Reaction-diffusion · minimax-m3.1-flash

Let me build it. Grid 256x256, Canvas 2D with ImageData, run several steps per frame.

Let me write index.html.

Parameters: Du=0.16, Dv=0.08, F=0.035, k=0.062, dt=1.0 (simulation units). Laplacian with 5-point stencil weights 0.2 each neighbor, -1 center (or -4 with 1.0 weights). Common: lap = (neighbors*0.2 - center). With dt=1.0.

Let's use grid 256, dt = 1.0, steps per frame = 8 maybe. 256*256*8 = 524k updates per frame — fine in JS.

Color mapping: V concentration -> color. Use classic: v = V clamped; map via some palette. Simple: r = 255*v, g = 255*v*? Let's do a nice mapping: color = mix of dark blue background to orange/yellow.

Let's write it.

Note: idx uses `(y<<8)|x` assuming N=256 — fine.

Now testin browser.

Frame 104 already (fast). But max V = 0.000 — the pattern died? The center seed vanished. mean U 0.523, which is odd — with V=0 everywhere, U should approach 1 (feed pushes U to 1). mean U 0.523 suggests U is not converging to 1. Hmm, but max V 0 means V died.

Issue: kill=0.062 with F=0.035 — the seed may be unstable and V decays. Classic Gray-Scott uses F=0.035, k=0.062 which is a known working combination (coral growth). But the decay: with dt=1 and Laplacian weight 0.2, Dv*lap max 0.08*0.8=0.064... The issue is stability of reaction: du = -uvv + F(1-u) with u=0.25,v=0.9: uvv = 0.2, dv = 0.2 - (0.097)(0.9)= 0.2-0.087=+0.113 positive. Should grow. Hmm.

Wait — maybe the bug: idx function `(y<<8)|x` fine. But maybe the issue is the canvas rendering doesn't show because V is small. max V = 0.000 rounded to 3 decimals, meaning max V < 0.0005. So V died.

Why? Perhaps because seed is only radius 8 and grid is 256 with dt=1: reaction terms. Let's check: at v=0.9, u=0.25: dv/dt = Dv*lap + uvv - (F+K)v = 0 + 0.2025 - 0.097*0.9=0.2025-0.0873=0.115 → grows. So it shouldn't die immediately... unless u got reduced fast.

Hmm, wait: du/dt = Du*lap - uvv + F(1-u) = 0 - 0.2025 + 0.035*0.75 = -0.176. So u drops to 0.074 quickly, then uvv small, v decays. That's normal — a small seed with low U decays unless U replenishes. But diffusion brings U in from surroundings. Diffusion rate: Du=0.16 with lap weight 0.2, over dt=1, that's a slow spread: diffusion length sqrt(Du*t) = sqrt(0.16*100)=4 cells. Hmm.

Actually the well-known behavior: standard Gray-Scott with dt=1, Du=0.16, Dv=0.08, F=0.035, k=0.062 works and produces "coral". But typically U starts at 1 everywhere and seed is V=1, U=0.5 in a small region and it grows. The key is that a small seed can die.

Actually with F=0.035, k=0.062: for the "mitosis/coral" regime. F/k ratio matters. Let's just test: seed radius larger, and check evolution. Also the initial seed decays because V=0.9 and U=0.25 gives u dropping fast.

Standard recipes: U=1, V=0, then square of V=0.5, U=0.25 (typically 10x10 or 20x20). And it grows if the region is big enough.

Let me instead compute in the browser: run step and print max V over frames to debug. Better: use console via browser inspect? The tool doesn't have eval. Hmm — only inspect/click etc. I can add a debug hook and read via inspect? inspect gives DOM text.

Alternative: make the app itself display max V over time. But I can reason analytically.

Let me analyze the dynamics of a seeded spot. Consider interface: inside u low, v high. Reaction uvv consumes u. u is replenished by F(1-u) which at u=0 is 0.035 per time unit — very slow. Diffusion of u: from outside u=1, flux = Du * grad. With weight 0.2 laplacian, effective diffusivity coefficient D_eff = 0.16*0.2 = 0.032 per step in units of cell². Over 100 steps, diffusion length sqrt(2*0.032*100)=2.5 cells. So u won't reach the center of a radius-8 seed quickly. Meanwhile v diffuses away: D_eff_v = 0.08*0.2=0.016, over 100 steps length sqrt(3.2)=1.8.

