From an interview with L. C. Birch:
“We found no need to
postulate density regulating factors in any of these case histories.” https://www.science.org.au/node/457707
Background and Major Contributions
Louis Charles Birch (1918-2009) was an influential Australian
ecologist. During World War II, he worked on the problem of whether insects
would destroy stored wheat crops. The silos in Australia were full, and wheat
was being stored in large piles on the ground. He found that insects could
thrive only on the surface of the wheat, since when they were deeper in the
pile, the insects themselves generated so much heat that they could not survive
there.
L. C. Birch’s two most famous
contributions to the field of ecology were:
1) External factors such as weather
affect the population and distribution of animals, as presented in The Distribution and Abundance of Animals (1954),
which he wrote with his advisor, H. G. Adrewartha, at the University of
Adelaide;
2) All living organisms have
consciousness; every species is affected in complex ways by every other species
(following A. N. Whitehead). He won the Templeton prize in 1990, for his work
promoting the intrinsic value of all life.
The
Intrinsic Rate of Natural Increase
Birch’s model for population growth
is density-independent, in contrast to the density-dependent models of Pearl’s
logistic equation and the Lotka-Volterra predator-prey equations. The great
ecologist is aware of the existence of the logistic curve, and of density
driven dynamics; however, he sees the need to establish a simple theoretical
basis for the rate of growth, before considering refinements.
 |
Sitophilus oryzae (rice weevil)
Photo credit: Olaf Leillinger
|
Following P. H. Leslie, Birch
considers a population growing exponentially with unlimited resources, no
predators, and unlimited space. The rice weevil Calandra (Sitophilus) oryzae, multiplying in a large stack of wheat, is close
to such a population.
Birch begins his 1948 paper by
acknowledging his debt to P. H. Leslie, for his use of actuarial tables in
calculating growth rates for voles and rats. He does not neglect to mention
Darwin and Malthus for the concept of exponential population growth, and A. J. Lotka
for the application of human demographic analysis to insect populations.
Biological
significance of the intrinsic rate of natural increase
Birch defines the intrinsic rate of
increase as “the rate of increase per head under specified physical conditions,
in an unlimited environment where the effects of increasing density do not need
to be considered.” This is represented as the constant ‘r’ in the differential
equation dN/dt = rN. The solution is Nt = N0ert, where N0 is
the starting population, Nt is the number of animals at time t, and
r is the infinitesimal (instantaneous) rate of increase. Again following
Leslie, Birch states that this is only true in population with a stable age
distribution. He further points out that ‘r’ is a maximum value, and therefore
represents “the true intrinsic capacity of the organism to increase.”
Since calculating an entire range values
of r for different physical conditions and population densities is outside the
scope of the paper, and since a detailed age distribution such as one obtained
from a human census has never been done for insects, Birch proposes to
calculate r for an insect population from age-specific rates of fecundity and
survival, measured under defined environmental conditions.
Calculation
of the intrinsic rate of natural increase
Birch uses data from his previous
1945 papers, along with an estimate of lifespans for adult females drawn from a
study of Tribolium confusum, to
create a life table (Table 1) for Calandra
oryzae. This table only shows females during their reproductive lifetimes. The weevils are kept at 290C,
in wheat with 14% moisture. The table has columns for x (age in weeks), lx
(probability at birth of being alive at age x), mx (age-specific
fecundity rates), and the product lxmx. The net
reproductive rate, R0, or the ratio of total female births in two
successive generations, is the sum of all the products lxmx
for the population, and R0 = NT/N0.
Next, he sets out to calculate the
mean length of a generation. He starts with the original equation, NT
= N0ert.
Substituting, R0 = erT. Taking the natural log of both sides, T =
ln(R0)/r. But T, or the mean
length of a generation, cannot be obtained without r. So he approximates T from
Table 1 as the mean of a frequency distribution, T = 8.3 weeks, from which he
gets r as 0.57, which he states is low.
Next, Birch uses a summation
approximation of the Lotka-Euler equation to calculate r from the data in Table
1, summarized in Table 2, arriving at r = 0.762. Table 3 shows the relative
contribution of each age group to the value of r. It is evident from the table
that the first two weeks of oviposition (egg-laying) accounts for 85.24% and
the first three weeks accounts for 93.84% of the value of r. Thus overall, the
rate of increase is mostly determined by the first 2-3 weeks of fecundity, even
when the adult females lay eggs for a total of 10 weeks.
The value of r is calculated again,
ignoring the adult life table. This time, r = 0.77. There is almost no
difference because most of the value of r is determined so early in the
reproductive life of the adult female, a time when survival is very high. Birch
has made the point early in the paper that the error introduced by estimating
adult mortality values from data from another species probably is not
significant. Now we see why: because the calculation of r would be almost the
same if the ages with high mortality were deleted from the table.
The
stable age distribution
Birch illustrates the stable age
distribution based on r = 0.76 in Table 4. He mentions equations from Dublin
& Lotka, but does not use them. Instead, he uses an equation from Leslie,
estimating the age-specific survival rates from the midpoint of each age group.
According to this age distribution, the vast majority of the population is
represented by the egg and larval stages (95.5%). Birch attributes this to the
high value of r, and notes that it would be easy to vastly underestimate insect
populations when most of the individuals are hidden inside the grains.
