Note that r/p(x) is constant at the ages before reproduction starts, declines after that until reproduction stops, and is zero postreproduction. However,r/m(x) always decreases with age if r > 0.
Assume that there is a fixed age ß after which reproduction ceases, so that m(x) = 0 for x > ß. Assume also that mutations affecting survival are always deleterious. Then for x > ß there is no selection to balance the mutational pressures, and it must follow that survival rate will evolve to zero. This is what I will call the salmon limit for mortality, because it corresponds to a catastrophic increase in mortality beyond age ß.
It is important to note that this result is unaffected by several changes in the assumptions. First, suppose that we have antagonistic pleiotropy, defined specifically by saying that some mutations have a beneficial effect early in life (i.e., before age ß) and also have deleterious effects late in life. Such mutations will be favored, but they will only accelerate the accumulation of postreproductive deleterious effects. Therefore the salmon limit is stable under such mutations.
Second, remove the assumption of a fixed age of last reproduction and allow a tradeoff between mortality and fertility at every age, so that a positive change in p(x) comes at the expense of a negative change in m(x). Such a tradeoff alters the values of the derivatives in Equations 2 and 3 but does not alter the fact that the selective pressure measured by [r/m(x)] declines with age x. Consequently, with deleterious mutations affecting survival there will always be some age ß past which the mutation-selection balance will be dominated by mutations and survival will fall catastrophically. This is precisely the pattern observed by Charlesworth (1990).
For a different limit, let us constrain total reproduction, so that, for example, the total lifetime reproduction is fixed. Now assume that there are both beneficial and deleterious mutations affecting survival and fertility. Because of the age pattern of the selection coefficients, mutations with beneficial effect on early fertility will be most strongly selected. In other words, selection will act to increase early reproduction even at the cost of late reproduction. As long as a supply of some mutations that can increase early reproduction exists, the mortality schedule will be compressed toward earlier ages, leading to a collapse into a one-age-class life cycle. This is what I call the bacterium limit.
Although it is certainly true that we can observe both salmon and bacteria