Transcript Theory and its correction

Cole’s paradox revisited
The key assumption in Cole's treatment was the assumption
that all lx in both strategies were = 1. Adding mortality
schedules have clarified some of the results.
The first attempt at correction made the apparent advantage of
iteroparity even smaller. Bryant (1971) showed that
under assumptions of 1) constant litter size and 2) a monotonically decreasing lx = e-mx, or, for that matter, any
monotonically decreasing lx, the value of 'r' for an iteroparous
species is not even increased as much as by adding 1 to the
semelparous litter size.
However, a second attempt was both more ‘realistic’ and more
successful.
Charnov and Schaffer(1973) used a more general approach to
correct for Cole's generous assumption about survivorship.
Their model includes, but separates, pre- and post-
mortality.
The comparison begins again between an annual and a
perennial species.
1) Assume that the litter sizes for the two types of species are
Ba for the annual species and Bp for the perennial species.
2) The survivorship pattern in the first year is l1 = C for both
types. Following reproduction in the first year, the
semelparous species dies, and the perennial species
continues with an annual proportional survivorship px = P.
Compare finite rates of increase, where =er, and t = 1:
N(t+1) =  N(t)
For an annual species, the number alive to breed is C x N(t),
and each of these has a litter of size Ba, or:
N(t+1) = C x N(t) x Ba
and
 a = Ba x C
For the perennial species there is adult survivorship and
repeated reproduction. Counting noses just before
reproduction, so that surviving adults haven't yet added the
additional litters, we get:
N(t+1) = Bp x C x N(t) + P x N(t)
and
 p = Bp x C + P
For an annual species to make up the advantage of iteroparity,
the growth rates must be equal, therefore:
Ba x C = Bp x C + P
Ba = Bp + P/C
In Cole’s model, with immortality both P and C are taken as 1,
their ratio is 1, and the result here is identical to that of Cole.
Bryant’s survivorship pattern is equivalent to saying P is fixed,
and less than C; the advantage under those conditions is < 1.
So, when is iteroparity favored? Consider the P and C in the
three Deevey survivorship patterns:
type I - you might think that P and C are ~ equal, but
remember most species show an initially moderately
high qx, (or equivalently an initial decline in lx) before
extended, flat, values through reproduction, i.e. C is
less than P. The advantage of iteroparity here is larger
than Cole suggested.
type II - for comparisons of real numbers we need to convert
back to a linear scale, so that survivorship is
curvilinear. P and C are more similar, but the curvature
makes it 'flatter' during than before reproduction. The
advantage is ~ what Cole calculated.
type III - its obvious that P is far larger than C, the early part
of the curve is very steep, and the curve is relatively flat
through reproduction. The advantage of iteroparity is
far larger than Cole’s suggestion.
If early survivorship declines more steeply than in the later
period, then the perennial habit is favored to that degree. That
is true of types I and III. Examples: long-lived trees, whales,
etc.
We can also extend this to a more real α > 1. To do it we use
Euler’s equation, knowing that the  = 1 for both iteroparous
and semelparous species. For simplicity, we’ll also use  (=
er).
Age of first reproduction α = k.
For the semelparous species:
1 = Bs x C/k
For the iteroparous species (with terms for each year of
reproduction, beginning with age k):
1 = Bi x C [1/k + P/ k+1 + P2/k+2 + …]
or, using a series sum…
1 = Bi x C/k [1/1-P/ ]
Making these two sums equal…
Bi /Bs = 1-P/
The life history equilibrium leans in favor of iteroparity when
the iteroparous litter can be much smaller than the
semelparous litter which produces the same population
growth rate.
How can that happen?
a) the adult survivorship px or P, is very high, and/or
b) the finite rate of increase (or annual population growth) is
very low.
Cole’s graph showed us (with the restrictions of his
assumptions) that the higher the adult proportional
survivorship the more the advantage to be gained from
iteroparity, since that means many more litters will
be produced over the lifespan. It also showed that the
advantage grew as litters decreased in size.
The terms and quantitative advantages are different in the
more realistic approach (no immortality, at least), but the
basic result is identical, even in terms of the life history
patterns that affect advantage.
