Growth Models for Tree Stems and Vines Alberto Bressan Department - PowerPoint PPT Presentation

Growth Models for Tree Stems and Vines Alberto Bressan Department of Mathematics, Penn State University Center for Interdisciplinary Mathematics bressan@math.psu.edu Alberto Bressan (Penn State) growth models 1 / 24

Stabilizing stem growth what kind of stabilizing feedback is used here? Alberto Bressan (Penn State) growth models 2 / 24

Growth in the presence of obstacles Are the growth equations still well posed, when an obstacle is present? What additional feedback produces curling around other branches? Alberto Bressan (Penn State) growth models 3 / 24

A model of stem growth (F. Ancona, A.B., O. Glass) New cells are born at the tip of the stem Their length grows in time, at an exponentially decreasing rate γ (t,t) k (t,s) γ (t,s) γ ( t , s ) = position at time t of the cell born at time s Unit tangent vector to the stem: k ( t , s ) = ∂ s γ ( t , s ) Alberto Bressan (Penn State) growth models 4 / 24

Stabilizing growth in the vertical direction stem not vertical = ⇒ local change in curvature e − β ( t − s ) = stiffness factor, ω = k ( t , σ ) × e 3 = angular velocity � s e − β ( t − σ ) � � � � ∂ t γ ( t , s ) = k ( t , σ ) × e 3 × γ ( t , s ) − γ ( t , σ ) d σ 0 γ (t,s) k e 3  γ( ,σ) t  γ ( t 0 , s ) = γ ( s ) s ∈ [0 , t 0 ]   � ω �   � γ ss ( t , s ) = 0  e 3 � s = t e 2 e 1 Alberto Bressan (Penn State) growth models 5 / 24

We say that the growth equation is stable in the vertical direction if for any initial time t 0 > 0 and every ε > 0 there exists δ > 0 such that | e 1 · ∂ s γ ( t 0 , s ) | ≤ δ for all s ∈ [0 , t 0 ] implies � � � � � e 1 · γ ( t , s ) � ≤ ε � e 1 · ∂ s γ ( t , s ) � ≤ ε , for all t > t 0 , s ∈ [0 , t ] P(t,s) e 2 P(t ,s) 0 e 1 x 1 0 ε Alberto Bressan (Penn State) growth models 6 / 24

Numerical simulations (Wen Shen, 2016) β = 0 . 1 β = 1 . 0 β = 2 . 5 stability is always achieved increasing the stiffness reduces oscillations Alberto Bressan (Penn State) growth models 7 / 24

Analytical results (F. Ancona, A.B., O. Glass, 2017) β = stiffening constant If β 4 − β 3 ≥ 4, then the growth is stable in the vertical direction (non-oscillatory regime: β ≥ β 0 ≈ 1 . 7485) √ If β > β ∗ = (48 + 9504) / 160, then growth is still stable in the (oscillatory regime, β ∗ ≈ 0 . 9093) vertical direction Stability apparently holds for all β > 0 (??) Alberto Bressan (Penn State) growth models 8 / 24

A linearized problem Key step: prove stability for the linearized system � ∞ e − β y u ( y ) dy u t + u x = − for x ∈ [0 , + ∞ [ x with Neumann boundary condition at x = 0 u x ( t , 0) = 0 u(t ) 0 u(t) u ( t , x ) ≈ e 1 · γ s ( t , t − x ) x t 0 t 0 Alberto Bressan (Penn State) growth models 9 / 24

Growth with obstacles Ω ⊂ R 3 Obstacle: open set, with smooth boundary At time t the stem γ ( t , · ) is a curve of length t remaining outside the obstacle γ (t,T) Ω γ (t,t) γ (t,s) Basic space: γ ( t , · ) ∈ H 2 ([0 , T ] ; R 3 ) For s ∈ ] t , T ] define γ ( t , s ) = γ ( t , t ) + γ s ( t , t )( s − t ) Time-dependent constraint: γ ( t , s ) / ∈ Ω for s ∈ [0 , t ] Alberto Bressan (Penn State) growth models 10 / 24

A push-out operator Ω Ω γ (s) ~ γ (s) γ(σ) ω ( σ ) = additional bending of the stem caused the obstacle, at the point γ ( σ ) � s γ ( s ) − γ ( s ) = � ω ( σ ) × ( γ ( s ) − γ ( σ )) d σ s ∈ [0 , t ] 0 Among all infinitesimal deformations that push the stem outside the obstacle, � t minimize the elastic energy: E = 1 e β ( t − σ ) | ω ( σ ) | 2 d σ 2 0 Alberto Bressan (Penn State) growth models 11 / 24

The evolution equation with constraints � � Ψ( σ ) = Ψ t , σ, γ ( t , σ ) , γ s ( t , σ ) = upward bending, as response to gravity � s � � without obstacle: γ t ( t , s ) = Ψ( σ ) × γ ( t , s ) − γ ( t , σ ) d σ 0 � s � � � � with obstacle: γ t ( t , s ) = Ψ( σ ) + ¯ ω ( t , σ ) × γ ( t , s ) − γ ( t , σ ) d σ 0 � t e β ( t − σ ) | ω ( σ ) | 2 d σ ω ( · ) = argmin ¯ ω ( · ) 0 Alberto Bressan (Penn State) growth models 12 / 24

