In the fish population problem, the matrix representing the
transition from one year’s population to the next was
[0.7 0.2]
[3 0 ]
representing a 70% adult survival rate from year to year, a 20% survival rate for young
fish, and the fact that an adult fish produces on average 3 young fish per year.
With these numbers, the population grows at an asymptotic rate of 20% (i.e., a factor
of 1.2) per year. Suppose that we decide to allow fishing, and allow a yearly fraction
of f adult fish to be caught (and we arrange fishing season so that it will not interfere
with th

Question

In the fish population problem, the matrix representing the
transition from one year’s population to the next was
 [0.7       0.2]
 [3            0 ]
representing a 70% adult survival rate from year to year, a 20% survival rate for young
fish, and the fact that an adult fish produces on average 3 young fish per year.
With these numbers, the population grows at an asymptotic rate of 20% (i.e., a factor
of 1.2) per year. Suppose that we decide to allow fishing, and allow a yearly fraction
of f adult fish to be caught (and we arrange fishing season so that it will not interfere
with th

anonymous · Answer

the reproduction of the fish). The new matrix describing this situation is:
A =  [0.7 − f           0.2]
        [3                    0 ]
If we pick f very small (close to zero) the population will still grow. If we pick f too
large, the population will die off. We want to pick f “just right” so that the population
will be stable. What must f be in order for this to happen?