a food chain model: double Hopf-bifurcation to chaos”, J. Appl. Math. Comput. 55 (2017) 513–547; doi:10.1007/s12190-016-1048-1.Google Scholar
Beddington, J. R., “Mutual interference between parasites or predators and its effect on searching efficiency”, J. Animal Ecol. 51 (1975) 331–340; doi:10.2307/3866.Google Scholar
Cosner, C., DeAngelis, D. L., Ault, J. S. and Olson, D. B., “Effects of spatial grouping on the functional response of predators”, Theor. Popul. Biol. 56 (1999) 65–75; doi:10.1006/tpbi.1999.1414.Google Scholar
Crowley, P. H. and Martin, E. K., “Functional responses and interference within and between year classes of a dragonfly population”, J. North Am. Benth. Soc. 8 (1989) 211–221; doi:10.2307/1467324.Google Scholar
DeAngelis, D. L., Goldstein, R. A. and O’eill, R. V., “A model for trophic interaction”, Ecology 56 (1975) 881–892; doi:10.2307/1936298.Google Scholar
Dennis, B., Desharnais, R., Cushing, J., Henson, S. and Costantino, R., “Can noise induce chaos?”, Oikos 102 (2003) 329–339; http://www.jstor.org/stable/3548035.Google Scholar
Gakkhar, S. and Naji, R., “Seasonally perturbed prey-predator system with predator-dependent functional response”, Chaos Solitons Fractals 18 (2003) 1075–1083; doi:10.1016/S0960-0779(03)00075-4.Google Scholar
Gakkhar, S. and Singh, A., “Complex dynamics in a prey predator system with multiple delays”, Commun. Nonlinear Sci. Numer. Simul. 17 (2012) 914–929; doi:10.1016/j.cnsns.2011.05.047.Google Scholar
Gao, J., Hwang, S. and Liu, J., “When can noise induce chaos?”, Phys. Rev. Lett. 82 (1999) 11–32; http://www.jstor.org/stable/3548035.Google Scholar
Gao, M., Shi, H. and Li, Z., “Chaos in a seasonally and periodically forced phytoplankton-zooplankton system”, Nonlinear Anal. Real World Appl. 10 (2009) 1643–1650; doi:10.1016/j.nonrwa.2008.02.005.Google Scholar
Guckenheimer, J. and Holmes, P., Nonlinear oscillations, dynamical systems and bifurcations of vector fields (Springer, Berlin, 1983).Google Scholar
Hale, J. K., Theory of functional differential equations, Volume 3 of Applied Mathematical Sciences (Springer, New York, 1977); doi:10.1006/jdeq.2000.3874.CrossRefGoogle Scholar
Hassard, B. D., Kazrinoff, N. D. and Wan, W. H., Theory and application of Hopf bifurcation, Volume 41 of London Mathematical Society Lecture Note Series (Cambridge University Press, Cambridge, 1981); doi:10.1137/1024123.Google Scholar
Hassell, M. P. and Varley, G. C., “New inductive population model for insect parasites and its bearing on biological control”, Nature 223 (1969) 1133–1136; doi:10.1038/2231133a0.Google Scholar
Huisman, G. and De Boer, R. J., “A formal derivation of the Beddington functional response”, J. Theor. Biol. 185 (1997) 389–400; doi:10.1006/jtbi.1996.0318.Google Scholar
Jana, D., Agrawal, R. and Upadhyay, R. K., “Top-predator interference and gestation delay as determinants of the dynamics of a realistic model food chain”, Chaos Solitons Fractals 69 (2014) 50–63; doi:10.1016/j.chaos.2014.09.001.Google Scholar
Jana, D., Agrawal, R. and Upadhyay, R. K., “Dynamics of generalist predator in a stochastic environment: effect of delayed growth and prey refuge”, Appl. Math. Comput. 268 (2015) 1072–1094; doi:10.1016/j.amc.2015.06.098.Google Scholar
Kuang, Y., Delay differential equations with applications in population dynamics (Academic Press, New York, 1993).Google Scholar