The
effect of temperature on ‘r’
Birch uses Table 5 to illustrate the
different values of r under different temperature conditions. At 23 C, r =
0.43; at 33.5, r = 0.12. In either case, the growth rate is reduced; one
wonders how the rates would have been affected if actual life table data had
been gathered at those temperatures. It is interesting that among the modern
methods of control for Sitophilus oryzae are
to reduce the temperature to -17.7 C for three days, or to increase the
temperature to 60C for 15 minutes.
Summary
As James H. Brown points out in the
introduction to the methodology section, this was “the first comprehensive
analysis of the demography of a population growing exponentially under
carefully controlled conditions.” Birch’s paper also showed a) that an abiotic
factor such as temperature could affect population growth, even when other
resources were not limiting; and b) (in a thought experiment) that decreasing
the age when females first deposited eggs could amplify the intrinsic rate of
natural increase.
Modern
Developments
Since 1948, increases in computing
speed have led to the ability to use equations with stochastic components.
Statistical analysis techniques have become much more powerful. However, the
basic method of estimating growth rates from life tables is still in widespread
use in 2015.
Questions
for Discussion
1. Why does Birch choose to focus so
intently on unlimited exponential growth in insects, rather than on limiting
factors?
2. Why does Birch insist so adamantly
that “intrinsic rate of natural increase” is a better term than “biotic
potential”?
3. How important are the life tables
to Birch’s calculations, and why?
4. Would “r” be higher or lower with
longer generation times? With shorter reproductive span? With lower age of
first reproduction?
I like that this paper combines the math theory of Leslie with an experimental check. While the experimental setup may not be testing “natural” conditions, it does a good job of testing some of the assumptions these models make. It is notable that again this work emphasizes the advantage of reproducing early in an organism’s life. This trend certainly contradicts the success of late reproducers like humans.
ReplyDeleteGreat point, Eric. In human populations, the propensity to live much longer than the span one's reproductive years presumably has an effect on the longevity and fecundity of the younger generation. In other animals, this effect seems not so pronounced--although not completely absent. Are you familiar with studies on this topic?
DeleteThis paper was a great segue from (or to) the Leslie paper. While the Leslie paper was very theoretical and not as applied, this paper took some of the theory of population growth and applied it to an actual population. To address the first of Julie's questions, I think that Birch chose to focus on unlimited exponential growth simply to see the result of an unchecked population. This may seem simplified, but so many of the studies up until this paper have focused on the limiting factors of population growth. While interesting and essential to know, they don't see the result of a population let to expand as much as possible. I think for that reason Birch was influential, and he offered a new perspective on the subject of population growth and related predictions.
ReplyDeleteI also enjoyed this paper. It struck me as almost a reinterpretation of the predator-prey dynamics papers, with a 'what happens if we remove the predators' approach. And although these concepts might seem simplistic or superfluous in the context of modern ecology, they're quite important to the understanding of paleoecology. Combined with life tables, intrinsic growth is really the only way to understand prehistoric community structure beyond what the fossil record can tell us directly.
ReplyDeleteI appreciated Birch’s paper because it was concise. This paper is a great follow up on Leslie’s matrix concept. I like the idea of using an experimental laboratory model to test the matrix, and I thought changing intrinsic rate of increase with temperature was a novel experimental design that relied on previous work, like Leslie and Lotka. Birch’s paper is most like what we see in modern ecology because of the experimental design and statistical aspects.
ReplyDeleteI think this paper is complement the work done by Lotka. Now we have density dependent and independent factor that constraint the rate of population increase in nature. This paper came right on time on right time after the contribution of Lotka and Leslies.
ReplyDeleteThis seems to be a nice use of population biology principles, including some of Leslie's, to a scenario with empirical data from nature. In answer to Julie's questions:
ReplyDelete1) Birch argues that the case of unlimited (density-independent) is of both theoretical and practical significance; theoretical because since it is a common parameter in population mathematics, and practical because it is useful and interested to observe the conditions under which the maximum intrinsic growth rate is reached, or the degree of departure under different conditions. A lot of theoretical models seem to operate this way - they are gross simplifications of reality, making a lot of assumption we know aren't necessarily true, but they nonetheless explain a significant amount of variation and operate as good baseline against which to evaluate other possible mechanisms and phenomenon.
2) Birch feels 'biotic potential' is imprecisely defined and may be interpreted more in physical environmental terms, whereas 'intrinsic rate of natural increase' implies maximum growth rate under given environmental conditions.
3) Life tables are generally useful in Birch's approach because they include age-specific survival and fecundity rates, but in cases where most of the contribution to r comes early in life (e.g. Table 2-3), the lifetable is less important.
4) r would decrease with increased generation time (T; see 4th equation on pg 19). A shorter reproductive period of life would decrease r, but not necessarily by much if fecundity peaks at young ages, contributing disproportionately to r (as in Table 2-3 eg). r would increase with early age of first reproduction, as we learned from Cole. To me none of these seem completely intuitive at first, but make sense with some thought. This is where the math helps!
Thank you, Ali, Kat, Sami, Carlos, and Dunbar, for your appreciation of the power of simplicity! It seems everyone agrees that Birch's pared-down approach of calculating growth rate for an unlimited, density independent population has provided a valuable contribution to modern ecology. And thank you, Dunbar, for addressing each of my questions. I agree that Birch probably felt that "biotic potential" was a messy term. It seems like a qualitative description of sudden expansions in insect populations that early ecologists must have observed.
ReplyDeleteMaybe in class today we can discuss the effect of varying life history traits on the value of the growth rate.