2024 MCSB bootcamp

This is a cool project from UCI MCSB designed by Dr. Jun Allard.

Let’s try to build some cool math model about Discrete logistic growth!!

Unbounded Growth Model

Suppose a rabbit colony has a population x(n) at month n, where x is measured in thousands. If the population were growing in an unbounded environment, the population obeys

The unbounded growth model is given by the equation:

where r is the per-capita growth rate.

Bounded Growth Model (Logistic Growth)

Suppose if instead the population is in a bounded environment (like an island), growth is limited, and the population obeys The bounded growth model, which includes a carrying capacity, is represented by the equation:

In these equations:

• ( r ) is the per-capita growth rate.
• ( K ) is the carrying capacity.
• ( x_n ) is the population at step ( n ).
• ( x_{n+1} ) is the population at the next step.

where K is a parameter we refer to as the carrying capacity. Suppose r = 2.5 and K = 0.6 Try to plot see if it 4 cycle function

% This is a simple MATLAB script
x = linspace(0, 2*pi, 100);
y = sin(x);
% Parameters
n =500;           % Number of years (time steps)
r = 2.5;          % Per-capita growth rate
K = 0.6;          % Carrying capacity
x0 = 0.2;         % Initial population

% Initialize population arrays
x_unbounded = zeros(1, n+1);
x_bounded = zeros(1, n+1);

% Set initial population
x_unbounded(1) = x0;
x_bounded(1) = x0;
%% Generate time series for unbounded and bounded population
for i = 1:n
    % Unbounded population model
    x_unbounded(i+1) = x_unbounded(i) + r * x_unbounded(i);

    % Bounded population model
    x_bounded(i+1) = x_bounded(i) + r * (1 - x_bounded(i)/K) * x_bounded(i);
end
% Plot the time series
figure;
subplot(2,1,1);
plot(0:n, x_unbounded, '-o');
title('Unbounded Population Growth x0=0.2 r=2.5 K=0.6');
xlabel('Time (Years)');
ylabel('Population');

subplot(2,1,2);
plot(0:n, x_bounded, '-*');
title('Bounded Population Growth x0=0.2 r=2.5 K=0.6');
xlabel('Time (Years)');
ylabel('Population');

Now play a little bit with r values, try to find 3-cycle ``` % This is a simple MATLAB script %% Experiment with different r values to find a 3-cycle r_values = [2.47, 2.48, 2.49, 2.6, 2.7]; for r = r_values x_bounded = zeros(1, n+1); x_bounded(1) = x0;

for i = 1:n
    x_bounded(i+1) = x_bounded(i) + r * (1 - x_bounded(i)/K) * x_bounded(i);
end

last_half = x_bounded(floor(n/2):end);
unique_values = unique(last_half);

if length(unique_values) == 3
    disp(['3-cycle found at r = ', num2str(r)]);
    disp('Unique values in the last half of the time series:');
    disp(unique_values);
end end %% Parameters K = 0.6;          % Carrying capacity x0 = 0.2;         % Initial population % Parameter sweep r_values = linspace(0, 3, 1000);  % Parameter sweep for r max_iter = 1000;  % Maximum iterations for each r value which is n last_iter = 500;  % Analyze the last 500 iterations

% Initialize matrix to store results cycle_values = [];

for r = r_values % Initialize population array x_bounded = zeros(1, max_iter+1); x_bounded(1) = x0;

% Calculate the population for each time step
for i = 1:max_iter
    x_bounded(i+1) = x_bounded(i) + r * (1 - x_bounded(i)/K) * x_bounded(i);
end

% Analyze the last few iterations
last_half = x_bounded(end-last_iter:end);
unique_values = unique(last_half);

% Store the results
cycle_values = [cycle_values; repmat(r, length(unique_values), 1), unique_values']; end

% Plot the bifurcation diagram figure; plot(cycle_values(:,1), cycle_values(:,2), ‘k.’, ‘MarkerSize’, 1); title(‘Bifurcation Diagram for 0 < r < 3.0 with K = 0.6’); xlabel(‘Growth Rate (r)’); ylabel(‘Population’);