van der Meer, J. and Ens, B. J., “Models of interference and their consequences for the spatial distribution of ideal and free predators”, J. Animal Ecol. 66 (1997) 846–858; http://www.jstor.org/stable/6000.CrossRefGoogle Scholar
Morozov, A., Petrovskii, S. and Li, B., “Bifurcations and chaos in a predator-prey system with the Allee effect”, Proc. R. Soc. Lond. Ser. B: Biol. Sci. 271 (2004) 1407–1414; doi:10.1098/rspb.2004.2733.Google Scholar
Pal, N., Samanta, S., Biswas, S., Alquran, M., Al-Khaled, K. and Chattopadhyay, J., “Stability and bifurcation analysis of a three-species food chain model with delay”, Internat. J. Bifur. Chaos 25 (2015) 1550123; doi:10.1142/S0218127415501230.Google Scholar
Pathak, S., Maiti, A. and Samanta, G. P., “Rich dynamics of a food chain model with Hassell–Varley type functional responses”, Appl. Math. Comput. 208 (2009) 303–317; doi:10.1016/j.amc.2008.12.015.Google Scholar
Ruan, S., “On nonlinear dynamics of predator-prey models with disc rete delay”, Math. Model. Nat. Phenom. 4 (2009) 140–188; doi:10.1051/mmnp/20094207.Google Scholar
Schmitz, O. J., “Exploitation in model food chains with mechanistic consumer-resource dynamics”, Theor. Popul. Biol. 41 (1992) 161–183; doi:10.1016/0040-5809(92)90042-R.Google Scholar
Skalski, G. T. and Gilliam, J. F., “Functional responses with predator interference: viable alternatives to the Holling type II model”, Ecology 82 (2001) 3083–3092; doi:10.1890/0012-9658(2001)082[3083:FRWPIV]2.0.CO;2.CrossRefGoogle Scholar
Song, Z., Zhen, B. and Xu, J., “Species coexistence and chaotic behavior induced by multiple delays in a food chain system”, Ecol. Compl. 19 (2014) 9–17; doi:10.1016/j.ecocom.2014.01.004.Google Scholar
Tanabe, K. and Namba, T., “Omnivory creates chaos in simple food web models”, Ecology 86 (2005) 3411–3414; doi:10.1890/05-0720.Google Scholar
Upadhyay, R. K. and Naji, R. K., “Dynamics of a three-species food chain model with Crowley–Martin type functional response”, Chaos Solitons Fractals 42 (2009) 1337–1346; doi:10.1016/j.chaos.2009.03.020.Google Scholar
Upadhyay, R. K., Raw, S. N. and Rai, V., “Dynamical complexities in a tritrophic hybrid food chain model with Holling type II and Crowley–Martin functional responses”, Nonlinear Anal. Model. Control 15 (2010) 361–375; http://www.lana.lt/journal/38/Upadhyay.pdf.CrossRefGoogle Scholar
Wang, K., “Periodic solutions to a delayed predator-prey model with Hassell–Varley type functional response”, Nonlinear Anal. Real World Appl. 12 (2011) 137–145; doi:10.1016/j.nonrwa.2010.06.003.Google Scholar
Yang, X., Chen, L. and Chen, J., “Permanence and positive periodic solution for the single species nonautonomous delay diffusive model”, Comput. Math. Appl. 32 (1996) 109–116; doi:10.1016/0898-1221(96)00129-0.Google Scholar
Yodzis, P. and Innes, S., “Body size and consumer resource dynamics”, Amer. Natural. 139 (1992) 1151–1175; http://www.jstor.org/stable/2462335.Google Scholar
Zhang, Y., Gao, S., Fan, K. and Wang, Q., “Asymptotic behavior of a non-autonomous predator-prey model with Hassell–Varley type functional response and random perturbation”, J. Appl. Math. Comput. 49 (2015) 573–594; doi:10.1007/s12190-014-0854-6.Google Scholar
Supplementary material: File

Agrawal et al. supplementary material

Agrawal et al. supplementary material 1

Download Agrawal et al. supplementary material(File)
File 275.3 KB