Mathematical-Model.

Population Growth Simulator

This Java program simulates population growth using the logistic growth model and the Lotka-Volterra model. It generates random parameters for the growth models and simulates the growth over a specified period of time.

Getting Started

To run the simulation, simply execute the PredictTheGrowth.java file in your Java environment. Ensure you have Java installed on your system.

javac PredictTheGrowth.java
java PredictTheGrowth

Overview

The program randomly generates parameters for two types of growth models:

Logistic Growth Model

• Growth rate (r): Between 0 and 0.5
• Carrying capacity (K): Between 0 and 2000
• Initial population size (N0): Between 0 and 100
• Time step (dt): Between 0 and 0.5
• Total simulation time: Between 0 and 500

It then simulates population growth using the logistic growth equation:

dN/dt = r * N * (1 - N / K)

Lotka-Volterra Model

• Prey birth rate (alpha): Between 0 and 0.5
• Predation rate coefficient (beta): Between 0 and 0.5
• Predator-prey interaction rate (delta): Between 0 and 0.5
• Predator death rate (gamma): Between 0 and 0.5

It then simulates population dynamics using the Lotka-Volterra equations:

dx/dt = alpha * x - beta * x * y
dy/dt = delta * x * y - gamma * y

Bayesian Inference Simulation

This Java program implements Bayesian inference to update beliefs about the probability of grass being wet given observations about rain and sprinkler activity. Here's how it works:

• Prior probabilities (pW, pR, pS): These represent the initial beliefs about the probability of the grass being wet, the probability of rain, and the probability of the sprinkler being on, respectively.
• Conditional probabilities (pWGivenR, pWGivenNR, pWGivenS, pWGivenNS): These represent the likelihood of observing wet grass given different combinations of rain and sprinkler activity.
• Observation (r, s): These are boolean variables representing whether it rained (r) and whether the sprinkler was on (s).
• The calculatePosteriorProbability function uses Bayesian inference to update the prior beliefs based on the observed data.
• Finally, the program outputs the final posterior probability of the grass being wet given the observations.

Black-Scholes

This Java program calculates the price of a European call option using the Black-Scholes model. Here's how it works:

• The blackScholesCall function implements the Black-Scholes formula for pricing a call option.
• It takes parameters such as the current stock price (s), the strike price (x), the risk-free interest rate (r), the time to expiration (t), and the volatility (v).
• Inside the main function, random values are generated for these parameters within certain ranges.
• The roundToFiveSignificantDigits function rounds a given value to five significant digits.
• Using the generated random data, the program calculates the price of a European call option using the Black-Scholes formula and outputs the result.

Output

The simulation results are printed to the console, displaying the population size at each time step for both growth models. Additionally, a plot of the results can be implemented using appropriate plotting libraries.

File Structure

• PredictTheGrowth.java: Contains the Java code for the population growth simulation.
• BayesianInferenceSimulation.java: Contains the Java code for the Bayesian inference simulation.
• BlackScholes.java: Contains the Java code for the Black-Scholes option pricing.

Author

This program was created by Samyam.81[https://github.com/samyam81].