Asymptotic behavior of a difference equation model in exponential form

Abstract

We investigate the global stability and the convergence rate of the exponential model:and u −2 and the parameters 𝜇 1 , 𝜆 1 , 𝜆 2 , 𝜇 3 , 𝜆 3 , 𝜇 2 , 𝜆 4 , and 𝜇 4 are non-negative real numbers. We also discuss the unboundedness, persistence, and boundedness of this system. Moreover, we introduce conditions for uniqueness and existence of the equilibrium. Finally, we give numerical explanations to verify our results. We can use the above system as a model for the growth of some perennial plants and their relationships with each other. KEYWORDS asymptotic global and local stability, boundedness, persistence, the rate of convergence, unboundedness MSC CLASSIFICATION 40A05, 39A10 INTRODUCTIONDifference equations (systems) play a crucial role in the development of a variety of disciplines (see, e.g., previous studies and references cited therein). The exponential form is one of the most diverse types of difference equations. Such structures have many topics in our life. For example, El-Metwally et al 8 worked out the difference equation for a population model: Ma et al. 33 examined the equation:where b, a ≥ 0 and the initials are positive numbers. Feng et al. 12 studied a model for the production of a biennial plant in the following form:where c 1 , c 3 ∈ (0, ∞) and c 2 ∈ (0, 1).