But the pattern should still be stable because known Gray-Scott uses exactly these with dt=1 and Laplacian with weights 1/4? Standard 9-point laplacian uses 0.05 corners, 0.2 edges, -1 center. That's the same edge weight 0.2 as mine. So diffusion coefficient effectively... in the standard formulation the laplacian is scaled so Du*lap with dt=1 gives pattern. So it should be the same.

So why did it die? Let me reconsider: maybe the issue is that with F=0.035,k=0.062 the seed decays unless seed is larger, and typical demos use seed of ~20x20 blocks (often multiple). Let me check equilibrium: nontrivial steady states exist for F/(F+k) ... For Gray-Scott, nontrivial states exist when sqrt(F/k) < ... The condition for existence: the system's non-trivial fixed points exist if F < ... Actually for gray-scott, V=0, U=1 always stable when? The "U=1,V=0" state is always a fixed point. Its stability: dV/dt near: v' = Dv lap v + F(1-u) approx at u=1: -uv² negligible, dv = -（F+K)v + F*(1-u) where 1-u perturbation. Linearization of u around 1: du = -Du lap u - v² du? Let u=1+a: du = -Da? Let's do: du/dt = -uv² + F(1-u) = -(1+a)v² - F a. dv/dt = Dv lap v + uv² - (F+K)v.

At a=0,v=0: du/dt = -F a → a decays with rate F; dv/dt = -（F+K)v → decays at F+K=0.097. So the trivial state is stable, and a small perturbation decays — meaning the seed dies unless nonlinear terms matter. Indeed for F=0.035,k=0.062, the ratio matters: known "coral" parameters F=0.0545,k=0.062; "mitosis" F=0.0367,k=0.0649; F=0.035,k=0.062 is in the "worms/chaos" region maybe. But the standard Pearson demo uses F=0.035, k=0.062 and it works when the seed is large enough (they use a 20x20 block and sometimes repeated).

Also crucially, u depletion: the seed u drops to near 0, so v has nothing to react with and decays at rate F+K=0.097 → v=0.9*exp(-0.097*t). Meanwhile u recovers via F(1-u): at u=0, du = F = 0.035 → u grows linearly 0.035t, reaching 0.35 at t=10, 0.7 at t=20. But v decays to 0.9*exp(-0.97)=0.34 by t=10. Then reaction: u*v² = 0.35*0.116=0.04 vs (F+K)v=0.033 → v grows! So there's a race; if the seed is big enough, the u recovery and v decay interplay works out.

For the seed to survive, the initial V region should be bigger so that diffusion of v doesn't deplete it, and the u recovery is happening in the region. Actually the v diffusion spreads out, reducing local v further. Hmm.

Common practice: after seeding V, the demo works. Let me just test empirically — I need a way to observe. Let me add a small debug panel element in the DOM showing max V and a history sparkline? Simpler: I can use the browser tool's inspect to read DOM, and add a hidden debug div with text like "maxV history: ...". Then inspect to read it.

Even better: write a headless Node test replicating the same algorithm to check parameter behavior quickly. Node may be available.

Let me do that: node script with same math, print maxV every 100 steps for seed radius 8, and for other seed sizes.

Write a node test script (temporary, outside index.html? I can put it in /tmp to avoid clutter — but cwd files fine, I'll delete after).

meanU stuck at 0.5224 — that's a stable nontrivial steady state! (u=0.5224? hmm mean U across grid is 0.52 while maxV=0?) Weird: mean U 0.52 with max V 0.000. If V=0 everywhere, U should → 1 (F(1-u) pushes u up). Unless... U=1 is not reached because F(1-u) term... at u=1, du=0 stable. Starting u=0.25 → grows to 1. Mean 0.52 means half the grid at u~1 and half at ~0? No—maxV=0 but u low somewhere? That contradicts: if u is low, v would grow since uvv term... but v=0 → uvv=0 → du = F(1-u) > 0 → u recovers. Unless u region is large and slow recovery: F=0.035 per step, over 1000 steps → fully recovered. Hmm mean stays exactly 0.5224 constant, suggesting a fixed point.

Oh! Bug: my swap. `[U,U2]=[U2,U]` — fine. But wait, `const xm=(x-1+N)&255` and `xc+x` with xc=y*N — since N=256, y*N = y<<8, and x added — no overflow issue. Fine.

