Paper III¶

The model¶

On a one-dimensional interval with no-flux boundary conditions, Paper III studies

\[ \begin{aligned} u_t &= u_{xx} - \chi_0\,\partial_x\!\left(\frac{u^m}{(1+v)^\beta}v_x\right) + au-bu^{1+\alpha}, \\ 0 &= v_{xx}-\mu v+\nu u^\gamma. \end{aligned} \]

The bifurcation parameter is the sensitivity strength \(\chi_0\). The paper locates the first loss of stability of a positive constant equilibrium, computes the coefficient selecting the local branch direction, treats the fixed-mass minimal case \(a=b=0\), and develops global continuation for the non-minimal model.

Two complementary numerical questions¶

Stationary continuation¶

Does the discrete branch leave the threshold with the sign and quadratic coefficient predicted by the local bifurcation calculation?

The solver prescribes a signed first-cosine amplitude \(A\), solves the stationary equations, and fits

\[ \chi_h(A)-\chi_h^*=c_{2,h}A^2+c_{4,h}A^4. \]

Time integration¶

Do selected semidiscrete trajectories decay below threshold and move toward opposite patterned states above threshold?

Two non-minimal manuscript figures answer this diagnostic question for fixed positive and negative seeds. They are supportive numerical observations, not proofs of PDE stability.

Stationary branch contract¶

For the four cases below, theory predicts \(c_2=\beta_{n_0}/\alpha_{n_0}\). The numerical value is the constrained fit on the finest mesh, N=160.

Parameter regime	Direction	Theory \(c_2\)	Numerical \(c_{2,160}\)	Observed order
Non-minimal `a=10, beta=0`	Supercritical	`0.0084254463`	`0.0084220572`	`2.009`
Non-minimal `beta=3`	Subcritical	`-19.66671032`	`-19.64492609`	`2.000`
Minimal `m=gamma=1`	Supercritical	`1.922989917`	`1.922366447`	`2.000`
Minimal `m=gamma=2`	Subcritical	`-1.148587331`	`-1.150522206`	`2.000`

The full design uses three meshes and eight symmetric nonzero amplitudes per case: 96 stationary states. All 780 acceptance gates pass, including solver, positivity, residual, amplitude, reflection, mass, sign, intercept, and mesh refinement checks.

Paper-to-data map¶

Paper-side role	Public object	Scope
Four local branch-direction checks	Immutable stationary continuation v1	Coefficient sign, value, symmetry, and mesh convergence
Quasi-linear supercritical figure	Non-minimal `a=10, beta=0` time archive	Below-threshold decay and two above-threshold trajectories
Nonlinear-mobility supercritical figure	`m=2, beta=1, gamma=2` time archive	Nonlinear mobility and signal-production diagnostic
Earlier minimal or branch-seeded runs	Historical Git objects	Provenance only; excluded from numerical claims

The manifest records the exact boundary and is checked on every push.

Revisions frozen by release 1.0.0¶

Paper III numerical science: fde25e17187bc3f247b36ce411f6f14eb93d52cf
Simulator and stationary generator: 7c2a09b24fdebb9000b9b996eb34150d6de5ed17
Immutable public data: e62ffa1e99122f8fbbeb3df7586f4050c4ff5c58
Font-embedded time figures: c0bfc431a19b81b1c45363dea472c29a745ad055

Interpretation boundary

The public archive supports comparisons with the spatially semidiscrete model. It is not, by itself, a proof of the continuous PDE statements.

Authors¶

Le Chen, Department of Mathematics and Statistics, Auburn University
Ian Ruau, Department of Mathematics and Statistics, Auburn University
Wenxian Shen, Department of Mathematics and Statistics, Auburn University