Need help in solving diffrential equation with initial conditions. Kindly help me correct my code.

My objective is to solve for X_1 and X_2 with inital conditon x1(0) and x2(0) for time t=0 to 1000

syms x1(t) syms x2(t)

%constants

k1=0.015;k2=0.025;k3=0.038;

alpha=0;k=7;a=2;

%Initial Conditions

x1(0)=0.45+0.05*alpha;

x2(0)=0.65-0.15*alpha;

t=0:1000;

K_lower=k1+alpha(k2-k1);

K_upper=k3-alpha(k3-k2);

X_1=dsolve(diff(x1,t)==(K_lower)*(x1)*(1-(x2/k))*(x1-a),x1(0)==0.45+0.05*alpha);

X_2=dsolve(diff(x2,t)==(K_upper)*(x2)*(1-(x1/k))*(x2-a),x2(0)==0.65-0.15*alpha);

Error using sym/subsasgn (line 951)

Invalid indexing or function definition. Indexing must follow MATLAB indexing. Function arguments must be

symbolic variables, and function body must be sym expression.

Error in Initial_Conditions_alpha (line 4)

x1(0)=0.45+0.05*alpha;

Answer 1

[Ameer Hamza]

You are defining the initial condition incorrectly. Check this code

syms x1(t) 
syms x2(t)
%constants
k1=0.015;k2=0.025;k3=0.038;
alpha=0;k=7;a=2;

K_lower=k1+alpha*(k2-k1);
K_upper=k3-alpha*(k3-k2);

Eq = [diff(x1,t)==(K_lower)*(x1)*(1-(x2/k))*(x1-a);
    diff(x2,t)==(K_upper)*(x2)*(1-(x1/k))*(x2-a)];
%Initial Conditions
conditions = [x1(0)==0.45+0.05*alpha;
    x2(0)==0.65-0.15*alpha];

X_1 = dsolve(Eq, conditions);

However, dsolve is used to find an analytical solution. It will return an empty vector because an analytical solution of your system does not exist. You need a numerical solver like ode45. Therefore, try this code next

k1=0.015;k2=0.025;k3=0.038;
alpha=0;k=7;a=2;

K_lower=k1+alpha*(k2-k1);
K_upper=k3-alpha*(k3-k2);

odefun = @(x) [(K_lower)*x(1)*(1-(x(2)/k))*(x(2)-a); ...
               (K_upper)*x(2)*(1-(x(1)/k))*(x(2)-a)];
tspan = [0 10];
conditions = [0.45+0.05*alpha; ...
              0.65-0.15*alpha];
          
[t, x] = ode45(@(t,x) odefun(x), tspan, conditions);
plot(t, y);

[syed]

It worked... I am very much grateful to you.

[Ameer Hamza]

Glad to be of help.