Mathematical models for cell movement Part II F ABIO A. C. C. C - - PowerPoint PPT Presentation

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Mathematical models for cell movement Part II F ABIO A. C. C. C - - PowerPoint PPT Presentation

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Mathematical models for cell movement Part II F ABIO A. C. C. C HALUB Centro de Matem atica e Aplicac oes Fundamentais Universidade de Lisboa Mathematical models for cell movementPart II p. 1 Overview Biological background

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Mathematical models for cell movement Part II

FABIO A. C. C. CHALUB

Centro de Matem´ atica e Aplicac ¸˜

  • es Fundamentais

Universidade de Lisboa

Mathematical models for cell movementPart II – p. 1

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Overview

Biological background Keller-Segel model Kinetic models Scaling up and down

Mathematical models for cell movementPart II – p. 2

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Overview – Today

The Keller-Segel model. Variations on the same theme. Models with global existence. Kinetic models

Mathematical models for cell movementPart II – p. 3

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The Keller-Segel Model

For x ∈ R2, we call the classical Keller-Segel model: ∂tρ = ∇ · (∇ρ − χρ∇S) , ∆S = −ρ , with ρ(·, 0) = ρI , with χ = χ0 = const.

Mathematical models for cell movementPart II – p. 4

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Keller-Segel Models

  • Corollary. In the two dimensional case, for the classical

Keller-Segel model, we have:

Mathematical models for cell movementPart II – p. 5

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Keller-Segel Models

  • Corollary. In the two dimensional case, for the classical

Keller-Segel model, we have: if M < 8π/χ: global existence of solutions,

Mathematical models for cell movementPart II – p. 5

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Keller-Segel Models

  • Corollary. In the two dimensional case, for the classical

Keller-Segel model, we have: if M < 8π/χ: global existence of solutions, if M > 8π/χ: finite-time-blow-up.

Mathematical models for cell movementPart II – p. 5

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Keller-Segel Models

Consider the following Keller-Segel model (with prevention of overcrowding) (Hillen-Painter model): ∂tρ = ∇ · (∇ρ − χ(ρ)ρ∇S) ∆S = −ρ , where χ(ρ) = 0 , ρ ≥ ¯ ρ > 0 .

Mathematical models for cell movementPart II – p. 6

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Keller-Segel Models

Consider the following Keller-Segel model (with prevention of overcrowding) (Hillen-Painter model): ∂tρ = ∇ · (∇ρ − χ(ρ)ρ∇S) ∆S = −ρ , where χ(ρ) = 0 , ρ ≥ ¯ ρ > 0 .

  • Theorem. (Hillen, Painter, 2002) Solutions of the HP

model exist globally.

Mathematical models for cell movementPart II – p. 6

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Keller-Segel Models

Define the non-local gradient

  • ∇R f(x, t) =

1 ωn−1Rn−1

  • Sn−1 f(x + yR)dy .

Mathematical models for cell movementPart II – p. 7

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Keller-Segel Models

Define the non-local gradient

  • ∇R f(x, t) =

1 ωn−1Rn−1

  • Sn−1 f(x + yR)dy .

Then the Hillen-Schmeiser-Painter model ∂tρ = ∇ ·

  • ∇ρ − χρ
  • ∇R S
  • ,

has global existence of solutions.

Mathematical models for cell movementPart II – p. 7

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Keller-Segel Models

Consider a sensitivity (Velazquez’ model): χ(ρ) = χµ(ρ) = ρ 1 + µρ ,

Mathematical models for cell movementPart II – p. 8

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Keller-Segel Models

Consider a sensitivity (Velazquez’ model): χ(ρ) = χµ(ρ) = ρ 1 + µρ ,

  • Theorem. (Velazquez, 2004) The V model has global

existence of solutions for any µ > 0.

Mathematical models for cell movementPart II – p. 8

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Keller-Segel Models

Consider a sensitivity (Velazquez’ model): χ(ρ) = χµ(ρ) = ρ 1 + µρ ,

  • Theorem. (Velazquez, 2004) The V model has global

existence of solutions for any µ > 0.

Mathematical models for cell movementPart II – p. 8

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Keller-Segel Models

Consider a sensitivity (Velazquez’ model): χ(ρ) = χµ(ρ) = ρ 1 + µρ ,

  • Theorem. (Velazquez, 2004) The V model has global

existence of solutions for any µ > 0. For t < T, limµ→0 ρµ = ρ0.

