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Modelling is a portrait of complex natural or real world processes which enhances the knowledge of the mechanisms and provides assistance for prediction. In biology and bio medicine, organ malfunctioning of micro and macro level living populations can be understood through modelling and it helps to develop different methods to control. This paper considers the mathematical model of two species interaction using Caputo type fractional difference operator. The presence of chaos and its influence are studied in this work. The prey-predator fractional map's dynamic behaviour is validated experimentally by means of time series, bifurcation diagrams, chaotic and coexisting attractors. Further, adaptive control law is proposed to stabilize the fractional map and a control scheme is introduced to enhance the synchronization of the fractional model. Finally the effectiveness of the theoretical findings are demonstrated with numerical examples. [ABSTRACT FROM AUTHOR]

In mathematical biology, discrete time models governed by difference equations are more suitable to study the evolutionary growths of species or taxonomic group of organisms over the course of time. These discrete time population models are appropriate for non-overlapping generations and are suitable to analyse the nonlinear dynamics and their chaotic behaviour. In this research article, the complexity of intra species competition between two species with carrying capacity is investigated in discrete time, under the assumption that average growth rates in prey and predator populations change at regular intervals of time. The method of piecewise constant arguments for differential equations is implemented to obtain the discrete time counterpart of continuous case. Results related to uniform boundedness of solutions, and existence and uniqueness of positive fixed points are established. The parametric conditions for local asymptotic stability of fixed points are obtained. Also, the existence criteria and direction of flip bifurcation and hopf bifurcation are derived. Moreover, feedback control strategy is employed to control the chaos under the influence of bifurcations. To validate the analytical results, numerical illustrations are presented. The existence of chaos is confirmed from calculating Maximal Lyapunov Exponent. [ABSTRACT FROM AUTHOR]

This paper aims to establish some new sufficiency conditions for the oscillatory behavior of the solutions of a forced non linear difference equation of fractional order with damping term Δ (Δ μ u (l)) + γ (l) Δ μ u (l) + ρ (l) Ψ (u (l)) = ϕ (l) , l ∈ ℕ 0 Δ (μ − 1) u (0) = u (0) = u 0 with the assistance of properties of Riemann-Liouville (RL) sum and difference operators. Suitable examples are provided to demonstrate the applicability of the derived outcomes. [ABSTRACT FROM AUTHOR]

This article takes harvesting effects into account, to investigate the qualitative behavior of 2D prey-predator model with Leslie - Gower functional response. Existence and uniqueness of the solutions of the system is examined. The prey population is continuously harvested by an harvesting agency and the rate of harvesting is presumed to be proportional to the prey population. Discrete version of the system is obtained to examine the dynamical behavior of the model. Existence of non-negative equilibrium positions is established and a comprehensive analysis on the local stability of the system is performed. Numerical simulations are illustrated to compliment the analytical outcomes. The system undergoes periodic doubling bifurcation under certain conditions. Time plots, phase portraits, limit cycles, bifurcation diagrams and Lyapunov exponent are helpful to study the dynamics of the system. Finally dependence on initial conditions is discussed. Under the influence of harvesting the prey – predator model exhibits rich complex behaviors. [ABSTRACT FROM AUTHOR]

This work considers discrete fractional nonlinear equations with Riemann-Liouville (R-L) type difference operator of the form Δ β [ v (l) − p (l , v (l)) ] = f (l + β , v (l + β)) , l ∈ N 0 , 0 < β ≤ 1 Δ β − 1 v (l) | l = 0 = v (0) = c. The discrete Arzela-Ascoli's theorem and Krasnoselkii's theorem are employed to study the asymptotic stability of the fractional equation. An example illustrating the main result is provided. [ABSTRACT FROM AUTHOR]