From Cain to Matignon: Exact Positive-Diagonal Stability in Dimension Three and a Nonlinear Caputo Ecological Realization
- 1Department of Mathematics, Faculty of Science, Islamic University of Madinah, Madinah 42351, Saudi Arabia
Abstract
Positive-diagonal D-stability asks whether stability persists under every positive diagonal left scaling. We replace the classical Hurwitz half-plane by the Matignon sector for commensurate Caputo systems and solve the resulting real three-dimensional problem on the full-dimensional strict-P(−A) stratum. Four scale-invariant quantities (β12,β13,β23,κ) reduce the complete positive-diagonal orbit to an open simplex, and for 2/3<α<1 we prove the exact criterion κ<Tα(β). At α=1, Tα collapses to Cain’s classical threshold, while Tα>T1 for every 2/3<α<1, producing an explicit genuinely fractional band. The threshold objective is strictly convex in logit coordinates, has a unique nondegenerate optimizer, and yields the minimum-dimension result: three is the minimum dimension in which the genuinely fractional multiplicative-D-stable class has nonempty full-dimensional interior. We then construct a nonlinear three-species Caputo Kolmogorov realization with double-Allee basal growth and intraguild predation. A natural competing-consumer architecture is proved unable to enter the fractional-only band, whereas the intraguild-predation architecture realizes an open biological subset of it. Varying only the Allee threshold moves the fixed biological system transversally across Cain’s classical D-stability boundary into the fractional-only Matignon region. A reproducible integration of the full nonlinear model at a certified orbit point displays decaying Caputo transients at α=0.9, while the same parameters, rate scaling and initial perturbation leave a fixed local neighborhood at α=1.
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BibTeX
@article{alraddadi2026from,
title = {{From Cain to Matignon: Exact Positive-Diagonal Stability in Dimension Three and a Nonlinear Caputo Ecological Realization}},
author = {Ibrahim Alraddadi},
journal = {Mathematics},
year = {2026},
month = nov,
doi = {10.3390/math14214087},
url = {https://www.mdpi.com/2227-7390/14/21/4087},
note = {CC BY 4.0},
}
RIS
TY - JOUR
TI - From Cain to Matignon: Exact Positive-Diagonal Stability in Dimension Three and a Nonlinear Caputo Ecological Realization
AU - Ibrahim Alraddadi
PY - 2026
DA - 2026/11/07
JO - Mathematics
DO - 10.3390/math14214087
UR - https://www.mdpi.com/2227-7390/14/21/4087
AB - Positive-diagonal D-stability asks whether stability persists under every positive diagonal left scaling. We replace the classical Hurwitz half-plane by the Matignon sector for commensurate Caputo systems and solve the resulting real three-dimensional problem on the full-dimensional strict-P(−A) stratum. Four scale-invariant quantities (β12,β13,β23,κ) reduce the complete positive-diagonal orbit to an open simplex, and for 2/3<α<1 we prove the exact criterion κ<Tα(β). At α=1, Tα collapses to Cain’s classical threshold, while Tα>T1 for every 2/3<α<1, producing an explicit genuinely fractional band. The threshold objective is strictly convex in logit coordinates, has a unique nondegenerate optimizer, and yields the minimum-dimension result: three is the minimum dimension in which the genuinely fractional multiplicative-D-stable class has nonempty full-dimensional interior. We then construct a nonlinear three-species Caputo Kolmogorov realization with double-Allee basal growth and intraguild predation. A natural competing-consumer architecture is proved unable to enter the fractional-only band, whereas the intraguild-predation architecture realizes an open biological subset of it. Varying only the Allee threshold moves the fixed biological system transversally across Cain’s classical D-stability boundary into the fractional-only Matignon region. A reproducible integration of the full nonlinear model at a certified orbit point displays decaying Caputo transients at α=0.9, while the same parameters, rate scaling and initial perturbation leave a fixed local neighborhood at α=1.
SN - 2227-7390
ER -