Mathematical models for cell movementPart II – p. 8

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Keller-Segel Models

Consider a sensitivity (Velazquez’ model): χ(ρ) = χµ(ρ) = ρ 1 + µρ ,

  • Theorem. (Velazquez, 2004) The V model has global

existence of solutions for any µ > 0. For t < T, limµ→0 ρµ = ρ0. This cannot be extended after T because ρ0 no longer exists (T is the blow up time).

Mathematical models for cell movementPart II – p. 8

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Keller-Segel Models

Consider a sensitivity (Velazquez’ model): χ(ρ) = χµ(ρ) = ρ 1 + µρ ,

  • Theorem. (Velazquez, 2004) The V model has global

existence of solutions for any µ > 0. For t < T, limµ→0 ρµ = ρ0. This cannot be extended after T because ρ0 no longer exists (T is the blow up time). For any µ > 0, ρµ exists for any time t.

Mathematical models for cell movementPart II – p. 8

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Kinetic Models

f(x, v, t) is the density of cell in space-time point (x, t) with velocity v (phase-space density).

Mathematical models for cell movementPart II – p. 9

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Kinetic Models

f(x, v, t) is the density of cell in space-time point (x, t) with velocity v (phase-space density). The cell goes in straight line for a certain characteristic time and then changes its direction from v′ to v (in a space-time point (x, t) in the presence of the substance S and cell density ρ) according to a certain turning kernel T[S, ρ](x, v, v′, t).

Mathematical models for cell movementPart II – p. 9

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Kinetic Models

f(x, v, t) is the density of cell in space-time point (x, t) with velocity v (phase-space density). The cell goes in straight line for a certain characteristic time and then changes its direction from v′ to v (in a space-time point (x, t) in the presence of the substance S and cell density ρ) according to a certain turning kernel T[S, ρ](x, v, v′, t). The set of all possible velocities is given by a compact, spherically symmetric set V .

Mathematical models for cell movementPart II – p. 9

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Kinetic Models

We define an equilibrium distribution F = F(v): F > 0 ,

  • V

Fdv = 1 ,

  • V

vFdv = 0 ,

if S = S0 = ⇒ T[S0, ρ](x, v, v′, t)F(v′) = T[S0, ρ](x, v′, v, t)F(v) .

Mathematical models for cell movementPart II – p. 10

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Kinetic Models

We define an equilibrium distribution F = F(v): F > 0 ,

  • V

Fdv = 1 ,

  • V

vFdv = 0 ,

if S = S0 = ⇒ T[S0, ρ](x, v, v′, t)F(v′) = T[S0, ρ](x, v′, v, t)F(v) .

Two possible turning kernels:

T[S, ρ](x, v, v′, t) = λ(S, ρ)(x, t)F(v) + a(S, ρ)F(v)v · ∇S(x, t) , T[S, ρ](x, v, v′, t) = ψ(S(x + vt, t) − S(x, t))F(v) .

Mathematical models for cell movementPart II – p. 10

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Kinetic Models

∂tf(x, v, t) + v · ∇f(x, v, t) =

  • V

(T[S, ρ](x, v, v′, t)f(x, v′, t) − T[S, ρ](x, v′, v, t)f(x, v, t))dv′ .

Mathematical models for cell movementPart II – p. 11

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Kinetic Models

Notation f = f(x, v, t) , f ′ = f(x, v′, t) , T[S, ρ] = T[S, ρ](x, v, v′, t) , T ∗[S, ρ] = T[S, ρ](x, v′, v, t).

Mathematical models for cell movementPart II – p. 12

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Kinetic Models

Notation f = f(x, v, t) , f ′ = f(x, v′, t) , T[S, ρ] = T[S, ρ](x, v, v′, t) , T ∗[S, ρ] = T[S, ρ](x, v′, v, t). Equation ∂tf + v · ∇f =

  • V

(T[S, ρ]f ′ − T ∗[S, ρ]f)dv′ .

Mathematical models for cell movementPart II – p. 12

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Kinetic Model

This is an example of a Boltzmann-type integro-differential equation (kinetic model).

Mathematical models for cell movementPart II – p. 13

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Kinetic Model

This is an example of a Boltzmann-type integro-differential equation (kinetic model). The “macroscopic” density ρ is related to the “microscopic” density f by ρ(x, t) =

  • V

f(x, v, t)dv .

Mathematical models for cell movementPart II – p. 13

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Kinetic Model

This is an example of a Boltzmann-type integro-differential equation (kinetic model). The “macroscopic” density ρ is related to the “microscopic” density f by ρ(x, t) =

  • V

f(x, v, t)dv . We should consider also an equation for S: ∂tS = D0∆S + ϕ(S, ρ) .

