In the following problems you are given a set of reaction terms \(R_{1}\left(c_{1}, c_{2}\right)\) and \(R_{2}\left(c_{1}, c_{2}\right)\). Determine whether or not a homogeneous steady state can be obtained and whether the system is capable of giving rise to diffusive instability. If so, give explicit conditions for instability to arise, and determine which modes would be most destabilizing. (a) Lotka-Volterra: $$ \begin{aligned} &R_{1}=a c_{1}-b c_{1} c_{2} \\ &R_{2}=-e c_{2}+d c_{1} c_{2} \end{aligned} $$ (b) Species competition: $$ \begin{aligned} &R_{1}=\mu_{1} c_{1}-a_{1} c_{1}^{2}-y_{12} c_{1} c_{2} \\ &R_{2}=\mu_{2} c_{2}-\alpha_{2} c_{2}^{2}-\gamma_{2} c_{1} c_{2} \\ &\left(\frac{\mu_{2}}{\mu_{1}}<\frac{\gamma_{21}}{\alpha_{1}}, \quad \frac{\mu_{2}}{\mu_{1}}<\frac{\alpha_{2}}{\gamma_{12}}\right) \end{aligned} $$ (c) Glycolytic oscillator: $$ \begin{aligned} &R_{1}=\delta-k c_{1}-c_{1} c_{2}^{2} \\ &R_{2}=k c_{1}+c_{1} c_{2}^{2}-c_{2} \end{aligned} $$ (d) Schnakenberg chemical system: $$ \begin{aligned} &R_{1}=c_{1}^{2} c_{2}-c_{1}+b \\ &R_{2}=-c_{1}^{2} c_{2}+a \end{aligned} $$ (e) Van der Pol oscillator: $$ \begin{aligned} &R_{1}=c_{2}-\frac{c_{1}^{3}}{3}+c_{1} \\ &R_{2}=-c_{1} \end{aligned} $$ (f) Phytoplankton-herbivore system (Levin and Segel, 1976 ): $$ \begin{aligned} &R_{1}=a_{c_{1}}+e c_{1}^{2}-b_{1} c_{1} c_{2} \\ &R_{2}=-d c_{2}^{2}+b_{2} c_{1} c_{2} \end{aligned} $$ (g) Tyson-Fife model for Belousov-Zhabotinsky reaction: $$ \begin{aligned} &R_{1}= \\ &\frac{1}{\epsilon} \frac{c_{1}\left(1-c_{1}\right)-b c_{1}\left(c_{2}-a\right)}{c_{2}+a} \\ &R_{2}=c_{1}-c_{2} \end{aligned} $$ (h) Meinhardt (1983) model: $$ \begin{aligned} &R_{1}=e c_{1}^{2} c_{2}-\mu c_{1} \\ &R_{2}=c_{0}-e c_{1}^{2} c_{2} \end{aligned} $$

Step by step solution

01

Title - Identify the Problem

The problems involve determining whether a homogeneous steady state exists and if the system can exhibit diffusive instability given a set of reaction terms for different models. Each model has its unique reaction terms.

02

Title - Finding Steady State Conditions

For each model, set the reaction terms equal to zero and solve for the concentrations \(c_{1}\) and \(c_{2}\). This gives the homogeneous steady states.

03

Title - Evaluate the Jacobian Matrix

Calculate the Jacobian matrix of the reaction terms with respect to the concentrations to analyze the stability of the steady states.

04

Title - Check for Diffusive Instability

Determine the conditions under which at least one eigenvalue of the Jacobian matrix has a positive real part when considering diffusion. This is done by checking the Turing instability criteria.

05

Title - Identify Most Destabilizing Modes

If diffusive instability occurs, identify which modes (patterns of concentration) are the most destabilizing by examining the eigenvalues of the Jacobian for the most positive real part.

06

Title - Apply Procedure to Individual Models

Repeat steps 2-5 for each individual model: Lotka-Volterra, Species competition, Glycolytic oscillator, Schnakenberg chemical system, Van der Pol oscillator, Phytoplankton-herbivore system, Tyson-Fife model for Belousov-Zhabotinsky reaction, and Meinhardt model.

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Key Concepts

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

Homogeneous Steady State

A homogeneous steady state is a condition where the concentrations of all species in a system remain constant over time. To find this state, we set the reaction terms equal to zero and solve for the concentrations. For example, in the Lotka-Volterra model, we set \(a c_{1} - b c_{1} c_{2} = 0\) and \(-e c_{2} + d c_{1} c_{2} = 0\), then solve for \(c_{1}\) and \(c_{2}\). This gives us the steady-state concentrations where the system is in equilibrium.

