Population Growth Model Development

2023 2 2
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更新时间：2024 5 18
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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.

5 个评论
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Torsten
Torsten 2023-2-2
It is necessary to show your code so far to get help in this forum.
Maya
Maya 2023-2-2
%% 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
Torsten 2023-2-2
编辑：Torsten 2023-2-2
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
Maya 2023-2-2
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
Torsten 2023-2-2
The part you could use is only the lower part of what I posted, not the upper line which is the mathematical problem description.

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回答（1 个）

Denish
Denish 2024-5-18

0 个投票

% 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;

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Maya
Maya
2023-2-2

回答：

Denish
Denish
2024-5-18

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