Mathematical models for cell movementPart II – p. 13

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Kinetic Model

This is an example of a Boltzmann-type integro-differential equation (kinetic model). The “macroscopic” density ρ is related to the “microscopic” density f by ρ(x, t) =

  • V

f(x, v, t)dv . We should consider also an equation for S: ∂tS = D0∆S + ϕ(S, ρ) .

Mathematical models for cell movementPart II – p. 13

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Kinetic Models

  • Theorem. (C., Markowich, Perthame, Schmeiser, 2004;

Hwang, Kang, Stevens, 2005) If ψ(y) ≤ Ay + B then solutions of the kinetic model exist globally.

Mathematical models for cell movementPart II – p. 14

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Kinetic Models

  • Theorem. (C., Markowich, Perthame, Schmeiser, 2004;

Hwang, Kang, Stevens, 2005) If ψ(y) ≤ Ay + B then solutions of the kinetic model exist globally. Proof: (Let us suppose n = 3, the case n = 2 is technically more complicated but similar.)

Mathematical models for cell movementPart II – p. 14

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Kinetic Models

  • Theorem. (C., Markowich, Perthame, Schmeiser, 2004;

Hwang, Kang, Stevens, 2005) If ψ(y) ≤ Ay + B then solutions of the kinetic model exist globally. Proof: (Let us suppose n = 3, the case n = 2 is technically more complicated but similar.) We divide S(x, t) = 1 4π

  • R3

1 |x − y|ρ(y, t)dy , in S = SS + SL, where SS = 1 4π| · |I{|x|<1} ∗ ρ , SL = 1 4π| · |I{|x|≥1} ∗ ρ .

Mathematical models for cell movementPart II – p. 14

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Kinetic Models

Mass conservation: ||ρ(·, t)||L1(R3) = ||f(·, ·, t)||L1(R3×V ) = ||f I||L1(R3×V ) .

Mathematical models for cell movementPart II – p. 15

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Kinetic Models

Mass conservation: ||ρ(·, t)||L1(R3) = ||f(·, ·, t)||L1(R3×V ) = ||f I||L1(R3×V ) . From Young’s inequality: ||SL(·, t)||L∞(R3) ≤ 1 4π||f I||L∞(R3×V ) .

Mathematical models for cell movementPart II – p. 15

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Kinetic Models

Mass conservation: ||ρ(·, t)||L1(R3) = ||f(·, ·, t)||L1(R3×V ) = ||f I||L1(R3×V ) . From Young’s inequality: ||SL(·, t)||L∞(R3) ≤ 1 4π||f I||L∞(R3×V ) . Possibly changing the bounds on the turning kernels, we can change S by SS.

Mathematical models for cell movementPart II – p. 15

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Kinetic Models

Now, we have that ∂tf + v · ∇f ≤

  • V

T[S(x, v, v′, t)f(x, v′, t)dv′ .

Mathematical models for cell movementPart II – p. 16

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Kinetic Models

Now, we have that ∂tf + v · ∇f ≤

  • V

T[S(x, v, v′, t)f(x, v′, t)dv′ . We write T[S(x, v, v′, t) ≤ C(1 + SS(x + v, t)), and then

f(x, v, t) ≤ f I(x − vt, t) + C t ρ(x − vs, t − s)ds + Cf 1(x, v, t) ,

where

∂tf(x, v, t) + v · ∇f(x, v, t) =

  • V

SS(x + v, t)f(x, v′, t)dv′ , f(x, v, 0) = 0 .

Mathematical models for cell movementPart II – p. 16

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Kinetic Models

f 1(x, v, t) = t S(x − vs + v, t − s)ρ(x − vs, t − s)ds .

Mathematical models for cell movementPart II – p. 17

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Kinetic Models

f 1(x, v, t) = t S(x − vs + v, t − s)ρ(x − vs, t − s)ds . ||f 1(·, ·, t)||Lp ≤ sup

s∈[0,t]

||SS(·, s)||Lp t ||ρ(·, t − s)||Lpds .

Mathematical models for cell movementPart II – p. 17

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Kinetic Models

f 1(x, v, t) = t S(x − vs + v, t − s)ρ(x − vs, t − s)ds . ||f 1(·, ·, t)||Lp ≤ sup

s∈[0,t]

||SS(·, s)||Lp t ||ρ(·, t − s)||Lpds . ||ρ(·, t)||Lp ≤ C(V )||f(·, ·, t)||Lp .

