Dynamical behaviour of discrete logistic equation with Allee effect in an uncertain environment

Abstract

The Allee effect refers to a natural occurrence where the average individual fitness of a population or species is correlated with population size or density. A demographic Allee effect having a threshold population size constitutes a strong Allee effect. On the contrary, it is called the weak Allee effect when it lacks an adequate population size. This paper revisits a single-species logistic growth model under the combined impacts of the strong Allee effect, discrete population dynamics, and an impreciseness phenomenon. Several existing literatures on models with deterministic and continuous time frames address the Allee effect’s significance. The population growth model may disobey the continuous time frame of dynamics. The difference equation may be a fruitful substitute for the traditional differential equation to trace a discrete logistic population growth model phenomenon. Also, initial information and the population growth model’s associated coefficient may sometimes be vague. Fuzzy logic is a suitable tool to describe such vagueness in mathematical terms. We have discussed the proposed model through the different fuzzy scenarios to realize the relative impact of different fuzzy parameters. This paper also includes the studies of equilibrium points and their corresponding stability analysis for each case. This paper differs from the existing literature on Allee effect in that it contributes an alternative approach to manifest the logistic growth model in a discrete time frame under uncertainty. Numerical results show the fuzzy impact of different parameters of the model. One of the significant insights of the outcome is that the extinction or highly growing population depends on the parameter’s values rather than the population size only.

Keywords

• Fuzzy set theory
• Difference equation
• Stability criteria for difference equation
• Allee effect