Lauffenburger and Kennedy (1983) suggest a model for the chemotaxis of phagocytes (white blood cells) towards high bacterial densities (part of the tissue inflammatory response to bacterial infection). A set of dimensionless equations that they studied are: $$ \begin{aligned} &\frac{\partial \partial}{\partial t}=\rho \frac{\partial^{2} v}{\partial x^{2}}+\frac{\gamma v}{1+v}-\frac{u v}{\kappa+0} \\ &\frac{\partial u}{\partial t}=\frac{\partial^{2} u}{\partial x^{2}}-\delta \frac{\partial}{\partial x}\left(u \frac{\partial v}{\partial \mathrm{x}}\right)+\alpha(1+\sigma v-u) \end{aligned} $$ where \(\begin{aligned} v= \text { dimensionless bacterial density, } \\ u= \text { dimensionless phagocyte density. } \\ \gamma=\text { ratio of maximum bacterial growth rate to maximum phagocyte } \\ \text { killing rate; } \\\ \sigma= \text { ratio of enhanced phagocyte emigration rate to normal "back- } \\\ \text { ground" emigration rate; } \\ \kappa= \text { ratio of inhibition effect of increasing bacteria density on bacterial } \\ \text { growth to inhibition effect of killing; } \\ \alpha= \text { ratio of phagocyte death rate to maximum phagocytic killing rate. } \end{aligned}\) (See problem 11 of Chapter 10.) (a) Show that the stability of a uniform steady state to uniform (i.e. space- independent) perturbations is governed by the Jacobian matrix $$ \mathbf{J}_{t}=\left[\begin{array}{cc} \frac{\gamma}{(1+v)^{2}}-\frac{\kappa u}{(\kappa+v)^{2}} & -\frac{v}{\kappa+v} \\\ \alpha \sigma & -\alpha \end{array}\right] $$ (b) What would be the corresponding matrix J governing stability of nonuniform perturbations? (c) For the steady state \((\bar{u}, \bar{v})=(1,0)\) show that the eigenvalues of the matrix in part (b) are $$ \lambda_{1}=-\rho q^{2}+\gamma-\frac{1}{\kappa}, \quad \lambda_{2}=-q^{2}-\alpha $$ (d) Which mode (i.e. which value of \(q\) ) is the most likely to cause instability? What is the implication? (e) For the second steady state of the equations, \(\bar{v}>0, \bar{u}=1+\sigma \bar{v}\), show that the stability matrix is $$ \mathbf{J}=\left[\begin{array}{cc} -\rho q^{2}+F(\bar{v}) & -H(\bar{v}) \\ \delta \bar{u} q^{2}+\alpha \sigma & -q^{2}-\alpha \end{array}\right] $$ where $$ F(v)=\frac{v(1+\sigma v)(1-\kappa)}{(1+v)(\kappa+v)^{2}} $$ and $$ H(v)=\frac{v}{\kappa+v} $$ and that eigenvalues have negative real parts provided the following inequalities are satisfied: $$ \begin{aligned} &\text { tr } \mathbf{J}_{t}-(1+\rho) q^{2}<0 \\ &\text { det } \mathbf{J}_{1}+\left(\rho q^{2}+\rho \alpha+\delta \bar{u} H(\bar{v})-F(\bar{u})\right) q^{2}>0 \end{aligned} $$ '(f) Further discuss how diffusive instability might arise in this model. You may wish to refer to the analysis in Lauffenburger and Kennedy (1983).

Step by step solution

01

Establish the Jacobian Matrix

Understand that the Jacobian matrix \( \mathbf{J}_{t} \) needs to be derived to analyze the stability of the system. The Jacobian matrix for part (a) is given by: \[ \mathbf{J}_{t} = \begin{bmatrix} \frac{\gamma}{(1+v)^{2}}-\frac{\kappa u}{(\kappa+v)^{2}} & -\frac{v}{\kappa+v} \ \ \alpha \sigma & -\alpha \ \end{bmatrix} \] which governs the stability of uniform steady states to uniform perturbations.

02

Determine the Jacobian for Nonuniform Perturbations

For nonuniform perturbations, the matrix \mathbf{J}\ incorporates the effect of spatial variations: \[ \mathbf{J} = \begin{bmatrix} -\rho q^{2}+F(\bar{v}) & -H(\bar{v}) \ \ \delta \bar{u} q^{2}+\alpha \sigma & -q^{2}-\alpha \end{bmatrix} \] where \(F(v)\text{ and }H(v)\text{ are defined as given.}\)

03

Steady State \((\bar{u}, \bar{v})=(1,0)\)

For the steady state \(\bar{u} = 1, \bar{v} = 0\), substitute these values into the Jacobian matrix. Simplify to obtain eigenvalues: \[ \lambda_{1} = -\rho q^{2} + \gamma - \frac{1}{\kappa}, \ \lambda_{2} = -q^{2} - \alpha. \]

04

Identify the Mode of Instability

To determine which mode is most likely to cause instability, consider the eigenvalues with respect to \( q \). Analyze which \( q \) value makes \( \lambda_{1} \) or \( \lambda_{2} \) positive.

