Eco-Evolutionary Dynamics of Bimatrix Games

The paper states slope criteria for when two interior saddles of the (constant-coefficient) 2×2 bimatrix replicator systems admit trapping regions under switching: a left–right configuration if |(a*−c*)/(b*−d*)| < min{√(β/α), √(β′/α′)} and an up–down configuration if it is > max{…}, with boundary-equality and mixed cases treated as well (their Eqs. (19)–(24)) . These conditions come from the linearization at each saddle, where the Jacobian is [0 α; β 0] and the eigendirections have slopes ±√(β/α) (Sec. 6) . The paper’s trapping-region logic is presented qualitatively via the “closed circular path” criterion along alternating stable/unstable boundary segments (Fig. 5 and discussion) . It also derives an autonomous first integral V(x,y)=x^v(1−x)^{u−v} y^{−q}(1−y)^{q−p} for each fixed environment (Sec. 7) . The candidate solution reaches the same slope conditions and phase-plane configurations but via a different, more constructive argument: it uses the first integral to identify the separatrix cross at each saddle and then shows (by monotone quadrant structure and endpoint side-switching) how the separatrix branches from the two environments intersect to form a four-sided trapping region; it then gives a forward-invariance argument for admissible switching and notes periodic switching yields a boundary cycle. This aligns with the paper’s pictures and claims, including equality degeneracies and mixed (non-convex) cases , and matches the nonlinear example’s behavior (Fig. 7) . Minor issues: the paper’s arguments are largely schematic (no formal proof of the “iff” trapping criterion), while the model slightly overstates uniqueness of certain intersections and briefly assumes signs (p,u>0) when discussing local Hessian signs before noting time-reversal symmetry. Net: both are substantively correct; the model gives a more explicit construction; the proofs differ (linearization-with-figures vs. first-integral/separatrix geometry).