Bifurcation and Stability of Two-Dimensional Activator–Inhibitor Model with Fractional-Order Derivative

Abstract

In organisms’ bodies, the activities of enzymes can be catalyzed or inhibited by some inorganic and organic compounds. The interaction between enzymes and these compounds is successfully described by mathematics. The main purpose of this article is to investigate the dynamics of the activator–inhibitor system (Gierer–Meinhardt system), which is utilized to describe the interactions of chemical and biological phenomena. The system is considered with a fractional-order derivative, which is converted to an ordinary derivative using the definition of the conformable fractional derivative. The obtained differential equations are solved using the separation of variables. The stability of the obtained positive equilibrium point of this system is analyzed and discussed. We find that this point can be locally asymptotically stable, a source, a saddle, or non-hyperbolic under certain conditions. Moreover, this article concentrates on exploring a Neimark–Sacker bifurcation and a period-doubling bifurcation. Then, we present some numerical computations to verify the obtained theoretical results. The findings of this work show that the governing system undergoes the Neimark–Sacker bifurcation and the period-doubling bifurcation under certain conditions. These types of bifurcation occur in small domains, as shown theoretically and numerically. Some 2D figures are illustrated to visualize the behavior of the solutions in some domains.

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:…”

“…This type of bifurcation takes place when a closed invariant curve arises from a fixed point in a discrete dynamical system. Then, the stability of the point changes via a pair of complex eigenvalues with unit modulus [24][25][26][27][28]. In this discussion, we study the considered system around the point P E .…”

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

“…x n (4) x n (5) Figure 1. A graph representing the global stability of system (15) (left) and system (16) (right).…”

“…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 .…”

Complex Dynamics of a Predator-Prey System With Gompertz Growth and Herd Behavior

“…This strategy facilitated the predator's ability to interact with the prey in the boundary region of the group. Numerous scholars have investigated the mechanisms of predator-prey relationships by utilizing square-root functional responses [14][15][16][17][18][19][20][21][22][23].…”