An Energetics View of Iteroparity vs. Semelparity
The various approaches used by Cole and and others to test
the advantage of iteroparity are all based on comparison of
population growth rates; they do not consider the position of
the individual committed to a strategy. A final view of the
contrast between iteroparity and semelparity, and when each
should be expected, considers instead the energetics of the
reproducing organism.
A surprising amount of modern ecological theory has come
from application and/or adaptation of ideas from economics.
One of those is the use of cost-benefit analyses to assess
strategies.
Gadgil and Bossert (1970) used cost/benefit analysis to
suggest patterns which lead to iteroparity. At any age the
amount of reproductive effort which should be expended is
determined by balancing the profit gained by reproduction
(measured as the reproductive value at that age against the
costs inevitably incurred.
To compare all possibilities, Gadgil and Bossert defined 3
shapes of profit and cost functions to be plotted against
reproductive effort. The 3 kinds of curves are concave, linear
and convex.
We’ll consider some examples:
Consider an example: cost and profit functions for
reproduction in migrating salmonids The initial effort required to spawn even one egg is the
enormous cost of migration from deep ocean to upstream
fresh water. Thus the curve rises very sharply at low
reproductive effort.
However, the additional cost to spawn more eggs is basically
the metabolic cost of the egg tissue, that is much more modest
and basically flat (i.e. each egg costs about the same amount,
independent of how many have been already formed).
The profit from increases in reproductive effort, i.e. the
number of offspring, increases at least in direct proportion to
the number of eggs. Schooling or other advantages of
offspring groups might make the profit curve not linear, but
slightly concave.
What does this look like?
Which curves might fit the mammalian situation?
The cost function could range from linear (if we consider
the direct costs, including parental care, there is no obvious
reason to suggest either declining or increasing costs per
offspring as litter size changes) to concave (if we also consider
indirect costs, e.g. homeothermic stress, increasing risk of
predator tracking with increased litter size). It probably isn't
strongly concave, but at least slightly non-linear.
The profit function is probably convex, in parallel with what
we've seen for the provisioning of seeds, from the decreasing
individual size of individual offspring with increasing seed
crop size. Again, the curvature is probably not dramatic, at
least in homeothermic animals (birds, mammals) since
homeothermy leads to some sharing of warmth among
members of a brood, and also sharing of teats, which limits
food intake no matter how hard the mother works.
Graphically, what does this look like as a strategy?
Where is the ratio of benefit to cost greatest?
That is the way we evaluate all possible strategies.
Semelparity should be the pattern unless there is some
intermediate reproductive effort that maximizes the benefitcost ratio.
Here is the original figure in its entirety…
profit
cost
cost concave here
profit convex here
When is it advantageous to withhold energy from
reproduction, i.e. to adopt an iteroparous strategy?
1. If the profit function is convex. In this case, above some
intermediate reproductive effort the profits cannot keep pace
with the proportional increase we might expect for
metabolic costs per offspring. With costs increasing faster
than profits, energy should not be spent inefficiently, but
retained for use in the next bout of reproduction. This is just
the 'law of diminishing returns' applied to reproductive
ecology.
The efficiency argument even applies when the ratio
remains constant. For that, the middle case, the total number
of offspring produced from a given amount of energy can be
maximized by reducing reproductive effort, even if benefitcost ratio has no maximization.
2. If the cost function is concave. In this case the cost per unit
gain in fitness becomes too high at high reproductive
effort. Retaining a portion of the energy available, the
animal can produce offspring (or increments in fitness) at a
lower cost per unit in later bouts of reproduction.
3. In all other circumstances animals should wait until the
maximum benefit-cost ratio has been reached, then put all
available energy into reproduction. Under these
circumstances the semelparity is optimum. That includes the
case where profit and cost are both linear. There is no
change in cost per unit with effort,and we now know the
advantage of early reproduction. There is no advantage to
restraint, especially given survivorship factors
4. If more complex functions for benefit and/or cost are
considered, e.g. sigmoid shapes, there will almost always be
some intermediate level of effort at which the curve goes
through an inflection point, i.e. a change from concave to
convex. In most cases that inflection will be the critical
point at which benefit-cost ratio will be maximized, and an
iteroparous strategy will result.