Hmm, but mean U = 0.5224 exactly constant... Let me check: is there a bug where U in the swap aliasing? U2 written each iteration fully. Fine.

Wait — maybe the issue is that v doesn't die but maxV computed wrongly? `if(V[i]>m)m=V[i]` fine.

Let me debug: print min/max/mean of U and V separately and a small snapshot of the center region.

Actually! I bet the bug: `const ym=((y-1+N)&255)*N` — y-1+N for y=0 gives 255, ok.

Hmm, another suspicion: dv term coefficient: standard GS reaction is V += u*v*v - (F+k)*v. Yes.

Let me actually print center values overtime.

Weird: U converges to 0.5224 everywhere even where V=0 and U started at 1?? Uedge went from 1 → 0.522 too! And U0 recovered to 0.5224 but stopped. That means the update has some issue: U reaching 0.52 and stuck.

At u=0.5224, v=0: du = Du*lap + F(1-u) = 0 + 0.035*0.4776=0.0167 > 0. Should keep rising to 1. Unless... the F term is wrong sign? du = ... + F*(1-u) → positive. So u should grow. Unless u is capped by something.

Hmm, U converged to 0.5224 = equilibrium where? Actually this looks like the "u cap". Wait — maybe my u update writes to U2 but then swap, and step 1 shows U0 = 0.0658 from 0.25: du = -uvv + F(1-u) = -(0.25*0.81) + 0.035*0.75 = -0.2025+0.026 = -0.176 → 0.074, plus clipping to... 0.0658 close enough with lap terms.

Then U rises slowly: at step 12 U0=0.1013, growing ~0.008/step = F*(1-u)≈0.031*... hmm 0.035*0.9=0.0315/step but observed 0.008/step. Then it stalls at 0.5224.

Something's off: du should be F(1-u). Observed growth rate much lower, and zero at 0.5224. This smells like the classic "1/N" averaging issue? Or... maybe the drawn values: U2 was reused? Let me check the swap: [U,U2]=[U2,U] destructuring — fine.

Hmm wait, but maybe my per-cell loop writes U2 for all i, fine.

Alternatively: laplacian with 0.2 weights and center -1 sums to zero — right.

Actually! I see it: `const xm=(x-1+N)&255` — for x=0, xm=255, correct wrap. Fine.

Hold on: maybe the bug is `F*(1-u)` with F=0.035 but the equilibrium of du at v=0 should be u=1. Getting stuck at 0.5224 means du=0 → F(1-u)=0 → u=1. Contradiction unless the F term isn't applied... Let me directly test: set u=0.5, v=0 everywhere, one step, print.

Could there be a name shadowing: I used `F` const and also... no. Let me test minimal.

Fine. So why does the sim stall at 0.5224? Because U2 contains stale values? Let's check: after swap, U=U2_old. U2 now holds old U. Next iteration we write all cells of U2. Fine.

Hmm, but the growth observed 0.008/step instead of 0.0175 (F(1-u) at u=0.5 = 0.0175). Wait observed between step 12 (0.1013) and 51 (0.4599): Δ=0.3586 over 39 steps = 0.0092/step, but F(1-u) with u~0.28 ≈ 0.025. And Uedge went 0.730→0.536 over 39 steps = -0.005/step, negative! U should increase from 0.73. So there IS a spurious term: U decreasing where V=0 and far from center... Du*lap: neighbors are all ~1 initially, so lap>0 positive. Hmm but decreasing.

OH. I see it! In the loop I compute `lu` using `U[ym+x]` etc. But `xm`/`xp` computed as `(x-1+N)&255` — x is a number 0..255 fine. ym is a row offset. OK.

Wait, but the whole grid U decreasing toward 0.52 while V=0 everywhere... The only negative term is -uvv. v=0 so 0. Unless V is not zero! maxV printed 0.0000 with 4 decimals... yes zero.

Hmm, is F applied? du = Du*lu - uvv + F*(1-u). Let me actually print for a specific far cell the components. Let me instrument.

Ohh! I wonder: is `U2` Float32Array but assignment `U2[i]=nu` — yes.

