Dynamical analysis of fractional-order modified logistic model

Abstract

a b s t r a c tIn this paper, we study a fractional differential equation model of the single species multiplicative Allee effect. First we study the stability of equilibrium points. Further we give some sufficient conditions ensuring the existence and uniqueness of integral solution.In the last section we perform several numerical simulations to validate our analytical findings.

Global dynamics of fractional-order systems with an extended generalized cell mapping method

“…The dynamics of these three systems was studied previously using only one or several initial points by the improved predictor-corrector algorithm [40][41][42]. Our results by using the EGCM method confirm their previous results on the one hand, and obtain the global properties in the interesting region including attractors, unstable solutions, basins of attraction and basin boundaries on the other hand.…”

“…5 that the yellow boundary is exactly covered with black symbol ×, which occurs only in the simple case of one fractional differential equation. These results by the EGCM method completely confirm that of [40] and furthermore demonstrates the ability to determine the domains and boundaries of attraction for fractionalorder systems.…”

“…The equilibrium points E 1 and E 3 are stable, and E 2 is unstable. When the order q = 0.5 and parameters r = 0.5, m = 1, k = 10, the authors of [40] found that the solution trajectories converge to the fixed point E 1 when initial points x 0 < 1, and to the E 3 for x 0 > 1. The simulation results are presented in Fig.…”

“…The system has three equilibrium points, namely E 1 = 0, E 2 = m, E 3 = k. In [40], the stability of the three fixed points has been analyzed theoretically and numerically. The equilibrium points E 1 and E 3 are stable, and E 2 is unstable.…”

“…And the model with fractional order has also been studied in detail [40]. The fractional-order logistic model can be described as…”