Mathematical models for cell movementPart II – p. 17

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Kinetic Models

f 1(x, v, t) = t S(x − vs + v, t − s)ρ(x − vs, t − s)ds . ||f 1(·, ·, t)||Lp ≤ sup

s∈[0,t]

||SS(·, s)||Lp t ||ρ(·, t − s)||Lpds . ||ρ(·, t)||Lp ≤ C(V )||f(·, ·, t)||Lp .

We put everything together and find for p ≥ 2 ||f(·, ·, t)||Lp ≤ ||f I||Lp+ C

  • 1 + sup

s∈[0,t]

||SS(·, s)||Lp t ||f(·, ·, s)||Lpds .

Mathematical models for cell movementPart II – p. 17

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Kinetic Models

In the previous equation, fix p = 2. In this case 1 4π|x|I{|x|≤1} ∈ L2 .

Mathematical models for cell movementPart II – p. 18

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Kinetic Models

In the previous equation, fix p = 2. In this case 1 4π|x|I{|x|≤1} ∈ L2 . Then, from Young’s inequality: ||SS(·, t)||L2 ≤ c||f I||L1 ,

Mathematical models for cell movementPart II – p. 18

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Kinetic Models

In the previous equation, fix p = 2. In this case 1 4π|x|I{|x|≤1} ∈ L2 . Then, from Young’s inequality: ||SS(·, t)||L2 ≤ c||f I||L1 , and from Gronwall’s inequality we conclude a bound for ||f(·, ·, t)||L2.

Mathematical models for cell movementPart II – p. 18

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Kinetic Models

Still, from Young’s inequality, we have ||SS(·, t)||L∞ ≤ c||f(·, ·, t)||L2 ≤ C(t) , and again from Young’s inequality, we have a bound for ||f(·, ·, t)||L∞.

Mathematical models for cell movementPart II – p. 19

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Kinetic Models

Let us consider the following turning kernels (with prevention of overcrowding): Tε[S, ρ] = λ(S, ρ)F + εa(S, ρ)Fv · ∇S , Tε[S, ρ] = ψ(S(x + εµ(ρ)v, t) − S(x, t))F with a(S, ρ) = 0 , µ(ρ) = 0 , ρ ≥ ¯ ρ > 0 , and ε > 0 is a small parameter.

Mathematical models for cell movementPart II – p. 20

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Kinetic Models

  • Theorem. (C., Rodrigues, 2005) The kinetic models

associated to these turning kernels have global existence

  • f solutions. Furthermore,

||ρ(·, t)||L∞(Rn) ≤ max{¯ ρ, ||ρI||L∞(Rn)} .

Mathematical models for cell movementPart II – p. 21

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Kinetic Models

  • Theorem. (C., Rodrigues, 2005) The kinetic models

associated to these turning kernels have global existence

  • f solutions. Furthermore,

||ρ(·, t)||L∞(Rn) ≤ max{¯ ρ, ||ρI||L∞(Rn)} . We prove only the first case, the second is similar.

Mathematical models for cell movementPart II – p. 21

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Kinetic Models

  • Theorem. (C., Rodrigues, 2005) The kinetic models

associated to these turning kernels have global existence

  • f solutions. Furthermore,

||ρ(·, t)||L∞(Rn) ≤ max{¯ ρ, ||ρI||L∞(Rn)} . We prove only the first case, the second is similar. We consider initial conditions given by f I = ρIF and S = 0 and that λ is constant.

Mathematical models for cell movementPart II – p. 21

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Kinetic Models

  • Lemma. First note that

sup

s∈[0,t]

||∇S(·, s)||L∞ ≤ c

  • sup

s∈[0,t]

||ρ(·, s)||L∞ + sup

s∈[0,t]

||ρ(·, s)||L1

  • .

Mathematical models for cell movementPart II – p. 22

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Kinetic Models

  • Lemma. First note that

sup

s∈[0,t]

||∇S(·, s)||L∞ ≤ c

  • sup

s∈[0,t]

||ρ(·, s)||L∞ + sup

s∈[0,t]

||ρ(·, s)||L1

  • .
  • Lemma. Now, consider a time t∗ > 0 such that

T[S, ρ] ≥ 0, ∀(x, v, v′, t) ∈ Rn × V × V × [0, t∗]. Then, sup

s∈[0,t∗]

||ρ(·, s)||L∞ ≤ max{||ρI||L∞, ¯ ρ} .

