Qualitative behavior of a two‐dimensional discrete‐time prey–predator model

Berkal, Messaoud; Navarro, Juan F.

doi:10.1002/cmm4.1193

Cited by 9 publications

(5 citation statements)

References 23 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In this section, we discuss the existence of a Neimark-Sacker bifurcation and a perioddoubling bifurcation [4,22,[24][25][26][27][28][29][30] for System (6) by taking h as a bifurcation parameter.…”

Section: Bifurcation Analysismentioning

confidence: 99%

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

Berkal

Almatrafi

2023

Fractal Fract

Self Cite

View full text Add to dashboard Cite

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.

show abstract

Section: Bifurcation Analysismentioning

confidence: 99%

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

Berkal

Almatrafi

2023

Fractal Fract

Self Cite

View full text Add to dashboard Cite

show abstract

“…( 27,[29][30][31][32]). Assume that the polynomial K(ω) = ω 2 − pω + q, where P (1) > 0, and ω 1 and ω 2 are the two roots of K(ω) = 0.…”

Section: Local Stability Of Equilibrium Pointmentioning

confidence: 99%

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

Almatrafi,

Berkal

2023

Int. J. Anal. Appl.

View full text Add to dashboard Cite

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.

show abstract

“…Nowadays, difference equations becomes a valuable tool in the modeling of many phenomena in various fields such as biology, epidemiology, physics, probability theory, etc. See for example [7,[27][28][29]. Accordingly, difference equations have attracted the attention of many researchers, see for example [1, 5, 6, 8, 12-18, 21-24, 26, 30, 31].…”

Section: Introductionmentioning

confidence: 99%