Phyllotaxis and patterns on plants

Patterns arising from the governing equationsBoth mechanisms share the same structures in the equations.∂w∂t= −(∇ 2 + P) 2 w + ɛw + N.L..In 1-dimensional, The linear growth rate for e ikx+σt isσ(k) = (−k 2 + P) 2 + ɛ.In 2-dimensional, ∇ 2 = ∂2 + 1 ∂∂r 2 r ∂r + 1 ∂ 2The linear growth rate forr 2 ∂α 2e ilα+imx+σt is σ(l, m) = −(−l 2 − m2 + P) 2 + ɛ.r 2Triad: ⃗ k m = (l m , m), ⃗ k m + ⃗ k n = ⃗ k m+n ⇒ l m + l n = l m+n , m + n = (m + n)Alan C. Newell, Zhiying Sun, Patrick D. Shipman<strong>Phyllotaxis</strong> <strong>and</strong> <strong>patterns</strong> on plants