As a start, you could define three delays, namely delay(1) = tau-gamma, delay(2) = tau and delay(3) = tau+gamma, and approximate the integral as "gamma * ( Z(:,1)/2 + Z(:,2) + Z(:,3)/2 )" (trapezoidal rule with three points).
In principle, you can approximate the integral arbitrarily close by choosing a sufficient number of delays:
tau = 1;
gamma = 0.5;
number_of_delays = 11; % should be odd
lags = linspace(tau-gamma,tau+gamma,number_of_delays);
tspan=[0 600];
sol=ddesd(@(t,y,Z)ddefunc(t,y,Z,lags),lags,[0.2; 0.08],tspan);
p=plot(sol.x,sol.y);
set(p,{'LineWidth'},{2;2})
title('y(t)')
xlabel('Time(days)'), ylabel('populations')
legend('x','y')
function yp = ddefunc(~,y,Z,lags)
a=0.1;
b=0.05;
c=0.08;
d=0.02;
yl1 = trapz(lags,Z(2,:));
yp = [a*y(1)-b*y(1)*yl1;
c*y(1)*y(2)-d*y(2)];
end
