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

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

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
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Abstract

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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 

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