Abstract

A 3-dimensional nutrient-prey-predator model with intratrophic predation is proposed and studied. Some elementary properties such as invariance of nonnegativity, boundedness and dissipativity of the system are presented. The purpose of this chapter is to study the existence and stability of equilibria along with the effects of intratrophic predation towards the positions and stability of those equilibria of the system. We also investigate the occurrence of Hopf bifurcation. In the case when there is no presence of predator organisms, intratrophic predation may not give impact on the stability of equilibria of the system. We also analysed global stability of the equilibrium point. A suitable Lyapunov function is defined for global stability analysis and some results of persistence analysis are presented for the existence of positive interior equilibrium point. Besides that, Hopf bifurcation analysis of the system are demonstrated.

Highlights

  • Chemostat, a piece of laboratory apparatus is frequently used in mathematical ecology

  • Local and global stability of other steady states were shown in the paper along with persistence analysis of the system

  • In the research of Yang et al [5], piecewise chemostat models which involve control strategy with threshold window are proposed and analysed. They investigated the qualitative analysis such as existence and stability of equilibrium points of the system and it is proved that the regular equilibria and pseudo-equilibrium cannot coexist

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Summary

Introduction

A piece of laboratory apparatus is frequently used in mathematical ecology. Another paper regarding chemostat model is by El-Sheikh and Mahrouf [3] that presented a 4-dimensional food chain in a chemostat with removal rates They studied local and global stability of equilibria along with elementary properties including boundedness of solutions, invariance of nonnegativity, dissipativity and persistence analysis. In the research of Yang et al [5], piecewise chemostat models which involve control strategy with threshold window are proposed and analysed They investigated the qualitative analysis such as existence and stability of equilibrium points of the system and it is proved that the regular equilibria and pseudo-equilibrium cannot coexist. The parameter b and term 1 þ 1þxDþ1Dx1by are added to differential equations x0 and y0 in the interest of studying the behaviour of the modified model By this motivation, we analysed the stability and Hopf bifurcation of the model with intratrophic predation, as intratrophic predation analysis is rarely considered in mathematical models of populations biologically [6]. We applied Hopf bifurcation theorems (see [9]) in the analysis of Hopf bifurcation

The model
Elementary properties of system (3)
Existence of equilibrium points
Stability analysis of equilibrium points E1, E2 and E3
C C C À 1C
À ζs þ D1f
Global stability and uniform persistence analysis
Hopf bifurcation
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