Modelling and analysis of prey-predator model involving predation of mature prey using delay differential equations

Pankaj Kumar; Shiv Raj

doi:10.3934/naco.2021035

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2022, Volume 12, Issue 4: 783-791. Doi: 10.3934/naco.2021035

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Modelling and analysis of prey-predator model involving predation of mature prey using delay differential equations

Pankaj Kumar^1, and
Shiv Raj^2, ,

1.
Department of Mathematics, Lovely Professional University, Phagwara, Punjab -144411, INDIA
2.
Ph.D. Scholar, Lovely Professional University, Phagwara, Punjab -144411, INDIA

^* Corresponding author: Shiv Raj
^* Corresponding author: Shiv Raj

This paper is handled by V. Sree Hari Rao as the guest editor

Received: November 30, 2020

Revised: July 31, 2021

Early access: September 2021

Published: December 2022

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Abstract

In this paper, the modelling and analysis of prey-predator model involving predation of mature prey is done using DDE. Equilibrium points are calculated and stability analysis is performed about non-zero equilibrium point. Delay parameter destabilizes the system and triggers asymptotic stability when value of delay parameter is below the critical point. Hopf bifurcation is observed when the value of delay parameter crosses the critical point. Sensitivity analysis has also been performed to look into the effect of other parameters on the state variables. The numerical results are substantiated using MATLAB.

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Figure 1. The equilibrium $ E^* $($ P_r^* $, $ P_d^* $) is stable in the absence of delay i.e. when $ \tau $ = 0

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Figure 2. The equilibrium $ E^* $($ P_r^* $, $ P_d^* $) is asymptotically stable when delay is less than critical value i.e. when $ \tau $ $ < $ 8.23

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Figure 3. Phase plane of equilibrium $ E^* $($ P_r^* $, $ P_d^* $) when the delay is less than critical value i.e. $ \tau $ $ < $ 8.23

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Figure 4. The equilibrium $ E^* $($ P_r^* $, $ P_d^* $) losses stability and shows Hopf-bifurcation when delay is crosses the critical value i.e. when $ \tau $ $ \geq $ 8.23

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Figure 5. Phase plane of equilibrium $ E^* $($ P_r^* $, $ P_d^* $) when the delay crosses the critical value i.e. $ \tau $ $ \geq $ 8.23

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Figure 6. Time series graph between partial changes in $ P_d $ (predator populations) for different values of the parameter $ \alpha $

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Figure 7. Time series graph between partial changes in $ P_r $ (Prey populations) for different values of the parameter $ \delta $

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Table 1. Description of the parameters of the system (1) – (2)

Parameter	Discription
b	Intrinsic birth rate of prey population
$ \alpha $	Intra specific competition rate among prey
$ \beta $	Inter specific competition rate
d	Death rate of predator population
$ \gamma $	Inter specific competition rate
$ \delta $	Intra specific competition rate among predator
$ \tau $	Delay parameter

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