Math 2.0

Partial differential equations Definition

Keller–Segel model

Builds on Parabolic partial differential equationLaplacianGradientDivergence

Definition

Let Ω⊆ℝn\Omega \subseteq \R^n be open and let χ>0\chi > 0, τ≥0\tau \ge 0, α≥0\alpha \ge 0 be constants. The Keller–Segel model is the system for a cell density u(x,t)≥0u(x, t) \ge 0 and a chemical concentration v(x,t)v(x, t): ∂tu=Δu−χ ∇⋅(u∇v),τ ∂tv=Δv−αv+u.\partial_t u = \Delta u - \chi\, \nabla \cdot (u \nabla v), \qquad \tau\, \partial_t v = \Delta v - \alpha v + u. Here Δ\Delta is the Laplacian, ∇\nabla the gradient and ∇⋅\nabla \cdot the divergence in xx. On a bounded Ω\Omega one usually imposes no-flux conditions ∂νu=∂νv=0\partial_\nu u = \partial_\nu v = 0 on ∂Ω\partial\Omega. For τ>0\tau > 0 the system is parabolic; the case τ=0\tau = 0 is the parabolic-elliptic Keller–Segel model.

Remarks

Example: on a bounded Ω\Omega with α>0\alpha > 0, the constants u≡αcu \equiv \alpha c, v≡cv \equiv c with c>0c > 0 form a stationary solution, since all derivatives vanish and −αc+αc=0-\alpha c + \alpha c = 0. Non-example: with χ=0\chi = 0 the cells do not respond to the chemical, so the system decouples into a heat equation for uu; it is excluded by χ>0\chi > 0. The total mass ∫Ωu dx\int_\Omega u\, dx is conserved under the no-flux conditions. Consider the parabolic-elliptic model on ℝ2\R^2 with τ=α=0\tau = \alpha = 0 and χ=1\chi = 1. Solutions with mass less than 8π8\pi and suitable integrability of the initial data exist for all time. Solutions with mass greater than 8π8\pi and finite second moment blow up in finite time.

Reading path

To understand this page from the ground up, read these 13 pages in order. The foundations come first.

Show the path as a graph
Euclidean space, object, no evidence levelEuclidean spaceobjectFunction, concept, no evidence levelFunctionconceptParabolic partial differential equation, concept, no evidence levelParabolic partial dif…conceptTotal derivative, concept, no evidence levelTotal derivativeconceptFunction of several variables, concept, no evidence levelFunction of several v…conceptVector-valued function, concept, no evidence levelVector-valued functionconceptDifferentiable function of several variables, concept, no evidence levelDifferentiable functi…conceptPartial derivative, concept, no evidence levelPartial derivativeconceptVector field, concept, no evidence levelVector fieldconceptGradient, concept, no evidence levelGradientconceptDivergence, concept, no evidence levelDivergenceconceptLaplacian, concept, no evidence levelLaplacianconceptKeller–Segel model, definition, no evidence levelKeller–Segel modeldefinition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Euclidean space object
  2. Function concept
  3. Parabolic partial differential equation concept
  4. Total derivative concept
  5. Function of several variables concept
  6. Vector-valued function concept
  7. Differentiable function of several variables concept
  8. Partial derivative concept
  9. Vector field concept
  10. Gradient concept
  11. Divergence concept
  12. Laplacian concept
  13. Keller–Segel model definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Chemotaxis
About this page
TagCM7D3ES438
URIm2://CM7D3ES438
Kinddefinition
WritingW0 stub
MSC35Q92, 92C17, 35K57
DefinesKeller–Segel model
UsesParabolic partial differential equation, Laplacian, Gradient, Divergence

Markdown · JSON · Edit on GitHub

What this page needs for W1

  • an inventory row: the row of the inventory that the node accounts for (0 of 1)
  • a source with a locator (0 of 1)

The writing levels say how complete the text of a page is. They do not measure its truth.

What links here

No page links to this page yet.