Please, I want help in adjusting the quality of the figure surface shape as in the attached picture (as in color, and colored lines in the horizontal plane)

options = odeset('RelTol',1e-6,'Stats','on'); 
% initial conditions     
Xo = [0.5; 0.7; 2];  
% choose parameters t and a
tspan = linspace(0,100);
a_vec = 0.01 : 0.005 : 0.03;
% ease the coding and readability
num_t = numel(tspan);
num_a = numel(a_vec);
num_x = numel(Xo);
% allocate output
X = zeros(num_t,num_x,num_a);
for idx = 1:num_a
    % Collect all X values into a 3D matrix. Additional parameters to
    % TestFunction can be added after ode45 options
    [t,X(:,:,idx)] = ode45(@TestFunction,tspan,Xo,options,a_vec(idx)); 
end
% make sure we have the proper dimmensions before calling surf
plot_t = repmat(t,1,num_a);
plot_a = repmat(a_vec,num_t,1);
plot_x = squeeze(X(:,1,:));
% create the surf plot
figure
surf(plot_t,plot_a,plot_x)
xlabel('t values')
ylabel('a values')
zlabel('x values')
grid on
function [dx_dt]= TestFunction(~,x,a) 
    r=0.05; k=0.1;  %a=0.02; 
    m=0.02; b=0.2; eta=0.06; h=1;
    dx_dt(1)=r.*x(1).*(1-(x(1)./k))-a.*x(1).*x(3)+x(3).*eta;
    dx_dt(2)=a.*x(1).*x(3)-m.*x(2)-b.*x(2)+h.*eta;
    dx_dt(3)=b.*x(2)-eta.*x(3)-r.*x(1);
    dx_dt = dx_dt';   
end

[Walter Roberson]

Please format as code. Click on the > button in the CODE section of the edit ribbon here, and paste your code into the area that results.

I went to do it for you but I saw you the function definition is all on one line and that parts of it are commented out; there is no way for me to know which parts are commented out or not commented out.

[T K]

options = odeset('RelTol',1e-6,'Stats','on'); 
% initial conditions     
Xo = [0.5; 0.7; 2];  
% choose parameters t and a
tspan = linspace(0,100);
a_vec = 0.01 : 0.005 : 0.03;
% ease the coding and readability
num_t = numel(tspan);
num_a = numel(a_vec);
num_x = numel(Xo);
% allocate output
X = zeros(num_t,num_x,num_a);
for idx = 1:num_a
    % Collect all X values into a 3D matrix. Additional parameters to
    % TestFunction can be added after ode45 options
    [t,X(:,:,idx)] = ode45(@TestFunction,tspan,Xo,options,a_vec(idx)); 
end
% make sure we have the proper dimmensions before calling surf
plot_t = repmat(t,1,num_a);
plot_a = repmat(a_vec,num_t,1);
plot_x = squeeze(X(:,1,:));
% create the surf plot
figure
surf(plot_t,plot_a,plot_x)
xlabel('t values')
ylabel('a values')
zlabel('x values')
grid on
function [dx_dt]= TestFunction(~,x,a) 
    r=0.05; k=0.1;  %a=0.02; 
    m=0.02; b=0.2; eta=0.06; h=1;
    dx_dt(1)=r.*x(1).*(1-(x(1)./k))-a.*x(1).*x(3)+x(3).*eta;
    dx_dt(2)=a.*x(1).*x(3)-m.*x(2)-b.*x(2)+h.*eta;
    dx_dt(3)=b.*x(2)-eta.*x(3)-r.*x(1);
    dx_dt = dx_dt';   
end

[T K]

@Walter Roberson The code mentioned above

Answer 1

[Walter Roberson]

It is not clear to me what changes you want.

If you want a plot that is smoother along the a axis then you ned to use more a_vec values.

options = odeset('RelTol',1e-6,'Stats','on'); 
% initial conditions     
Xo = [0.5; 0.7; 2];  
% choose parameters t and a
tspan = linspace(0,100);
a_vec = 0.01 : 0.005 : 0.03;
% ease the coding and readability
num_t = numel(tspan);
num_a = numel(a_vec);
num_x = numel(Xo);
% allocate output
X = zeros(num_t,num_x,num_a);
for idx = 1:num_a
    % Collect all X values into a 3D matrix. Additional parameters to
    % TestFunction can be added after ode45 options
    [t,X(:,:,idx)] = ode45(@TestFunction,tspan,Xo,options,a_vec(idx)); 
end

34 successful steps
0 failed attempts
205 function evaluations
34 successful steps
0 failed attempts
205 function evaluations
34 successful steps
0 failed attempts
205 function evaluations
34 successful steps
0 failed attempts
205 function evaluations
34 successful steps
0 failed attempts
205 function evaluations

% make sure we have the proper dimmensions before calling surf
plot_t = repmat(t,1,num_a);
plot_a = repmat(a_vec,num_t,1);
plot_x = squeeze(X(:,1,:));
% create the surf plot
figure
surf(plot_t,plot_a,plot_x)
xlabel('t values')
ylabel('a values')
zlabel('x values')
grid on

%adjust color
colormap(jet)

%another way of adjusting color
lighting GOURAUD
shading interp

%colored lines on the horizontal plane
ax = gca;
ax.XAxis.Color = 'b';
ax.YAxis.Color = 'r';

%but maybe this is what you mean about colored lines...

figure();

surfc(plot_t,plot_a,plot_x)
xlabel('t values')
ylabel('a values')
zlabel('x values')
grid on

function [dx_dt]= TestFunction(~,x,a) 
    r=0.05; k=0.1;  %a=0.02; 
    m=0.02; b=0.2; eta=0.06; h=1;
    dx_dt(1)=r.*x(1).*(1-(x(1)./k))-a.*x(1).*x(3)+x(3).*eta;
    dx_dt(2)=a.*x(1).*x(3)-m.*x(2)-b.*x(2)+h.*eta;
    dx_dt(3)=b.*x(2)-eta.*x(3)-r.*x(1);
    dx_dt = dx_dt';   
end

[T K]

Thank you very much for the tip, yes it has been clarified