One last evaluation, how long should the period  to  be?
That depends on the security of survivorship (px), and on the
variability in reproductive success. If survival is relatively
assured, no single bout of reproduction is under severe
pressure to produce success. Else, if survivorship is low or
variable, then the pressure for success from any single bout (or
a few) is much higher, and more effort (and a shorter  -  is
likely.
Unpredictability in annual success rate is exactly what we
might expect of species in mature communities. A good
example might be a climax forest. The only places young are
successful is where openings develop due to death or chance
destruction of mature individuals. Such clearings are rare,
chance events. So each year a tree puts out a crop of seeds, but
it may be many years of suppressed growth before even
one seed is successful in such a chance clearing.
In terms of profit-cost functions, the profit function is,
therefore, convex, i.e. the chance of a clearing happening
cannot be missed, so its necessary to put out a seed crop each
year, but little advantage in putting out a huge crop. Instead,
energy is retained to improve survivorship and make possible
a larger number of seed crops.
If some reasonable level of success is relatively assured, i.e.
variance in success is low, then the reproductive span is
shortened, ultimately to semelparity.
Murphy summarized this approach in a 2 x 2 table:
long-lived
short-lived
reproductive success
steady (assured)
reproductive success
variable
is this possible (?)
iteroparous
semelparous
strategies
not possible
The upper right hand box, semelparous strategies, is a
necessary result of the association of lx and mx. If a species is
short-lived, quantifiable as a small e0, then the only pattern to
reproduction which will permit persistence is steady, assured
success. Else, extinction will follow.
The lower, right hand box is also simply explained. This is
what happens to a short-lived species does not have assured
success. Extinction is the inevitable result when a species
attempts reproduction only once (or possibly a few times)
while the variability in reproductive success is high. That
variability ensures that at some time a few bad years will
follow in succession, and prevent an entire mature population
from producing any surviving offspring, i.e. local extinction.
The closest real species come to this is to have facultative
(that is have the potential for) dormancy in offspring. In this
way the low survivorship of parents and low annual success in
offspring can be mitigated through appearance of offspring
(release from dormancy) when chances of offspring success
are high.
In terms of the success of offspring, rather than annual
success rates, this strategy becomes a special case of the upper
right hand box, short adult life, but predictable offspring
success. This strategy is fairly common in weeds, whose
seeds may remain dormant for >100 years, and in some desert
plants.
The box in the upper left has a ?. There are a number of
approaches to indicate why that box should not be occupied by
observed reproductive strategies. The most intuitive of them
considers the effect of such a strategy on species interactions.
This is a species which is long-lived and has predictable
reproductive success. That should make this population very
successful.
If the species in question is a prey item, its predator will
evolve to more extensively utilize a species which is
predictable, either through numerical or functional
responses. The result of increased predation on this prey,
whose life history we are following, will be either a reduced
lifespan or greater variability in reproductive success.
The effective result is to move the strategy for the species out
of the ? box and into either of the more usual strategies.
Example(s) of this move:
A number of long-lived trees (apple trees are an example of
one kind of response) have variable reproductive success;
some years a heavy seed crop is produced (so-called mast
years) and in other years the crop is much smaller.
This can be considered an adaptation to restrict insect (or
other) herbivores by starvation, though it can also be a
necessary response to energetic demands of reproduction. In
this latter sense, it is a cost of reproduction. Having
reproduced heavily in one year, the 'cost' is a very restricted
reproduction in the following year.
There are always a variety of explanations for a pattern.
Variability in reproductive success has also been used as an
explanation for the strategy called ‘bet-hedging’. If there is
variation in reproductive effort and success, and if
reproductive effort and adult survivorship trade-off, then it
pays to reduce reproductive effort in order to live longer and
reproduce more times. Here’s a way of plotting that for
bird clutches (sorry, plant plot unavailable):
First, the data...
Why should there be bet-hedging?
Relative fitness of differing clutch sizes peaks at an
intermediate size. The energy not applied to making the larger
clutches, applied instead to growth and maintenance, i.e.
survivorship makes additional clutches more likely to be
produced.