Let me just instrument a single cell far from center over steps.

lu is hugely negative (-0.19) while neighbors equal u — meaning the sum 0.2*(4*0.968) = 0.7744 vs u=0.968 → 0.7744-0.968 = -0.1936. Right! The Laplacian formula: 0.2*(sum of 4 neighbors) - u is wrong; the correct 5-point with 0.2 weights and -1 center requires the center weight -1 and neighbors 0.2 each → sum of weights = 0.8-1 = -0.2 ≠ 0. The standard formulation: lap = (0.2*N + 0.05*diagonals) - u... that's for 9-point: center -1, edges 0.2, corners 0.05: sum = 0.8+0.2-1 = 0. ✓.

So with only 4 neighbors at 0.2, correct is 0.25 each with center -1, or use 9-point with corners. I'll use the 9-point: center -1, edges 0.2, corners 0.05. That's the classic Pearson. Let me fix.

Then du = Du*lap + ... with lap scaled properly (this 9-point lap approximates the discrete Laplacian with the grid spacing; effectively diffusion coefficient is as in standard demos with dt=1).

Let's fix and re-test innode.

Missing closing braces for run(). Let me write fileinstead.

Now it grows (cell count increasing). Good — the 9-point Laplacian works and the pattern expands. Growth is slow-ish (1292 cells at step 4000) — that's spots spreading. With stepsPerFrame=8 and 60fps → 480 steps/s → 4000 steps in ~8s. Fine.

Also note the 5-point version had wrong weights. Update index.html to 9-point Laplacian, and maybe increase steps per frame default (e.g., 10) to speed evolution.

Also I should verify the interaction: click adds V near pointer and affects next frames — with brush radius default 10 that works.

Also fix: the color mapping thresholds; V ~0.4 typical → amber-ish. Fine.

Let me update index.html Laplacian.

Note: `ym | xm` — ym is a multiple of 256 so `ym|xm` = ym+xm ✓ (xm<256). And `ym | xp` ✓.

Wait: `ym | x` where ym = row offset; ok.

But careful: `(y-1+N)&255)<<8` — ym is up to 255*256=65280, `ym | xm` fine.

Now also increase default steps per frame to 10? 8 is fine. Let me reloadand check.

Working. Pattern grows. Now let's test interactions: click elsewhere on canvas and confirm a new blob appears and affects subsequent frames.

Canvas coordinates: canvas element spans x 184..1014 in the screenshot? The holder width is 830 CSS px displayed (canvas 256x256 scaled). Canvas top ~110, bottom beyond 800 (height 830). Let me click at a visible point, e.g. (400, 300) → in-canvas.

Let's click at (350, 250).

The click created a new blob (bright yellow-green) and the center pattern evolved into mitosis-like structures. Interesting that the center pattern turned into maze/mitosis structures — good.

But wait: frame 3832 after ~40s of real time? That's ~95 steps/s... hmm the browser tool waited. Anyway.

Interesting: the center seed turned into maze — with F=0.035, k=0.062 we got mitosis-like. Fine, organic patterns visible.

The clicked blob is yellow-green — V is high (0.9) initially. Colors work.

Now test pause/resume and reset. Click Pause button at (197, 257)? The controls are below the canvas which is tall (830px). Need to scroll. Let's scroll down.

The page seems weirdly scrolled; the canvas is huge (830px) causing long page. Perhaps I should limit canvas display size (max-width) so page is shorter. The canvas-holder width = stage width ~830. Let me set canvas max-width 640 and center, so the layout is compact.

Continue scrolling to reachthe buttons.

Great, nice pattern. Now test Pause button (217,629), then check frame stops, then Reset.

Also note: the earlier clicked blob merged/grew. Good.

Also I notice the page is long due to tall canvas; I'll reduce canvas display size to 560px max for better layout.

Let's click Pause.

Paused, button shows Resume. Now click Reset to verify restore of initial central seed. Reset button at (300,629).

Note: the pattern filled the whole grid (wrapping) — nice, demonstrates wrapping.

Also note: the color is teal/green mostly; V typical 0.4 → in my palette, v=0.4 falls in the 0.35–0.55 band → r=53+160*0.25=93, g=162+60*0.25=177, b=130-60*0.25=115 → greenish. That's OK but "contrasting" would be nicer with more variety. Fine.

Click Reset.

Reset works: back to a single central seed blob.

