Females vary more in fitness (top left case in Table 1)

r1= 1
r2 = 1/2 -k1/(2 k2 W2m/W1m)

parameters={W1m->1,W2m->1/3,W1f->1,W2f->1,k1->1/5,k2->4/5};

popsex=k1 r1+k2 r2/.k2->1-k1//Factor

r2/.parameters

{W2m/W1m,W2f/W1f,k1/k2}/.parameters

popsex/.parameters

If we weight the sex ratio of each patch by the total fitness of that patch:

weightedpopsex=(k1 r1*(W1m+W1f)+k2 r2*(W2m+W2f))/(k1 (W1m+W1f)+k2(W2m+W2f))/.k2->1-k1//Factor

weightedpopsex/.parameters

If we weight the sex ratio of each patch by the total male fitness of that patch (males are the rarer sex):

Maleweightedpopsex=(k1 r1*(W1m)+k2 r2*(W2m))/(k1 (W1m)+k2(W2m))/.k2->1-k1//Factor

Females vary more in fitness (top right case in Table 1)

r1= 1/2 +(k2 W2f/W1f)/(2 k1)
r2 = 0

parameters={W1m->1,W2m->1,W1f->1/3,W2f->1,k1->4/5,k2->1/5};

popsex=k1 r1+k2 r2/.k2->1-k1//Factor

r1/.parameters

{W2m/W1m,W2f/W1f,k1/k2}/.parameters

popsex/.parameters

If we weight the sex ratio of each patch by the total fitness of that patch:

weightedpopsex=(k1 r1*(W1m+W1f)+k2 r2*(W2m+W2f))/(k1 (W1m+W1f)+k2(W2m+W2f))/.k2->1-k1//Factor

weightedpopsex/.parameters

If we weight the sex ratio of each patch by the total female fitness of that patch (females are the rarer sex):

Femaleweightedpopsex=(k1 r1*(W1f)+k2 r2*(W2f))/(k1 (W1f)+k2(W2f))/.k2->1-k1//Factor

Males vary more in fitness (bottom left case in Table 1)

r1= 0
r2 = 1/2 +k1/(2 k2 W2f/W1f)

parameters={W1m->1,W2m->1,W1f->1,W2f->1/3,k1->1/5,k2->4/5};

popsex=k1 r1+k2 r2/.k2->1-k1//Factor

r2/.parameters

{W2m/W1m,W2f/W1f,k1/k2}/.parameters

popsex/.parameters

If we weight the sex ratio of each patch by the total fitness of that patch:

weightedpopsex=(k1 r1*(W1m+W1f)+k2 r2*(W2m+W2f))/(k1 (W1m+W1f)+k2(W2m+W2f))/.k2->1-k1//Factor

weightedpopsex/.parameters

If we weight the sex ratio of each patch by the total female fitness of that patch (females are the rarer sex):

Femaleweightedpopsex=(k1 r1*(W1f)+k2 r2*(W2f))/(k1 (W1f)+k2(W2f))/.k2->1-k1//Factor

Males vary more in fitness (bottom right case in Table 1)

r1=  1/2 -(k2 W2m/W1m)/(2 k1)
r2 =1

parameters={W1m->1/3,W2m->1,W1f->1,W2f->1,k1->4/5,k2->1/5};

popsex=k1 r1+k2 r2/.k2->1-k1//Factor

r2/.parameters

{W2m/W1m,W2f/W1f,k1/k2}/.parameters

popsex/.parameters

If we weight the sex ratio of each patch by the total fitness of that patch:

weightedpopsex=(k1 r1*(W1m+W1f)+k2 r2*(W2m+W2f))/(k1 (W1m+W1f)+k2(W2m+W2f))/.k2->1-k1//Factor

weightedpopsex/.parameters

If we weight the sex ratio of each patch by the total male fitness of that patch (males are the rarer sex):

Maleweightedpopsex=(k1 r1*(W1m)+k2 r2*(W2m))/(k1 (W1m)+k2(W2m))/.k2->1-k1//Factor

Middle column

In the middle column, the sex ratio is constrained from evolving to the sex ratio that would be even if we weighted the patches according to the frequency of the rarer sex.  [Could use more thought...]

r1= 1
r2 =0

Extreme case considered in the discussion:

parameters={W1m->1,W2m->1/3,W1f->1,W2f->1,k1->1/3,k2->2/3};

popsex=k1 r1+k2 r2/.k2->1-k1//Factor

{W2m/W1m,W2f/W1f,k1/k2}/.parameters

popsex/.parameters

If we weight the sex ratio of each patch by the total fitness of that patch:

weightedpopsex=(k1 r1*(W1m+W1f)+k2 r2*(W2m+W2f))/(k1 (W1m+W1f)+k2(W2m+W2f))/.k2->1-k1//Factor

weightedpopsex/.parameters

If we weight the sex ratio of each patch by the total male fitness of that patch (the rarer sex):

Maleweightedpopsex=(k1 r1*(W1m)+k2 r2*(W2m))/(k1 (W1m)+k2(W2m))/.k2->1-k1//Factor

Maleweightedpopsex/.parameters

Extreme case considered in the discussion:

parameters={W1m->1,W2m->0,W1f->1,W2f->1,k1->1/4,k2->3/4};

popsex=k1 r1+k2 r2/.k2->1-k1//Factor

{W2m/W1m,W2f/W1f,k1/k2}/.parameters

popsex/.parameters

If we weight the sex ratio of each patch by the total fitness of that patch:

weightedpopsex=(k1 r1*(W1m+W1f)+k2 r2*(W2m+W2f))/(k1 (W1m+W1f)+k2(W2m+W2f))/.k2->1-k1//Factor

weightedpopsex/.parameters

If we weight the sex ratio of each patch by the total male fitness of that patch:

Maleweightedpopsex=(k1 r1*(W1m)+k2 r2*(W2m))/(k1 (W1m)+k2(W2m))/.k2->1-k1//Factor

Maleweightedpopsex/.parameters