Global stability and period-doubling bifurcations of a discrete Kolmogorov predator-prey model with Ricker-type prey growth

The uploaded paper proves global asymptotic stability of the unique positive fixed point for the discrete Kolmogorov predator–prey map with Ricker prey via a nullcline-based geometric construction. Its main global-stability theorem (Theorem 3.9) assumes (i) existence of the interior fixed point, (ii) r ≤ 1 (to secure monotonicity of the prey update in the absorbing region), and (iii) the inequality γζ(r)^2 + 2ζ(r) > (1−s + c b0 r)/(c b0 − (1−s)γ), where ζ(r) = x* is the prey-coordinate of the fixed point; under these, p* is globally asymptotically stable in the positive quadrant. This matches the candidate’s conclusion and method: they use the same nullclines S(x) = (e^{r−x}−1)/c and V(x) = (1/c)(cb0/(1−s)·x/(1+γx) − 1), their inverses, and the associated order-preserving return maps to build nested invariant rectangles and prove global convergence to p*. The paper establishes dissipativity via a compact absorbing set (Lemma 2.1), uniqueness/existence of p* (Lemma 3.4), the monotonicity needed when r ≤ 1, and then the nullcline ordering that is encoded in the inequality (3.22) in Theorem 3.9; see these elements in the PDF. Minor discrepancies are that the candidate mis-cites the theorem and inequality numbers (calling them Theorem 3.16 and “(33)”) and overstates necessity; the paper presents the condition as a sufficient “criterion,” not as an iff characterization. Substantively, however, the proofs align and rely on the same geometric mechanism. Citations: existence/uniqueness of p* (Lemma 3.4 and (3.6)–(3.10) in the paper ), absorbing set (Lemma 2.1 ), r ≤ 1 ensures G1 is increasing in x on [0,K1] (used in the global proof ), and the global-stability criterion (Theorem 3.9 with (3.22) ). The paper itself frames Theorem 3.9 as a “criterion” (discussion section) rather than a necessary-and-sufficient condition .