05

Analyze the Second Steady State

For the steady state \(\bar{v} > 0, \bar{u} = 1 + \sigma \bar{v}\), use the definitions: \[ \mathbf{J} = \begin{bmatrix} -\rho q^{2} + F(\bar{v}) & -H(\bar{v}) \ \ \delta \bar{u} q^{2} + \alpha \sigma & -q^{2} - \alpha \end{bmatrix} \text{where}\ F(v) = \frac{v(1 + \sigma v)(1 - \kappa)}{(1 + v)(\kappa + v)^{2}}, \ H(v) = \frac{v}{\kappa + v}. \]

06

Ensure Eigenvalues Have Negative Real Parts

Show that eigenvalues have negative real parts by satisfying: \[ \text{tr} \mathbf{J}_{t}-(1+\rho) q^{2}<0 \, \ \text{det} \mathbf{J}_{1}+(\rho q^{2}+\rho \alpha+\delta \bar{u} H(\bar{v})-F(\bar{u})) q^{2}>0. \]

07

Discussion on Diffusive Instability

Discuss how diffusive instability might arise. Refer to the Lauffenburger and Kennedy (1983) analysis for details on how specific parameters influence the stability and instability in the system.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

stability analysis

Stability analysis is a crucial method used to determine the behavior of a system near its steady state. In the context of chemotaxis models, we examine whether small perturbations or disturbances around the steady state grow or decay over time.
When analyzing stability, we often start by finding the steady states of the system. These are the points where the system remains unchanged over time if it starts there. Once we have the steady states, we introduce small perturbations and study their evolution. This involves calculating the Jacobian matrix of the system at the steady state and analyzing its eigenvalues.
If all the eigenvalues of the Jacobian matrix have negative real parts, the system is stable at that steady state. If any eigenvalue has a positive real part, the system is unstable at that steady state.

Jacobian matrix

The Jacobian matrix is a fundamental tool in stability analysis. It provides a linear approximation of a nonlinear system around a given point, typically a steady state. In the context of the chemotaxis model, the Jacobian matrix for a uniform steady state is given by:
$$ \mathbf{J}_{t} = \begin{bmatrix} \frac{\gamma}{(1+v)^{2}}-\frac{\kappa u}{(\kappa+v)^{2}} & -\frac{v}{\kappa+v} \ \alpha \sigma & -\alpha \end{bmatrix} $$
This matrix contains partial derivatives of the system's equations with respect to its variables. Each element in the Jacobian matrix represents how a slight change in one variable affects the entire system.
By analyzing the Jacobian matrix at a steady state, we can determine the system's stability. Specifically, we compute the eigenvalues of the Jacobian matrix.

eigenvalues and eigenvectors

Eigenvalues and eigenvectors are key concepts in linear algebra used to analyze the behavior of linear systems. In the stability analysis of dynamical systems, eigenvalues help determine whether perturbations grow or decay over time.
To find the eigenvalues of a matrix, we solve the characteristic equation:
$$ \text{det}(\mathbf{J} - \lambda \mathbf{I}) = 0 $$
where \( \mathbf{J} \) is the Jacobian matrix and \( \lambda \) represents the eigenvalues.
For the chemotaxis model, the eigenvalues of the Jacobian matrix provide information about the stability of the steady state:
For the steady state \( ( \bar{u}, \bar{v} ) = (1,0) \), the eigenvalues are:
$$ \lambda_{1} = -\rho q^{2} + \gamma - \frac{1}{\kappa}, \ \lambda_{2} = -q^{2} - \alpha $$
Analyzing these eigenvalues helps predict which modes (i.e., values of \( q \) ) are most likely to cause instability.

steady state

A steady state in a dynamical system is a situation where all variables remain constant over time. In the context of the chemotaxis model, steady states represent conditions where the bacterial and phagocyte densities do not change.
For instance, one of the steady states in the model is \( ( \bar{u}, \bar{v} ) = (1,0) \). This means that when the phagocyte density is 1 and the bacterial density is 0, the system remains in equilibrium.
Another steady state is characterized by \( \bar{v} > 0 \ \bar{u} = 1 + \sigma \bar{v} \). This represents a scenario where both bacterial and phagocyte densities are non-zero and balanced in such a way that the system stays in equilibrium.
Studying the stability of these steady states involves calculating the Jacobian matrix at these points and analyzing the resulting eigenvalues.

perturbations

Perturbations are small deviations from a steady state, used to study the stability of that state. In stability analysis, we introduce perturbations to see how the system responds.
For the chemotaxis model, uniform perturbations do not depend on spatial variations, while nonuniform perturbations incorporate spatial differences.
The stability to uniform perturbations is governed by the Jacobian matrix:
$$ \mathbf{J}_{t} = \begin{bmatrix} \frac{\gamma}{(1+v)^{2}}-\frac{\kappa u}{(\kappa+v)^{2}} & -\frac{v}{\kappa+v} \ \alpha \sigma & -\alpha \end{bmatrix} $$
For nonuniform perturbations, the matrix incorporates spatial variations:
$$ \mathbf{J} = \begin{bmatrix} -\rho q^{2} + F(\bar{v}) & -H(\bar{v}) \ \delta \bar{u} q^{2} + \alpha \sigma & -q^{2} - \alpha \end{bmatrix} $$
By studying how these perturbations affect the system, we can determine its stability or instability under different conditions and parametric values.