Abstract
This paper aims to study the effect of time-delay on a tritrophic food chainmodel with Michaelis-Menten type ratio-dependent functional responses. Boundednessof the time-delayed system is established. A simple criterion for deterministic extinctionis derived. It has been shown that the time-delay may introduce instability in the systemthrough Hopf bifurcation. Computer simulations are carried out to explain the analyticalfindings. It is discussed how these ideas illuminate some of the observed properties ofreal populations in the field, and explores practical implications.
Highlights
The use of mathematics in the modelling of biological phenomena has become more prevalent in recent years
We have studied the effect of discrete time-delay on a ratio-dependent tritrophic food chain model
It is shown that the time-delayed system is uniformly bounded, which, in turn, implies that the system is biologically well behaved
Summary
The use of mathematics in the modelling of biological phenomena has become more prevalent in recent years. Three-species systems like nutrient-bacteria-ciliate, plant-herbivoreparasitoid, plant-pest-predator, et cetera are emerging in different branches of biology in their own right. These are the examples of three-species “food chains”. Kuang [35] mentioned that animals must take time to digest their food before further activities and responses take place and any model of species dynamics without delays is an approximation at best. Theoretical ecologists have laid down immense importance on the effect of time-delay on realistic models. The present paper is designed to study the effect of discrete time-delay on a tritrophic food chain model with ratio-dependent functional responses.
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