The instantaneous minimization problem γ (t,t) n γ (t,s) Ω n � t 1 e β ( t − σ ) | ω ( σ ) | 2 d σ ω ( · ) = argmin ¯ 2 ω ( · ) 0 subject to the unilateral constraints at points of contact: � γ t ( t , s ) , n ( t , s ) � ≥ 0 whenever γ ( t , s ) ∈ ∂ Ω , If the tip of the stem touches the obstacle, one also needs � γ s ( t , t ) + γ t ( t , t ) , n ( t , t ) � ≥ 0 Alberto Bressan (Penn State) growth models 13 / 24

Necessary conditions for optimality The solution to the constrained minimization problem � t e β ( t − σ ) | ω ( σ ) | 2 d σ ¯ ω ( · ) = argmin ω ∈A 0 admits the representation: �� � � t e − β ( t − s ) n ( γ ( t , s ′ )) d µ ( s ′ ) ω ( s ) = ¯ − × γ s ( t , σ ) d σ [ σ, t ] s where µ is a positive measure, supported on the contact set � � χ ( t ) . s ′ ∈ [0 , t ] ; γ ( t , s ′ ) ∈ ∂ Ω = Alberto Bressan (Penn State) growth models 14 / 24

Discontinuous evolution problems Growth with obstacles yields an evolution equation with discontinuous right hand side Ω Ω Ω γ (t ) γ (t ) 3 2 γ (t ) 1 Can be reformulated as a differential inclusion with u.s.c. right hand side Alberto Bressan (Penn State) growth models 15 / 24

A cone of admissible reactions γ (t,s) v (s) χ ( t ) . = { s ′ ∈ [0 , t ] ; γ ( t , s ′ ) ∈ ∂ Ω } Ω = contact set Cone of admissible velocities produced by the obstacle reaction: � . � v : [0 , t ] �→ R 3 ; K ( s , s ′ ) d µ ( s ′ ) Λ( γ ( t )) = v ( s ) = � for some positive measure µ supported on χ ( t ) � s � � differential inclusion: γ t ( t , s ) ∈ Ψ( σ ) × γ ( t , s ) − γ ( t , σ ) d σ + Λ( γ ( t )) 0 Alberto Bressan (Penn State) growth models 16 / 24

Well-posedness of the stem growth model with obstacle Theorem (A.B. - M.Palladino, 2016-17) Solutions exist and are unique except if a (highly non-generic) breakdown configuration occurs. Ω Ω Ω Ω bad good good bad (B) The tip of the stem touches the obstacle perpendicularly, namely γ ( t 0 ) ∈ ∂ Ω , γ s ( t 0 ) = − n ( γ ( t 0 )) . (1) Moreover, γ ss ( s ) = 0 for all s ∈ ]0 , t [ such that γ ( s ) / ∈ ∂ Ω . (2) Alberto Bressan (Penn State) growth models 17 / 24

Geometric interpretation d dt γ ( t ) ∈ F ( γ ( t )) + Λ( γ ( t )) γ / ∈ S Λ 3 Λ 2 Λ 1 Ω Ω Ω γ 3 γ γ γ γ 2 F 1 2 3 S γ 1 Alberto Bressan (Penn State) growth models 18 / 24

Well posedness of evolution equations with constraints d dt z ( t ) ∈ f ( z ( t )) + Γ( z ( t )) z / ∈ S Γ (z) v 2 Γ z f Γ v 2 z z 2 2 z z v 1 v 1 S 1 1 S S If f is Lipschitz and Γ( z ) = N S ( z ) = outer normal cone to S at a boundary point z , then d � � � ≤ C � � � z 1 ( t ) − z 2 ( t ) � z 1 ( t ) − z 2 ( t ) (1) � dt Main idea: introduce an equivalent “Riemann metric” so that the cones Γ( z ) become perpendicular to the boundary of S Alberto Bressan (Penn State) growth models 19 / 24

Numerical simulations (Wen Shen, 2016) Alberto Bressan (Penn State) growth models 20 / 24

Vines clinging to tree branches (A.B., M.Palladino, W.Shen) The stem bends toward the obstacle, at points which are sufficiently close (i.e., at a distance < δ 0 from the obstacle) γ (t,s) η (s) α δ 0 γ( ,σ) t Ω s δ 0 � � . ψ ( x ) = η d ( x , Ω) In the case of a vine that clings to a branch of another tree, the evolution equation contains an additional term (= ⇒ bending toward the obstacle) � s e − β ( t − σ ) � � � � γ t ( t , s ) = ∇ ψ ( γ ( t , σ )) × γ s ( t , σ ) × γ ( t , s ) − γ ( t , σ ) d σ + · · · 0 Alberto Bressan (Penn State) growth models 21 / 24

Numerical simulations (Wen Shen) 5.5 5.5 5.5 5 5 5 4.5 4.5 4.5 4 4 4 3.5 3.5 3.5 3 3 3 2.5 2.5 2.5 2 2 2 1.5 1.5 1.5 1 1 1 0.5 0.5 0.5 0 0 0 -1 0 1 2 -1 0 1 2 -1 0 1 2 Alberto Bressan (Penn State) growth models 22 / 24

Alberto Bressan (Penn State) growth models 23 / 24

References [1] F. Ancona, A. Bressan, O. Glass, and W. Shen, Feedback stabilization of stem growth, J. Dynam. Diff. Equat. , submitted. [2] A. Bressan, M. Palladino, and W. Shen, Growth models for tree stems and vines, J. Differential Equations , to appear. [3] A. Bressan and M. Palladino, Well-posedness of a model for the growth of vines in the presence of obstacles, Discr. Cont. Dyn. Syst. , to appear. MATLAB source codes: http://math.psu.edu/shen w/STEM-VINE-SIM/ Alberto Bressan (Penn State) growth models 24 / 24