Abstract
We investigate the global asymptotic stability of the solutions of \begin{document}$X_{n+1}=\frac{β X_{n-l} + γ X_{n-k}}{A + X_{n-k}} $\end{document} for \begin{document}$n=1,2, ...$\end{document} , where \begin{document}$l$\end{document} and \begin{document}$k$\end{document} are positive integers such that \begin{document}$l≠ k$\end{document} . The parameters are positive real numbers and the initial conditions are arbitrary nonnegative real numbers. We find necessary and sufficient conditions for the global asymptotic stability of the zero equilibrium. We also investigate the positive equilibrium and find the regions of parameters where the positive equilibrium is a global attractor of all positive solutions. Of particular interest for this generalized equation would be the existence of unbounded solutions and the existence of prime period two solutions depending on the combination of delay terms ( \begin{document}$l$\end{document} , \begin{document}$k$\end{document} ) being (odd, odd), (odd, even), (even, odd) or (even, even). In this manuscript we will investigate these aspects of the solutions for all such combinations of delay terms.
Highlights
Consider the difference equation xn+1 =βxn−l + γxn−k, A + xn−k n = 0, 1, . . . , (1)where l and k are positive integers such that l = k, the parameters β, γ, and A are positive real numbers, and the initial conditions are arbitrary non-negative real numbers.A special case where l = 1 and k = 2 is studied in [7]
Of particular interest for this generalized equation would be the existence of unbounded solutions and the existence of prime period two solutions depending on the combination of delay terms (l, k) being, or
It was shown that when A ≥ β + 1 all solutions converge to the zero equilibrium
Summary
The parameters are positive real numbers and the initial conditions are arbitrary nonnegative real numbers. We investigate the positive equilibrium and find the regions of parameters where the positive equilibrium is a global attractor of all positive solutions. Of particular interest for this generalized equation would be the existence of unbounded solutions and the existence of prime period two solutions depending on the combination of delay terms (l, k) being (odd, odd), (odd, even), (even, odd) or (even, even). In this manuscript we will investigate these aspects of the solutions for all such combinations of delay terms
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