A CAPUTO FRACTIONAL LOTKA-VOLTERRA MODEL FOR COMPETITIVE OPINION DYNAMICS: STABILITY ANALYSIS AND DATA-DRIVEN VALIDATION

Phase portraits of the fractional-order Lotka–Volterra opinion dynamics model for different values of $q$. The coexistence equilibrium $E^{*}$ is marked by a black dot. (a) For $q=1$, trajectories form closed orbits around $E^{*}$, corresponding to the classical neutrally stable behavior. (b) For $q=0.9$, (c) for $q=0.8$, and (d) for $q=0.7$, trajectories spiral inward toward $E^{*}$, consistent with the local asymptotic stability result for $0 < q < 1$. The parameter values are $a=0.7$, $b=0.5$, $c=0.6$, and $d=0.8$, so that $E^{*}\approx(1.333, 1.4)$. The initial conditions are $(0.5, 2.0)$, $(2.0, 0.5)$, $(1.8, 1.8)$, and $(0.8, 0.8)$. The numerical scheme is the predictor–corrector Adams–Bashforth–Moulton method described in Section 4.1.

Time-series plots of the opinion prevalences $x(t)$ and $y(t)$ for the initial condition $(x_0, y_0)=(0.5, 2.0)$. (a) For $q=1$, the solutions exhibit sustained oscillations around the equilibrium. (b) For $q=0.9$, (c) for $q=0.8$, and (d) for $q=0.7$, the oscillations become damped and the solutions approach the equilibrium values $x^{*}\approx1.333$ and $y^{*}\approx1.4$. The parameter values are $a=0.7$, $b=0.5$, $c=0.6$, and $d=0.8$. The numerical scheme is the predictor–corrector Adams–Bashforth–Moulton method described in Section 4.1.

Normalized Google prevalence: Comparison between the pseudo-temporal sequence and model trajectories. Under the adopted aggregation procedure, the fractional model $(q\approx0.787)$ reproduces the constructed sequence somewhat more closely than the classical model $(q=1)$.

Normalized Microsoft prevalence: Comparison between the pseudo-temporal sequence and model trajectories. The fractional model $(q\approx0.787)$ provides a slightly closer descriptive fit than the classical model $(q=1)$ for the constructed pseudo-temporal sequence.

Pointwise error curves $e(k)$ for the classical model $(q=1)$ and the fractional model $(q\approx0.787)$. The fractional model gives smaller errors over much of the pseudo-temporal index under the adopted aggregation procedure.