Incommensurate conformable-type three-dimensional Lotka–Volterra model: discretization, stability, and bifurcation

Abstract

The classic Lotka–Volterra model is a two-dimensional system of differential equations used to model population dynamics among two-species: a predator and its prey. In this article, we consider a modified three-dimensional fractional-order Lotka–Volterra system that models population dynamics among three-species: a predator, an omnivore and their mutual prey. Biologically speaking, population models with a discrete and continuous structure often provide richer dynamics than either discrete or continuous models, so we first discretize the model while keeping one time-continuous dependent variable in each equation. Then, we analyze the stability and bifurcation near the equilibria. The results demonstrated that the dynamic behaviors of the discretized model are sensitive to the fractional-order parameter and discretization parameter. Finally, numerical simulations are performed to explain and validate the findings, and the maximum Lyapunov exponents is computed to confirm the presence of chaotic behavior in the studied model.

Keywords

• Conformable fractional derivative
• prey–predator– omnivore system
• discretization
• stability
• bifurcations
• chaos