Bifurcation Analysis and Chaos Control for Prey-Predator Model With Allee Effect

Abstract

The main purpose of this work is to discuss the dynamics of a predator-prey dynamical system with Allee effect. The conformable fractional derivative is applied to convert the fractional derivatives which appear in the governing model into ordinary derivatives. We use the piecewise-constant approximation method to discritize the considered model. We also investigate the occurrence of positive equilibrium points. Moreover, we analyse the stability of the equilibrium point using some stability theorems. This work also explores a Neimark-Sacker bifurcation and a period-doubling bifurcation using the theory of bifurcations. The distance between the obtained equilibrium point and some closed curves is examined for various values for the considered bifurcation parameter. The chaos control is nicely analysed using the hybrid control approach. Furthermore, we present the maximum Lyapunov exponents for different values for the bifurcation parameter. Numerical simulations are utilized to ensure that the obtained theoretical results are correct. The used techniques can be applied to deal with predator-prey models in various versions.

Qualitative Behavior for a Discretized Conformable Fractional-Order Lotka-Volterra Model With Harvesting Effects

“…This part will discretize the proposed model using the piecewise constant argument approach [23]. We also employ the concept of the conformable fractional derivative [11,24] (Definition 1.1). Using this concept and simplify the obtained results, model (1.1) can be converted into the following system:…”

Global Behavior of Solutions to a Higher-Dimensional System of Difference Equations with Lucas Numbers Coefficients

“…It is easy to show that system (3) has 2 θ fixed points. We shall study the stability of the fixed point E = x (1) , x (2) , . .…”

“…Difference equations and systems of difference equations are of great importance in the field of mathematics as well as in other sciences. The applications of the difference equations appear as discrete mathematical models of many phenomena such as in biology, economics, ecology, control theory, physics, engineering, population dynamics and so forth [1][2][3][4][5][6]. This is the reason why, recently, many scientists have devoted their work to the study of the theory of difference equations, the boundedness, the periodicity and the global asymptotic stability of their solutions .…”