First, we will solve this problem in the presence of red squirrels.
We have to figure out m0, the number of offspring born in the next year to individuals that were just born. This will equal the probability of surviving until the next year (0.25*0.5) times the expected number of offspring (4), so m0=0.5 in the absence of red squirrels.
m1, the number of offspring born in the next year to individuals that are currently one-year olds, is equal to the probability of surviving until the next year (0.5) times the expected number of offspring (8), so m1=4.
Finally, the probability of surviving from age class 0 to age class 1 is 0.25*0.5 = 0.125, in the absence of red squirrels. All of the individuals that were in age class 1 previously will have died over winter (0.5) or immediately after breeding (the remainder), so the probability of surviving from age class 1 to age class 2 is zero.
The Leslie matrix in this case is: {{0.5, 4},{0.125,0}}.
The eigenvalues of this matrix are 1 and -0.5. The largest eigenvalue is 1 and therefore the population will remain roughly constant in size over time. The eigenvector associated with this eigenvalue is {0.89, 0.11} (normalized to sum to one). You conclude that 89% of the individuals in the summer census should be newborn.
Now, you analyse the case with red squirrels absent. The parameters of the Leslie matrix become:
m0 = 4*0.75*0.5 = 1.5
m1 = 8*0.5 = 4
p0 = 0.75*0.5 = 0.375
The Leslie matrix in this case is: {{1.5, 4},{0.375,0}}.
The leading eigenvalue is now 2.19 and its eigenvector would be {0.85,0.15}. This suggests that the population would increase dramatically (doubling in size every year) if the red squirrels were removed and that a higher proportion of yearlings would be seen in the population.