Additive Allee effect of top predator in a mathematical model of three species food chain
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ORIGINAL ARTICLE
Additive Allee effect of top predator in a mathematical model of three species food chain Supriya Mandal1 • Fahad Al Basir3 • Santanu Ray2 1 2 3
Department of Mathematics, Visva-Bharati University, Santiniketan 731235, India Systems Ecology & Ecological Modeling Laboratory, Department of Zoology, Visva-Bharati University, Santiniketan 731235, India Department of Mathematics, Asansol Girls’ College, Asansol-4, West Bengal-713304, India
Received: 24 August 2020 / Revised: 10 October 2020 / Accepted: 22 October 2020 Ó The Joint Center on Global Change and Earth System Science of the University of Maryland and Beijing Normal University 2020
Abstract In the present paper, the Hastings and Powell model of three species food chain model is modified incorporating Allee effect. Additive type Allee effect has been used in order to modify the three-species food chain model. We investigate the impact of Allee in the food chain model. Dynamical behaviours such as existence of steady states and their local stability analysis, bifurcation analysis of the proposed three species model have been discussed. Numerical examples are provided to justify the analytical results. Regions of stability of different equilibrium points have been plotted which are in coherence with the results derived analytically. From this study, we find that chaos in the three species food chain can be controlled when prey is subject to the strength of the Allee effect. Bifurcation diagrams show that the model turns from stable to periodic doubling and periodic doubling to chaos due to the strength of Allee effect. We observe that this modified HP model is lesser chaotic in case of increase in severity of Allee effect. Keywords Additive Allee Hastings and Powell model Stability Bifurcation Chaos
1 Introduction In population biology, Allee effect plays an important role in describing population growth rate. Although the concept of Allee effect was first coined by Warder Clyde Allee in & Santanu Ray [email protected] Fahad Al Basir [email protected]
1930s, the term Allee principle was first introduced in 1950 (Courchamp et al. 2008; Allee and Bowen 1932). The concept indicates the idea that many species can be benefited from the presence of conspecifics. The Allee effect is visible at low population size as population growth rate is reduced. Various kinds of biological and environmental factors such as difficulties in finding mates, social felicitation of reproduction, low probability of successful mating, depletion in inbreeding rate, antipredator aggression and predator avoidance due to evolutionary change cause Allee effect. There are substantial and rich studies dedicated to describing the phenomenon of Allee effect which states positive relationship between any measurable component of individual fitness and population size, quantified by the number or density of species. In real world, it has been observed that many animals and plant species go through a decrease of the per capita growth rate as their populations re
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