Spatiotemporal Modeling and Simulation Exercises

PLEASE NOTE THAT THE DISCRIPTIONS ARE AI GENERATED

This repository contains solutions to a series of exercises focused on spatiotemporal modeling and simulation. Below are descriptions and implementation details for each exercise.

Exercise 5: Implementing Cell List and Verlet List

Objective:

Implement and compare the cell list and Verlet list methods for efficiently managing particles in a simulation.

Description:

• Cell List: Partition the simulation space into a grid of cells. Each particle is assigned to a cell, and interactions are only considered within neighboring cells.
• Verlet List: Maintain a list of nearby particles for each particle.

Implementation Details:

1. Cell List:

   • Create a grid structure based on the simulation domain.
   • Assign each particle to the corresponding cell.
   • For each particle, consider interactions with particles in the same and neighboring cells.

2. Verlet List:

   • For each particle, maintain a list of neighboring particles within a specified cutoff radius.
   • Use the Verlet list to determine potential interactions.

Exercise 6: Modeling Diffusion with Random Walk (RW) and Particle Strength Exchange (PSE)

Objective:

Model the diffusion process using Random Walk (RW) and Particle Strength Exchange (PSE) methods.

Random Walk (RW):

Description:

• Simulate particle movement as a series of random steps.

Implementation Details:

1. Initialization:

   • Initialize particles with random positions in a given domain.

2. Movement:

   • At each time step, move each particle randomly in one of the possible directions.

3. Tracking:

   • Record the positions of particles over time to analyze the diffusion process using binning.

Exercise 7: Modeling the Brusselator Reactions with Diffusion using PSE

Objective:

Simulate the Brusselator reaction-diffusion system using the Particle Strength Exchange (PSE) method.

Description:

The Brusselator is a theoretical model for a type of autocatalytic reaction. This exercise involves combining reaction kinetics with diffusion to model the spatiotemporal behavior of the system.

Implementation Details:

Reaction Kinetics:

The Brusselator model is governed by the following reaction equations:

• ( A \rightarrow X )
• ( 2X + Y \rightarrow 3X )
• ( B + X \rightarrow Y + D )

These reactions can be expressed as differential equations:

• ( \frac{dX}{dt} = A + X^2Y - (B + 1)X )
• ( \frac{dY}{dt} = BX - X^2Y )

Diffusion:

Use the Particle Strength Exchange (PSE) method to simulate the diffusion of reactants.

Steps:

1. Initialization:

   • Initialize particle positions randomly within a given domain.
   • Initialize particle concentrations (X and Y) randomly.

2. Reaction Update:

   • At each time step, update the concentrations of X and Y based on the reaction kinetics.

3. Diffusion Update:

   • Use the PSE method to simulate the diffusion of X and Y.

4. Combine Steps:

   • At each time step, perform both reaction and diffusion updates to simulate the complete reaction-diffusion system.

Exercise 8: Modeling Reaction-Diffusion in the Quorum Sensing Model

Objective:

Simulate the quorum sensing mechanism using a reaction-diffusion model.

Description:

Quorum sensing is a process by which bacteria communicate based on their population density. This exercise models the production, diffusion, and sensing of signaling molecules within a bacterial population.

Implementation Details:

Production and Decay:

• Production: Bacteria produce signaling molecules at a certain rate.
• Decay: Signaling molecules naturally decay over time.

Diffusion:

• Use the Particle Strength Exchange (PSE) method to simulate the diffusion of signaling molecules through the medium.

Sensing and Response:

• Sensing: Bacteria sense the concentration of signaling molecules.
• Response: Bacteria change behavior (e.g., increase production of signaling molecules) when the concentration exceeds a threshold.

Steps:

1. Initialization:

   • Initialize particle positions randomly within a given domain.
   • Initialize signaling molecule concentrations to zero.

2. Production and Decay Update:

   • At each time step, increase the concentration of signaling molecules due to production by bacteria.
   • Decrease the concentration of signaling molecules due to natural decay.

3. Diffusion Update:

   • Use the PSE method to simulate the diffusion of signaling molecules.

4. Sensing and Response Update:

   • At each time step, check the concentration of signaling molecules at each particle's location.
   • If the concentration exceeds a threshold, modify the behavior of the bacteria (e.g., increase production rate).