how to modify code for distributed delay

I have a code, which gives a solution of a delay logistic equation with discrete delay.

tau = 1;
tspan = [0 20];
y0 = 0.5;
sol = dde23(@ddefunc, tau, y0, tspan);
% Plot the solution
figure;
plot(sol.x, sol.y, 'LineWidth', 2);
xlabel('Time (days)');
ylabel('Population');
legend('y');

% Define the delay differential equation
function g = ddefunc(t, y, Z)
r = 1.5; 
y_tau = Z;
g = r * y * (1 - y_tau);
end

Now I want to modify my code for distributed delay as attached below.

Can someone guide me how to deal with distributed delay

Answer 1

[Torsten]

r = 1.5;
fun = @(t,x)[r*x(1)*(1-x(2));x(1)];
x0 = [0.5;0];
tspan = [0 20];
[t,x] = ode45(fun,tspan,x0);
plot(t,x(:,1))
grid on

[William Rose]

@Muhammad, @Torsten, nice answer.

[Muhammad]

@William Rose,@Torsten Thank you for your response! However, I think there might be some confusion. My original question was about how to modify the delay logistic equation code to account for a distributed delay rather than a simple delay.

[Torsten]

@Muhammad

This is the solution for equation (1.6) with r=1.5 and x(0) = 0.5.

If you have something different in mind, you must post it.

Note that the second equation

dy/dt = x, y(0) = 0

gives

y(t) = integral_{tau=0}^{tau=t} x(tau) dtau

as solution, thus the integral in (1.6).

[Muhammad]

Thank you for your clear response!