Abstract
The key objective of this paper is to study the imprecise biological complexities in the interaction of two species pertaining to harvesting threshold. It is explained by taking the prey–predator model with imprecise biological parameters and fractional order generalized Hukuhara (fgH) differentiability. In this vain, different possible systems of the model are constructed, according to the increasing and decreasing behavior of population growth. Feasibility and stability analyses of equilibrium points of the stated models are also discussed by means of variational matrix with Routh–Hurwitz conditions. In addition, the numerical elaborations are carried out by taking parametric expansion of fuzzy fractional Laplace transform (FFLT). This significantly helps the researchers in using a novel approach to analyze the constant solutions of the dynamical systems in the presence of fractional index. This would allow the avoidance of any intricacy that occurs while solving fractional order derivatives. Furthermore, this attempt also provides numerical and pictorial results, obtained through some well-known methods, namely fifth-forth Runge–Kutta method (FFRK), Grunwald–Letnikov’s definition (GL) and Adams–Bashforth method (ABM) that are deemed appropriate to scrutinize the dynamics of the system of equations.
Highlights
There has been considerable work done by researchers in associating real ecological situations with mathematical models to make it palpable
The model for population growth of biological species, named the Lotka–Volterra equation, played an important role in mathematical biology [1, 2]. These equations, with interspecific competition, have been considered to be a prerequisite for those which are associated with biology
These equations aid the perceptions about the outcomes of competitive interactions between different species [3,4,5,6]. Diversity of factors, such as environmental change, consumer-resource interactions, and disease, not covered in the model can affect the upshot of competitive interactions by affecting the dynamics of the respective populations
Summary
There has been considerable work done by researchers in associating real ecological situations with mathematical models to make it palpable. The model is classified with increasing and decreasing behavior of population growth of the species, which may transpire due to the climate change, with the help of fgHdifferentiability [31,32,33] This notion clearly describes all the possible cases of differentiability of the fuzzy functions by breaking it into multiple systems of equations, where each system elaborates a different scenario. We explore a comparative analysis between FFRK, Grunwald–Letnikov’s definition (GL) [15], and Adams–Bashforth method (ABM) [38] These techniques are widely known approximators to fractional and integer order differential equations. These assessments are examined on some illustrative examples and lucratively depicted increase or decrease in the populations, phase trajectories, and limit cycles etc., of the systems.
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