Remaining polish:
- Canvas display too tall (830px). Limit max-width to ~600px so page is compact.
- The color palette: the seed appears yellow-green (v high). Pattern mostly green/teal. Maybe increase contrast: make high V more orange/cream and mid teal. Current mapping: at v=0.6 → band4: r=213+42*(0.14)=219, g=222-40*.14=216, b=70+130*.14=88 → yellow-green. Hmm the seed center shows yellow-green, mid band green at 0.4. It's fine but let me make band boundaries give more blue→orange contrast: e.g., map v: 0-0.1 dark navy; 0.1-0.25 blue; 0.25-0.4 teal/green; 0.4-0.55 amber; >0.55 cream/yellow. Let's adjust thresholds to use the actual typical range (0.2-0.5).

Let me adjust the palette so typical v=0.4 → amber and v=0.2 → teal.

New mapping:
- v<0.08: (5,7,12)→(8,25,55) dark blue
- 0.08-0.2: blue→teal: t: r 8→18, g 25→120, b 55→140
- 0.2-0.35: teal→green: r 18→40, g 120→200, b 140→90
- 0.35-0.5: green→amber: r 40→230, g 200→140, b 90→40
- >0.5: amber→cream: r 230→255, g 140→245, b 40→200

That gives contrast. Let's implement.

Also resize canvas display: add max-width 600px, margin auto.

Also add "brush preview"? Not needed.

Also reset should restore frame to 0 — it does (frame reset) and then runs.

Let me also make the initial seed radius 8 and V=0.9/U=0.25 — already.

Apply edits.

Also update the legend swatch gradient to match roughly: linear-gradient(90deg,#05070c,#123a5c,#1f7fa8,#f0a13a,#fff3c4) — close enough.

Also, the pattern takes a long time to develop (18576 steps). With steps/frame=8 at 60fps → 480 steps/s → 40s. The browser seemed slower (~200 steps/s). Let's raise default steps per frame to 12 for faster evolution, still fine performance-wise (256²*12 = 786k cell updates per frame ≈ fine).

Also add a "Speed" indicator? Fine.

Let me also verify performance: measure frame rate via a quick check — not easy without eval. I'll trust; but let's set default steps to 12 and check responsiveness by clicking (interaction latency). The clicks responded fine.

Update default steps value in HTML and JS (both 8 →12).

Layout good now. Seed blob is teal ring (v~0.35 teal/green). Let's let it run and check the palette contrast later.

Now test drag: drag from (450,250) to (700,300) and confirm a streak appears and evolves.

Let me wait some frames then drag.

Drag created a dotted trail of green blobs that are evolving (mitosis-like). The center seed became a maze pattern.

Colors: mostly green/teal — I'd like more contrast (orange/cream for high V). maxV is 0.703 but most structure is ~0.4. The green band 0.35-0.5 → amber at 0.5. Structure shows bright green ~ maybe v 0.45 → r=160,g=176,b=73 greenish. Acceptable but let's shift thresholds lower so typical 0.4 → orange. Change band boundaries: 0.30-0.45 → green→amber, and 0.45+ amber→cream. Also 0.2-0.3 teal→green.

Let me tune: v<0.05 navy; 0.05-0.15 blue; 0.15-0.28 teal(0,120,150)→green(20,200,60); 0.28-0.42 green→orange(255,140,30); >0.42 orange→cream(255,250,220).

That gives more color variety. Implement.

Also the pattern is quite sparse; growth continues. Fine.

Let me update the palette.

Now the seed ring is green (v~0.35). Let's let it run and click to see contrast with orange. Let me wait by taking another screenshot after some frames, and click in a corner.

Also potential improvement: growth is slow; the classic "mitosis" for these params grows over thousands of steps. It's fine.

Let me click at (400,200) and screenshot later.

Nice: the clicked region at top-left shows a ring (blue interior = lower V, green rim) — V was injected so max V should be high initially, but at frame 5892 it relaxed. Colors look decent (green with blue fringe).

Everything works. Final checks:
- Pause/resume verified earlier.
- Reset verified earlier.
- Console clean.

One more: verify reset restores identical initial scene pixel-wise vs initial. Already saw the same single central ring.

Clean up test_sim.js? It's in the working dir; the deliverable is index.html. I'll delete the test file to keep dir clean. Actually keeping it might be fine but instructions say write app to index.html. I'll remove test file.

Also add a small note in index.html about the Laplacian. Fine.

One concern: `image-rendering: pixelated` with canvas scaled 600/256 → fine.

Also pointerleave ends painting; with pointer capture, moves outside still paint — ok.

Delete test file and finish.

Final reload to confirm clean console and initial state.