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.
In this article, we discuss the qualitative behavior of a two‐dimensional discrete‐time prey–predator model. This system is the result of the application of a nonstandard difference scheme to a system of differential equations for a prey–predator model including intraspecific competition of prey population. In particular, we evaluate the fixed points of the system and study their local asymptotic stability. We also prove the existence of a Neimark–Sacker bifurcation.
In this paper we give a representation formula for the general solution to the following two-dimensional system of difference equations x n+1 = y n−1 x n−2 y n (a + by n−1 x n−2) , y n+1 = x n−1 y n−2 x n (a + bx n−1 y n−2) , n ∈ N 0 where parameters a, b and initial values x −2 , x −1 , x 0 , y −2 , y −1 , y 0 are real numbers. We also give some theoretical explanations related to the representation.
In this paper we represent the well-defined solutions of the system of the higher-order rational difference equationsin terms of Fibonacci and Lucas sequences, where the initial valuesdo not equal -3. Some theoretical explanations related to the representation for the general solution are also given.
In this paper, we derive the forbidden set and determine the solutions of the difference equation that contains a quadratic term \begin{equation*} x_{n+1}=\frac{x_{n}x_{n-p}}{ax_{n-(p-1)}+bx_{n-p}},\quad n\in\mathbb{N}_0, \end{equation*} where the parameters $a$ and $b$ are real numbers, $p$ is a positive integer and the initial conditions $x_{-p}$, $x_{-p+1}$, $\cdots$, $x_{-1}$, $x_{0}$ are real numbers.
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