This chapter considers the discrete counterpart of a fractional order prey-predator ODE system involving Allee effect. Several more realistic models were proposed to describe nonlinear interactions between species by introducing different types functional responses and Allee effect. Non-local property of fractional differential equations is useful in modeling population interactions possessing memories. The model under investigation has three steady states, and the positive steady state exists under certain condition. Dynamic nature of the model is discussed through local stability analysis. Enquiry into the qualitative behavior of the model reveals rich and complex dynamics exhibited by the discrete-time model. Moreover, this model undergoes Neimark Sacker bifurcation when the chosen parameter passes through a critical value. The analytical results are strengthened with appropriate numerical examples. The computation of maximal Lyapunov exponents confirms the existence of chaos. Chaos control is achieved by linear feedback control and hybrid control methods.
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Over the past few centuries the explosion in scientific knowledge and technology has been a contributing factor and also to some extent a direct consequence in advancement of mathematical modeling. The development of more contemporary form of mathematical models has further enhanced to understanding of the world around us. A periodic motivation for those who create or develop mathematical models is the desire to form prediction. Ecology is a branch of biology which deals with the analysis of how species relate with each other and their environment essentially the inter and intra relationships of the species. In particular it encompasses aspects such as population growth, competition, symbiotic ecologic relationships (mutualism), trophic relations, migration and physical environment interactions. Such population dynamics have enthused biologists and mathematicians alike who collectively enhanced the study of these interactions of species by developing mathematical models.
(David Logan, 2011) A set of equations or an equation, that describes some physical phenomenon or problem that has its origin in engineering, science, economics, or some other area are called Mathematical Modeling. The subject of mathematical modeling involves formulation of equations, physical intuition, analysis and solution methods. During the past centuries, many types of models have been developed to investigate the natural and social processes that enlarge over time. Now-a-days, these models are referred as dynamical systems. (Frederick, 2006) Dynamical systems are divided into two general categories, i.e. deterministic models and stochastic models respectively. Deterministic models are employed when the number of quantities involved in the process being modeled is relatively small and all the underlying scientific principles are fairly well understood.
The tools of dynamical systems enable the researchers to better understand and investigate the different nonlinear characteristics, which exhibit new phenomena of the systems from various disciplines. Particularly, the tools of the dynamical systems such as the applied bifurcation theories are successfully employed to investigate the qualitative behaviors of nonlinear systems. This includes the enquiry in to existence of equilibrium and their stability and bifurcations of periodic orbits, quasi-periodic behavior, chaotic attractors, chaos control and synchronization.
There are numerous papers regarding the dynamics of the predator-prey system with and without different kinds of functional responses. In fact, the Allee effect is a phenomenon in biology named after ecologist Warder Clyde Allee (Allee, 1931) who brought attention to the possibility of a positive relationship between aspects of fitness and population size over fifty years ago (Stephens et al, 1999). In 1931, Allee introduced the Allee effect, a reproductive opportunities that cause negative growth rate of a population of biological species at densities below a critical value. It is well known that Allee effect made significant contribution in the stability analysis of equilibrium points of a population dynamics model (Ak Gumus et al, 2012). Biological facts on the Allee function are given below (Kangalgil, 2017):
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When the population density is zero, Allee function is zero.
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The derivatives of the Allee function with respect to independent variables are always strictly positive for all positive values of the population density.
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Allee effect vanishes at high densities.
Many researchers contributed to the study of stability of predator–prey system in the presence/absenceof Allee effect (Pk & Ghorais, 2017). Rarely bifurcation analysis of these systems with the Allee effect has been investigated. Hence, in this chapter, predator–prey system enforced with an Allee effect is considered.(Lakshmanan et al, 2012)Fractional-order systems with order less than three have the ability to exhibit chaos. Hence, investigation of chaos in fractional-order models has become a hot topic of research, and therefore, it has been studied by many biologists and mathematicians.