Mathematical models for cell movementPart II – p. 22

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Kinetic Models

Proof: First consider initial conditions such that ||ρI||L∞ ≤ ¯ ρ.

Mathematical models for cell movementPart II – p. 23

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Kinetic Models

Proof: First consider initial conditions such that ||ρI||L∞ ≤ ¯ ρ. We define ˜ f = ¯ ρF − f , ˜ ρ =

  • V

˜ fdv = ¯ ρ − ρ , ˜ S = ¯ ρt − S , ˜ a( ˜ S, ˜ ρ) = a(S, ρ) ρ ¯ ρ − ρ .

Mathematical models for cell movementPart II – p. 23

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Kinetic Models

( ˜ f, ˜ S) is solution of

∂t ˜ f + v · ∇ ˜ f = λF ˜ ρ + a( ˜ S, ˜ ρ)v · ∇ ˜ S˜ ρ − λ ˜ f , ∆ ˜ S = −˜ ρ .

with initial conditions given by ˜ ρI = (¯ ρ − ρI)F > 0 and ˜ S = 0.

Mathematical models for cell movementPart II – p. 24

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Kinetic Models

( ˜ f, ˜ S) is solution of

∂t ˜ f + v · ∇ ˜ f = λF ˜ ρ + a( ˜ S, ˜ ρ)v · ∇ ˜ S˜ ρ − λ ˜ f , ∆ ˜ S = −˜ ρ .

with initial conditions given by ˜ ρI = (¯ ρ − ρI)F > 0 and ˜ S = 0. The turning kernels is ˜ T[ ˜ S, ˜ ρ] = λF + ε˜ a( ˜ S, ˜ ρ)Fv · ˜ S ≥ 0 , ∀(x, v, v′, t) ∈ Rn × V × V × [0, t∗]. We conclude the positivity of ˜ f, then 0 ≤ ¯ ρF − f, which implies ρ ≤ ¯ ρ.

Mathematical models for cell movementPart II – p. 24

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Kinetic Models

Now consider x such that ρI(x) > ¯ ρ in a neighbourhood U of x we have ∂tf + v · ∇f = λFρ − λf .

Mathematical models for cell movementPart II – p. 25

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Kinetic Models

Now consider x such that ρI(x) > ¯ ρ in a neighbourhood U of x we have ∂tf + v · ∇f = λFρ − λf . This implies

eλtf(x, v, t) = f (x − vt, v, t) + t eλsλFρ (x − v(t − s), s) ds ,

Mathematical models for cell movementPart II – p. 25

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Kinetic Models

Now consider x such that ρI(x) > ¯ ρ in a neighbourhood U of x we have ∂tf + v · ∇f = λFρ − λf . This implies

eλtf(x, v, t) = f (x − vt, v, t) + t eλsλFρ (x − v(t − s), s) ds ,

and then eλtρ(x, t) ≤ ||ρI||L∞ + t eλsλ||ρ(·, s)||L∞(U)ds .

Mathematical models for cell movementPart II – p. 25

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SLIDE 59

Kinetic Models

and then eλtρ(x, t) ≤ ||ρI||L∞ + t eλsλ||ρ(·, s)||L∞(U)ds .

Mathematical models for cell movementPart II – p. 26

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SLIDE 60

Kinetic Models

and then eλtρ(x, t) ≤ ||ρI||L∞ + t eλsλ||ρ(·, s)||L∞(U)ds . Finally, using Gronwall’s lemma: ||ρ(·, t)||L∞(U) ≤ ||ρI||L∞

Mathematical models for cell movementPart II – p. 26

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Kinetic Models

Proof: (of the theorem) We put together these lemmas:

Mathematical models for cell movementPart II – p. 27

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SLIDE 62

Kinetic Models

Proof: (of the theorem) We put together these lemmas: ||∇S||L∞ is bounded by ||ρ||L∞.

Mathematical models for cell movementPart II – p. 27

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SLIDE 63

Kinetic Models

Proof: (of the theorem) We put together these lemmas: ||∇S||L∞ is bounded by ||ρ||L∞. Whenever the turning kernel is positive, ||ρ||L∞ is uniformly-in-time bounded.

Mathematical models for cell movementPart II – p. 27

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SLIDE 64

Kinetic Models

Proof: (of the theorem) We put together these lemmas: ||∇S||L∞ is bounded by ||ρ||L∞. Whenever the turning kernel is positive, ||ρ||L∞ is uniformly-in-time bounded. This implies the ||∇S||L∞ is uniformly in time bounded.

