### Boundedness and global attractivity of a higher-order nonlinear difference equation.

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In this paper, a discrete version of continuous non-autonomous predator-prey model with infected prey is investigated. By using Gaines and Mawhin's continuation theorem of coincidence degree theory and the method of Lyapunov function, some sufficient conditions for the existence and global asymptotical stability of positive periodic solution of difference equations in consideration are established. An example shows the feasibility of the main results.

The aim of this paper is to extend the classical linear condition concerning diagonal dominant bloc matrix to fully nonlinear equations. Even if assumptions are strong, we obtain an explicit condition which exactly extend the one known in linear case, and the setting allows also to consider bicontinuous operator instead of the schift and as particular case, we receive periodic or almost periodic solutions for discrete time equations.

Based on the fixed-point theorem in a cone and some analysis skill, a new sufficient condition is obtained for the existence of positive periodic solutions for a class of higher-order functional difference equations. An example is used to illustrate the applicability of the main result.

In this paper, we determine the forbidden set and give an explicit formula for the solutions of the difference equation $${x}_{n+1}=\frac{a{x}_{n}{x}_{n-1}}{-b{x}_{n}+c{x}_{n-2}},\phantom{\rule{1.0em}{0ex}}n\in {\mathbb{N}}_{0}$$ where $a$, $b$, $c$ are positive real numbers and the initial conditions ${x}_{-2}$, ${x}_{-1}$, ${x}_{0}$ are real numbers. We show that every admissible solution of that equation converges to zero if either $a<c$ or $a>c$ with $(a-c)/b<1$. When $a>c$ with $(a-c)/b>1$, we prove that every admissible solution is unbounded. Finally, when $a=c$, we prove that every admissible solution converges to zero.

In this paper, we introduce an explicit formula and discuss the global behavior of solutions of the difference equation $${x}_{n+1}=\frac{a{x}_{n-3}}{b+c{x}_{n-1}{x}_{n-3}}\phantom{\rule{0.166667em}{0ex}},\phantom{\rule{2.0em}{0ex}}n=0,1,\cdots $$ where $a,b,c$ are positive real numbers and the initial conditions ${x}_{-3}$, ${x}_{-2}$, ${x}_{-1}$, ${x}_{0}$ are real numbers.

In order to study the impact of fishing on a grouper population, we propose in this paper to model the dynamics of a grouper population in a fishing territory by using structured models. For that purpose, we have integrated the natural population growth, the fishing, the competition for shelter and the dispersion. The dispersion was considered as a consequence of the competition. First we prove, that the grouper stocks may be less sensitive to the...

In this paper we investigate the global convergence result, boundedness and periodicity of solutions of the recursive sequence $${x}_{n+1}=\frac{{a}_{0}{x}_{n}+{a}_{1}{x}_{n-1}+\cdots +{a}_{k}{x}_{n-k}}{{b}_{0}{x}_{n}+{b}_{1}{x}_{n-1}+\cdots +{b}_{k}{x}_{n-k}},\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}n=0,1,\cdots \phantom{\rule{0.166667em}{0ex}}\phantom{\rule{4pt}{0ex}}$$ where the parameters ${a}_{i}$ and ${b}_{i}$ for $i=0,1,\cdots ,k$ are positive real numbers and the initial conditions ${x}_{-k},{x}_{-k+1},\cdots ,{x}_{0}$ are arbitrary positive numbers.