For anyone interested: I solved to problem by rescaling r
syms a r
r1 = 5*10^(-11);
r2 = 0.01;
r3 = 20;
r4 = 0.1;
r5 = 1.386;
r8 = 0.0723;
r10 = 0.5;
r11 = 0.0001;
n_alpha = 1*10^(3);
r6 = 0.045;
lo_a = 0;
up_a = 10000;
lo_r = 0;
up_r = 1;
scale = 10000;
[A,R] = meshgrid(lo_a:100:up_a,lo_r:0.1:up_r);
a_dot = -r6.*A + r4*n_alpha*r10*r11./(r10*r11+r1*r2*A.^2.*R.^2) + r3*n_alpha*r1*r2*A.^2.*R.^2./(r10*r11+r1*r2*A.^2.*R.^2);
r_dot = -r5.*R + r4*r10*r11./(r10*r11+r1*r2*A.^2.*R.^2) + r3*r1*r2*A.^2.*R.^2./(r10*r11+r1*r2*A.^2.*R.^2);
streamslice(A,R*scale,a_dot,r_dot*scale,2);
xlabel('a')
ylabel('r')
yticks([0*scale 0.5*scale 1*scale])
yticklabels(num2str(yticks'/scale))
xlim([lo_a up_a])
ylim([lo_r*scale up_r*scale])