In this paper, we study the eventual periodicity of the following fuzzy max-type difference equationwhere {α n } n 0 is a periodic sequence of positive fuzzy numbers and the initial values z −d , z −d+1 , . . . , z −1 are positive fuzzy numbers with d = max{m, r}. We show that if max(supp α n ) < 1, then every positive solution of this equation is eventually periodic with period 2m.

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In this paper, we study the following max-type system of difference equations:where A n , B n ∈ (0, +∞) are periodic sequences with period 2 and the initial values x -1 , y -1 , x -2 , y -2 ∈ (0, +∞). We show that every solution of the above system is eventually periodic.

We study in this paper the following max-type system of difference equations of higher order:= max{ , − / −1 } and = max{ , − / −1 }, ∈ {0, 1, 2, . . .}, where ≥ > 0, ≥ 1, and the initial conditions − , − , − +1 , − +1 , . . . , −1 , −1 ∈ (0, +∞). We show that (1) if > 1, then every solution of the above system is periodic with period 2 eventually. (2) If = 1 > , then every solution of the above system is periodic with period 2 or 2 eventually. (3) If = = 1 or < 1, then the above system has a solution which is not periodic eventually.

In this paper, we study the following max-type system of difference equations of higher order: $$ \textstyle\begin{cases} x_{n} = \max \{A ,\frac{y_{n-t}}{x_{n-s}} \}, \\ y_{n} = \max \{B ,\frac{x_{n-t}}{y_{n-s}} \},\end{cases}\displaystyle \quad n\in \{0,1,2,\ldots \}, $${xn=max{A,yn−txn−s},yn=max{B,xn−tyn−s},n∈{0,1,2,…}, where $A,B\in (0, +\infty )$A,B∈(0,+∞), $t,s\in \{1,2,\ldots \}$t,s∈{1,2,…} with $\gcd (s,t)=1$gcd(s,t)=1, the initial values $x_{-d},y_{-d},x_{-d+1},y_{-d+1}, \ldots , x_{-1}, y_{-1}\in (0,+ \infty )$x−d,y−d,x−d+1,y−d+1,…,x−1,y−1∈(0,+∞) and $d=\max \{t,s\}$d=max{t,s}.

Difference equations are often used to create discrete mathematical models. In this paper, we mainly study the dynamical behaviors of positive solutions of a nonlinear fuzzy difference equation: x n + 1 = x n A + B x n − k ( n = 0 , 1 , 2 , … ) , {x}_{n+1}=\frac{{x}_{n}}{A+B{x}_{n-k}}\hspace{0.33em}\left(n=0,1,2,\ldots ), where parameters A , B A,B and initial value x − k , x − k + 1 , … , x − 1 , x 0 {x}_{-k},{x}_{-k+1},\ldots ,{x}_{-1},{x}_{0} , k ∈ { 0 , 1 , … } k\in \{0,1,\ldots \} are positive fuzzy numbers. We investigate the existence, boundedness, convergence, and asymptotic stability of the positive solutions of the fuzzy difference equation. At last, we give numerical examples to intuitively reflect the global behavior. The conclusion of the global stability of this paper can be applied directly to production practice.

In this paper, we present a predator-prey system with mutual interference and distributed time delay and study its dynamical behavior. Based on the existence and universality of mutual interference among species, it is necessary to further study an impulsive food web system. By using stability theory, slight perturbation technique, and comparison theorem, we obtain some theoretical results of the system, such as boundedness and permanence. Moreover, numerical experiments are used to verify the theoretical results and to explore the dynamical behavior of the system, which exhibits rich dynamical behavior such as chaotic oscillation, periodic oscillation, symmetry-breaking bifurcations, chaotic crises, and period bifurcation. Finally, we give some practical guidelines for biological systems based on the theoretical results and numerical experiments of the system.