Predator Prey Scavenger Model using Holling’s Functional Response of Type III and Physics-Informed Deep Neural Networks

Core subsystem results in the paper (predator–prey and scavenger–prey stability tests; predator–scavenger saddle at the positive point) agree with standard linearization/trace–determinant criteria and are correctly stated (e.g., 2a0 e (k−x0)/(d k) < 1 for predator–prey, and 2b0 j (k−x0)/(g k) < 1 for scavenger–prey) , and the yz-subsystem equilibrium product of eigenvalues is negative (saddle) . For the full system, the paper also correctly invokes the cubic Routh–Hurwitz test and provides a degree‑12 polynomial for x* (equation (4.5) with coefficients in the Appendix) . However, there are important issues: (i) the stated formula for y* omits a factor h in the denominator (paper: h + i0 z*2 − i z*; correct: h(1 + i0 z*2) − i z*) ; (ii) the bound extracted from dx/dt = 0 for z* misses a factor 1/k (paper has r(1 + b0 x*2)(k − x*)/(b x*), but logistic growth contributes (1 − x*/k), hence division by k is required) ; and (iii) the paper states z* < i e/(h f) as an “existence” constraint, but positivity of h − i z*/(1 + i0 z*2) actually yields a lower bound z* > i P/(h f) with P = e − d x*2/(1 + a0 x*2), so z* > i e/(h f) is a uniform sufficient condition, not an upper bound . There is also an internal inconsistency in 4.4 where, despite computing 2a0 e (k − x0)/(d k) < 1, the text concludes “stability conditions are not met” for E[4] (predator–prey with z≡0) . The model solution resolves these slips, reproduces the planar criteria cleanly, derives the correct elimination formulas for z*2 and y*, and justifies the degree‑12 equation and the Routh–Hurwitz cubic test.