Mathematical models for cell movementPart II – p. 27

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SLIDE 65

Kinetic Models

Proof: (of the theorem) We put together these lemmas: ||∇S||L∞ is bounded by ||ρ||L∞. Whenever the turning kernel is positive, ||ρ||L∞ is uniformly-in-time bounded. This implies the ||∇S||L∞ is uniformly in time bounded. Then, Tε[S, ρ] = λF + εa(S, ρ)Fv · ∇S is positive for small ε, for any time.

Mathematical models for cell movementPart II – p. 27

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SLIDE 66

Kinetic Models

Proof: (of the theorem) We put together these lemmas: ||∇S||L∞ is bounded by ||ρ||L∞. Whenever the turning kernel is positive, ||ρ||L∞ is uniformly-in-time bounded. This implies the ||∇S||L∞ is uniformly in time bounded. Then, Tε[S, ρ] = λF + εa(S, ρ)Fv · ∇S is positive for small ε, for any time. We do everything again!

Mathematical models for cell movementPart II – p. 27

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Kinetic Models

  • Theorem. (C., Kang) With

Tε,µ[S, ρ] = ψ

  • S
  • x +

ε 1 + µρv

  • − S (x, t)
  • the solution exists globally.

Mathematical models for cell movementPart II – p. 28

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SLIDE 68

General Picture

Consider a kinetic model Mε with a certain non-dimensional parameter ε > 0.

Mathematical models for cell movementPart II – p. 29

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SLIDE 69

General Picture

Consider a kinetic model Mε with a certain non-dimensional parameter ε > 0. Consider the solution Ψε := (fε, Sε) (microscopic variables),

Mathematical models for cell movementPart II – p. 29

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SLIDE 70

General Picture

Consider a kinetic model Mε with a certain non-dimensional parameter ε > 0. Consider the solution Ψε := (fε, Sε) (microscopic variables), and consider Φε := (ρε, Sε) := (

  • V fεdv, Sε)

(macroscopic variables).

Mathematical models for cell movementPart II – p. 29

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SLIDE 71

General Picture

Consider a kinetic model Mε with a certain non-dimensional parameter ε > 0. Consider the solution Ψε := (fε, Sε) (microscopic variables), and consider Φε := (ρε, Sε) := (

  • V fεdv, Sε)

(macroscopic variables). Let us define the limit Φ := lim

ε→0(ρε, Sε) .

Mathematical models for cell movementPart II – p. 29

slide-72
SLIDE 72

General Picture

Consider a kinetic model Mε with a certain non-dimensional parameter ε > 0. Consider the solution Ψε := (fε, Sε) (microscopic variables), and consider Φε := (ρε, Sε) := (

  • V fεdv, Sε)

(macroscopic variables). Let us define the limit Φ := lim

ε→0(ρε, Sε) .

Question: Which is the set of equations that Φ obey?

Mathematical models for cell movementPart II – p. 29

slide-73
SLIDE 73

General Picture

Model ε > 0 Limit model ε → 0

Mathematical models for cell movementPart II – p. 30

slide-74
SLIDE 74

General Picture

Model ε > 0 Limit model ε → 0 Initial conditions ΨI

ε

− → ΦI := limε→0 ΦI

ε

Mathematical models for cell movementPart II – p. 30

slide-75
SLIDE 75

General Picture

Model ε > 0 Limit model ε → 0 Initial conditions ΨI

ε

− → ΦI := limε→0 ΦI

ε

↓ ↓ Time evolution Mε[Ψε] = 0 M[Φ] = 0

Mathematical models for cell movementPart II – p. 30

slide-76
SLIDE 76

General Picture

Model ε > 0 Limit model ε → 0 Initial conditions ΨI

ε

− → ΦI := limε→0 ΦI

ε

↓ ↓ Time evolution Mε[Ψε] = 0 M[Φ] = 0 ↓ ↓ Final state Ψε(T) ? Φ(T)

Mathematical models for cell movementPart II – p. 30

slide-77
SLIDE 77

General Picture

Model ε > 0 Limit model ε → 0 Initial conditions ΨI

ε

− → ΦI := limε→0 ΦI

ε

↓ ↓ Time evolution Mε[Ψε] = 0 M[Φ] = 0 ↓ ↓ Final state Ψε(T) ? Φ(T) If Φ(t) = lim

ε→0 Φε(t) , t < T

(in some sense) then M is the limit model of Mε.

Mathematical models for cell movementPart II – p. 30