In this paper, our aim is to study the following fuzzy system: $x_{n+1}=\frac{A{x}_{n-1}{x}_{n-2}+B{x}_{n-3}}{D+C{x}_{n-4}},n=0,1,2,\cdots ,$ where $\{x_n\}$ is a sequence of positive fuzzy numbers and the initial conditions $x_{-4},x_{-3},x_{-2},x_{-1},x_0$ and the parameters $A,B,C,D$ are positive fuzzy numbers. Firstly, the existence and uniqueness of positive fuzzy solutions of the fuzzy system are proved. Secondly, the dynamic behavior of the equilibrium points for the fuzzy system are studied by means of the fuzzy sets theory, linearization method and mathematical induction. Finally, the results obtained in this paper are simulated by using the software package MATLAB 2016, and the numerical results not only show the dynamic behavior of the solutions for the fuzzy system, but also verify the effectiveness of the proposed results.

The purpose of this paper is to give the conditions for the existence and uniqueness of positive solutions and the asymptotic stability of equilibrium points for the following high-order fuzzy difference equation:xn+1=Axn−1xn−2/B+∑i=3kCixn−i n=0,1,2,…,wherexnis the sequence of positive fuzzy numbers and the parametersA,B,C3,C4,…,Ckand initial conditionsx0,x−1,x−2,x−ii=3,4,…,kare positive fuzzy numbers. Besides, some numerical examples describing the fuzzy difference equation are given to illustrate the theoretical results.

In this work, we present a novel four-species periodic diffusive predator–prey model, which incorporates delay and feedback control mechanisms, marking substantial progress in ecological modeling. This model offers a more realistic and detailed portrayal of the intricate dynamics of predator–prey interactions. Our primary objective is to establish the existence of a periodic solution for this new model, which depends only on time variables and is independent of spatial variables (we refer to it as a spatially homogeneous periodic solution). By employing the comparison theorem and the fixed point theorem tailored for delay differential equations, we derive a set of sufficient conditions that guarantee the emergence of such a solution. This analytical framework lays a solid mathematical foundation for understanding the periodic behaviors exhibited by predator–prey systems with delayed and feedback-regulated interactions. Moreover, we explore the global asymptotic stability of the aforementioned periodic solution. We organically combine Lyapunov stability theory, upper and lower solution techniques for partial differential equations with delay, and the squeezing theorem for limits to formulate additional sufficient conditions that ensure the stability of the periodic solution. This stability analysis is vital for forecasting the long-term outcomes of predator–prey interactions and evaluating the model’s resilience against disturbances. To validate our theoretical findings, we undertake a series of numerical simulations. These simulations not only corroborate our analytical results but also further elucidate the dynamic behaviors of the four-species predator–prey model. Our research enhances our understanding of the complex interactions within ecological systems and carries significant implications for the conservation and management of biological populations.

This article focuses on the stability of periodic solution of a delayed nonautonomous reaction-diffusion predator-prey model. The fine combination of upper and lower solution methods and Lyapunov stability theory is used to transform the study of the stability problem of delayed reaction-diffusion equations into the stability problem of their corresponding delayed ordinary differential equations. Some sufficient conditions are given to ensure the globally asymptotically stability of the periodic solution for this model. Unlike existing results, the stable solution which are obtained in this article is a time-periodic solution rather than a constant periodic solution or a solution for a steady-state system. We extend a stability theorem on predator-prey model introduced by V. Ortega and C. Rebelo in 2023 to nonautonomous delayed reaction-diffusion model. Finally, in order to show the application of the theoretical results, the proposed conditions are numerically validated over a 2-periodic delayed nonautonomous reaction-diffusion predator-prey model.

A 3-species nonautonomous ratio-dependent diffusive predator-prey system is considered in this article. Firstly, by utilizing a comparison principle and fixed point theorem, the existence of solution which is space homogenous strictly positive and periodic for the above system is obtained. And the obtained conditions ensuring the existence of solution can be very easily verified. At the same time, we develop some new analysis techniques as a byproduct. Furthermore, with the help of the upper and lower solutions (UALS) approach for the parabolic partial differential equations and Lyapunov theory, we aim at the globally asymptotically stability problems of the solutions, and some judgment criteria are achieved. Finally, we give some numerical simulations results which validate the theoretical findings of this article.