An Effective Sign Switching Dark Energy: Lotka-Volterra Model of Two Interacting Fluids

The paper formulates three two-fluid Lotka–Volterra-type ODE models (unfair competition, fair competition, conversion) for sign-switching dark energy and supports its claims primarily by continuity arguments, nullclines, and numerical phase portraits. Key claims—existence of an AdS-to-dS transition in the unfair model with a logistic cap; smooth phantom crossings in the fair model when x+y=0 and ρ≠0; the special r1=r2 case with constant w; and a stable spiral attractor enabling phantom crossings in the conversion model—are all explicitly stated or illustrated in the text and figures. These appear correct within the stated approximation of slowly varying H (treated as constant) and with the paper’s variable scalings and sign conventions (e.g., x=γρΛ1/r2, y=γρΛ2/r1) . The candidate solution reproduces the models and provides a more formal dynamical-systems treatment (forward-invariant regions, Jacobians, trace/determinant, discriminant for node/spiral), derives the same fixed points (including the conversion model’s spiral center), and gives clean criteria for smooth phantom crossing and for positivity of the late-time density. These strengthen the paper’s heuristic arguments and are methodologically consistent. One minor error in the candidate solution is the blanket claim that the line x=−K1 is inward-pointing for both the fair and conversion models: it holds for the fair model but not in general for the conversion model (where dx/dt|_{x=−K1}=r1K1(2−y), which need not be ≥0). Aside from this fixable point, the candidate’s analysis aligns with the paper and adds rigor (e.g., explicit node/spiral condition and ρ(∞)>0 inequality). The paper’s identification of the logistic “resource” term with a quadratic self-interaction in the fluid equation is also correctly echoed by the candidate .