Jacobian Matrix

The Jacobian matrix helps us analyze the stability of the steady state by considering small perturbations. It is a matrix of partial derivatives of the reaction terms with respect to the species concentrations. For a two-species system, the Jacobian matrix \(J\) can be written as:
\[ J = \begin{pmatrix} \frac{\partial R_{1}}{\partial c_{1}} & \frac{\partial R_{1}}{\partial c_{2}} \ \frac{\partial R_{2}}{\partial c_{1}} & \frac{\partial R_{2}}{\partial c_{2}} \end{pmatrix} \]
By evaluating this matrix at the steady state, we can determine the system's response to perturbations.

Turing Instability

Turing instability occurs when a system that is stable in the absence of diffusion becomes unstable due to diffusion. This instability leads to the formation of spatial patterns. To determine if Turing instability can arise, we need to check the eigenvalues of the Jacobian matrix when considering diffusion terms. If at least one eigenvalue has a positive real part, the system is said to be Turing unstable. This results in the growth of patterns from a homogeneous state.

Lotka-Volterra Model

The Lotka-Volterra model describes the dynamics of biological systems with predator-prey interactions. The reaction terms are:
\[ R_{1} = a c_{1} - b c_{1} c_{2} \] \[ R_{2} = -e c_{2} + d c_{1} c_{2} \]
To find the homogeneous steady state, we set \(R_{1} = 0\) and \(R_{2} = 0\). By analyzing the Jacobian matrix at this steady state, we can check for Turing instability and determine the conditions under which patterns form.

Species Competition

The species competition model describes interactions where species compete for the same resources. The reaction terms are:
\[ R_{1} = \mu_{1} c_{1} - a_{1} c_{1}^{2} - y_{12} c_{1} c_{2} \] \[ R_{2} = \mu_{2} c_{2} - \alpha_{2} c_{2}^{2} - \gamma_{2} c_{1} c_{2} \]
Homogeneous steady state is found by setting these terms to zero. Analyzing the Jacobian matrix at this state lets us determine the stability and if Turing instability can occur for this system.

Glycolytic Oscillator

The glycolytic oscillator is a model describing oscillations in biological systems. The reaction terms are:
\[ R_{1} = \delta - k c_{1} - c_{1} c_{2}^{2} \] \[ R_{2} = k c_{1} + c_{1} c_{2}^{2} - c_{2} \]
Setting \(R_{1} = 0\) and \(R_{2} = 0\) gives the homogeneous steady state. The Jacobian matrix, evaluated at this state, helps us understand stability and identify conditions for Turing instability.

Schnakenberg Chemical System

The Schnakenberg model is a simple chemical reaction system that can exhibit complex behavior. The reaction terms are:
\[ R_{1} = c_{1}^{2} c_{2} - c_{1} + b \] \[ R_{2} = -c_{1}^{2} c_{2} + a \]
To determine the homogeneous steady state, we set these terms to zero. Evaluating the Jacobian matrix at this steady state allows us to check for stability and Turing instability.

Van der Pol Oscillator

The Van der Pol oscillator describes non-linear oscillations and is used in diverse fields like biology and electronics. The reaction terms are:
\[ R_{1} = c_{2} - \frac{c_{1}^{3}}{3} + c_{1} \] \[ R_{2} = -c_{1} \]
Set \(R_{1} = 0\) and \(R_{2} = 0\) to find the steady state. The Jacobian matrix at this state tells us about the system's stability and if diffusive instability (Turing instability) can occur.

Phytoplankton-Herbivore System

This model describes interactions between phytoplankton and herbivores. The reaction terms are:
\[ R_{1} = a_{c_{1}} + e c_{1}^{2} - b_{1} c_{1} c_{2} \] \[ R_{2} = -d c_{2}^{2} + b_{2} c_{1} c_{2} \]
To find the homogeneous steady state, we set these terms to zero. The Jacobian matrix at the steady state shows us if the system is stable and if there’s potential for Turing instability.

Belousov-Zhabotinsky Reaction

The Tyson-Fife model for the Belousov-Zhabotinsky reaction describes a well-known chemical oscillation. The reaction terms are:
\[ R_{1} = \frac{1}{\epsilon} \frac{c_{1}(1-c_{1}) - b c_{1}(c_{2}-a)}{c_{2}+a} \] \[ R_{2} = c_{1} - c_{2} \]
Setting these terms to zero finds the homogeneous steady state. Evaluating the Jacobian at this state helps us understand stability and the potential for Turing instability.

Meinhardt Model

The Meinhardt model represents pattern formation in biological systems. The reaction terms are:
\[ R_{1} = e c_{1}^{2} c_{2} - \mu c_{1} \] \[ R_{2} = c_{0} - e c_{1}^{2} c_{2} \]
Setting \(R_{1} = 0\) and \(R_{2} = 0\) gives the homogeneous steady state. The Jacobian matrix, analyzed at this state, shows if the system is stable and whether Turing instability can generate patterns.