Is there an easy way to make numerical simulations of the ODE of the form dx1/dt=x1(2-x1-x2), dx2/dt=x2(3-x1-x2-x3) for any xn?

I have tried doing for 'n=5' and here is my code;

dx = @(t,x) [x(1); 
    x(1)*(2-x(1)-x(2));
    x(2); x(2)*(3-x(1)-x(2)-x(3));
    x(3); x(3)*(3-x(2)-x(3)-x(4));
    x(4); x(4)*(3-x(3)-x(4)-x(5));
    x(5); x(5)*(2-x(4)-x(5))];
tspan=[0 15];
x0=[0 2 0 3 0 4 0 6 0 8];
[t,x] = ode45(@(t,x) dx(t,x), tspan, x0);
figure(1)
plot(t, x)

Can someone please check if this code is correct, if not, how can I improve if and for any 'n'?. Thanks in advance.

Answer 1

[Abraham Boayue]

 See code comments for answers.  

clear variables
close all;
% dx must have 5 terms for n = 5
% the number terms in system equations
% must be the same as the number of 
% elements in your dx vector.
% Remember your dot before *
dx = @(t,x) [x(1);x(1).*(2-x(1)-x(2)); 
             x(2).*(3-x(1)-x(2)-x(3)); 
             x(3).*(3-x(2)-x(3)-x(4)); 
             x(4).*(3-x(3)-x(4)-x(5))]; 
tspan=[0 15]; 
x0=[0 2 0 3 0]; % same for the initial vector.
[t,x] = ode45(dx, tspan, x0); 
figure(1)
plot(t, x);

[Editor]

Thank you very much Abraham Boayue for your response and your code too does what I want. I have noted the mistakes I had made in my earlier code. However, is there a way in which this code can be be generalised to compute for any 'xn', say 'n=10'?.