Dynamics of a second-order nonlinear difference system with exponents

Abstract

In this paper, we study the persistence, boundedness, convergence, invariance and global asymptotic behavior of the positive solutions of the second-order difference system $$\begin{aligned} x_{n+1}&= \alpha _1 + a e ^{-x_{n-1}} + b y_{n} e ^{-y_{n-1}},\\ y_{n+1}&= \alpha _2 +c e ^{-y_{n-1}}+ d x_{n} e ^{-x_{n-1}} \quad n=0,1,2,\ldots \end{aligned}$$ x n + 1 = α 1 + a e - x n - 1 + b y n e - y n - 1 , y n + 1 = α 2 + c e - y n - 1 + d x n e - x n - 1 n = 0 , 1 , 2 , … where $$\alpha _1, \alpha _2, a, b , c,d$$ α 1 , α 2 , a , b , c , d are positive real numbers and the initial conditions $$x_{-1},x_0, y_{-1}, y_0$$ x - 1 , x 0 , y - 1 , y 0 are arbitrary nonnegative numbers.

Dynamics of a second order three species nonlinear difference system with exponents

“…Now we state a lemma which is an alteration of Theorem 1.16 in [16] and an extension of Lemma 5 in [26]. The proof is similar to the proof in [26] and hence omitted.…”

“…As seen in (1.1), (1.2), (1.3), (1.4), (1.5) and (1.6) much work has happened related to three species first order systems. In [26] and [27] we have analyzed a two species second order system wherein we assume interaction between species of exponential form, a choice which has prominent importance in population biology. Not much work has happened related to the three species second order systems.…”

“…In this paper, we propose a second order three species competitive model (0.1) which is an extension of [26]. Here x n , y n , z n can be considered as population densities of three species such that population density of x n , y n , z n depends on the growth of y n , z n , x n respectively with growth rate b, d, j respectively.…”

Qualitative Behavior of Sixth Order of Rational Difference Equation

“…Nevertheless, recent efforts have introduced several strategies to transform challenging NLDEs into linear forms, thereby revealing recognized solution forms. Notably, this approach has successfully led to the closed-form solutions of a significant class of NLDEs, as demonstrated in references [5][6][7][8]. Many scholars have studied the behavior of equations of difference equations (DEs) and investigated the behavior of systems of solved DEs.,for instance : Khaliq and Elsayed [25] examined the DEs periodic solutions existence and dynamics:…”