Bistability in a one-dimensional model of a two-predators-one-prey population dynamics system

The paper’s core claims (map form, fixed-point/flip curve k=b^2/(b+2), negativity of the Schwarzian for k<-4, wedge delineated by γ1 and γ2, and uniqueness of the 2-cycle) are consistent with its formulas and standard 1D dynamics results. However, it repeatedly asserts that the specific absorbing intervals J−=[f2(xmax),f(xmax)] and J+=[f(xmin),f2(xmin)] are positively invariant and globally attracting in the outer regions (b<min{b1,b2} and b>max{b1,b2}) and even claims that J− contains xmax; these statements are false as written. For example, at (k,b)=(-5,-4) (which satisfies b<min{b1(k),b2(k)}), one has J−≈[-1.3783,-1.3444] while f(J−)≈[-1.3847,-1.3783]⊄J−, and xmax≈-0.9624 is not in J−. The paper explicitly states positive invariance and this containment (see the text around Lemma 4 and its corollaries), so the proof as written is incomplete/incorrect on this point . The candidate solution identifies this gap and supplies corrected positively invariant traps (e.g., J_left=[min{x*,f2(xmax)}, f(xmax)]), which restore the outer-region arguments; the other parts of the model’s reasoning (negative Schwarzian, Singer/de Melo–van Strien consequences, and period-doubling analysis) align with the paper’s formulas and standard references .