Population Growth Model Development

Research in cell and tissue engineering often involves growing cells in the lab in a dish. Imagine

having a single 120 cm2 dish that has been seeded with 1500 cells/cm2. The dish can only

sustain 9x107 cells. With this seeding of the dish, the maximum that the dish can sustain

(carrying capacity) is reached in about 20-25 days. The growth rate, r, of this cell type is known

to be 0.75 cells per day (𝑏𝑖𝑟𝑡ℎ 𝑟𝑎𝑡𝑒 ― 𝑑𝑒𝑎𝑡ℎ 𝑟𝑎𝑡𝑒).

This cell population follows a logistic population growth model:

𝑑𝑃(𝑡)

𝑑𝑡 = 𝑟𝑃(𝑡)(1 ― 𝑃(𝑡)/𝐾 ),

where P(t) is the size of the population at time, t, K is a constant corresponding to the

saturation level (carrying capacity) and r > 0 is the birth rate.

1. Write a MATLAB script for the numerical solution of this cell population problem

utilizing the Euler differential equation solver as demonstrated in class.

[Torsten]

It is necessary to show your code so far to get help in this forum.

[Maya]

%% Problem One

% Research in cell and tissue engineering often involves growing cells in the lab in a dish.

% Imagine having a single 120 cm2 dish that has been seeded with 1500 cells/cm2. The dish can

% only sustain 9x107 cells. With this seeding of the dish, the maximum that the dish can sustain

% (carrying capacity) is reached in about 20-25 days. The growth rate, r, of this cell type is

% known to be 0.75 cells per day (birth rate-death rate).

clear; close all; clc;

%Graph 1: Time in Hours (Derivative)

dt=0.01;

Y0=1;

tlast=700;

iterations=round(tlast/dt);

Yall=zeros(1,iterations);

Y=Y0;

for k=1:iterations

Yall(k)=Y;

dYdt=Y/exp(Y);

Y=dt*dYdt+Y;

end

t=dt*(0:iterations-1);

figure('Color','White')

plot(t,Yall,'r','linewidth',2)

xlabel('Time (hours)','FontSize',14)

ylabel('Population','FontSize',14)

title('Cell Population vs Time','Fontsize',18)

%Graph: P(t)

clear

clc

close all

[Torsten]

If a differential equation reads

dy/dt = f(t,y),  y(0) = y0

the explicit Euler method approximates the solution as

y(1) = y0;                             % Initial value at time t(1)
for i = 1:N-1
    y(i+1) = y(i) + dt*f(t(i),y(i))   % Compute approximation for the solution at times t(2),t(3),...,t(N)
end

Can you transfer this loop to your problem ?

[Maya]

I keep trying to input your code, but it consistently gives me a Parse error at '=' and at ')' for line 29 (in my script, line 29 is dy/dt = f(t,y);)

[Torsten]

The part you could use is only the lower part of what I posted, not the upper line which is the mathematical problem description.

Answer 1

[Denish]

% Years from 1950 to 2020 years = 1950:2020;

% World population data (in billions) population = [2.557 2.768 2.981 3.165 3.333 3.552 3.692 3.842 4.035 4.442 ... 4.853 5.308 5.726 6.114 6.496 6.866 7.346 7.814 8.301 8.827 ... 9.357 9.847 10.325 10.771 11.173 11.515 11.837 12.123 12.374 12.606 ... 12.816 13.021 13.227 13.426 13.621 13.815 14.008 14.197 14.384 14.570 ... 14.756 14.942 15.128 15.315 15.503 15.691 15.879 16.069];

% Plotting plot(years, population, 'LineWidth', 2); title('World Population Over Time'); xlabel('Year'); ylabel('Population (in billions)'); grid on;