The Components of Predation as Revealed by a Study of Small-Mammal Predation of the European Pine Sawfly, by C. S. Holling (1959)

C. S. Holling proposes to create a comprehensive theory that breaks down predation into two basic components: 1) functional response, or change in number of prey consumed per predator as prey density rises; and 2) numerical response, or change in density of predators as prey density rises. The two main variables he considers are prey density and predator density.

Holling calls the calculation of rate of growth without considering limiting factors “whimsy.” He comments that several researchers (including Nicholson, of the 1934 paper we read on parasite-host dynamics) have focused too myopically on various aspects of predation, and states a need for a more general theory.

Approach: Holling combines field and laboratory experiments. His model system has the advantage that it avoids many complications. In the field, three small mammals prey on one cocooned insect in an even layer of pine needles, under a uniform canopy of pines. The three main predators are the masked shrew (Sorex), the short-tail shrew (Blarina), and the deer mouse (Peromyscus). In the laboratory, variables that are constant in the field could be varied to extend the scope of the results.

In the field, Holling and his associates sample and estimate mammal and cocoon numbers from areas of different prey densities caused by spraying viruses of differing concentrations. In the lab, they vary prey density and the amount and type of alternate food available. The number of prey eaten, in addition to the identity of each predator, can be determined from scrutinizing the marks on the opened cocoons.

Results for basic components: In Figure 1, the functional responses of the three predator species are plotted against prey density. As the density of prey rises, the number of cocoons opened increases in an “S” curve for each predator, which levels out at different densities. The rate of increase is greatest for Blarina, least for Peromyscus, and Sorex is between the two. Analysis of Peromyscus stomach contents in the field, as well as functional response in the lab, support these data.

Figure 3 plots numerical response, or predator density against prey density. Increasing prey density apparently has an effect on prey density for two species. Holling states that for these two species, Sorex and Peromyscus, he has demonstrated that predator density is a “response” to prey density, but as he does not mention any correlation calculations or p-values, we would be justified in remaining skeptical about whether a causal relationship has been established.

The effect of predator density is tested briefly. Different densities do not result in different functional responses, so predator density is not taken into account in the totals for Figure 4.

In Figure 4, Holling combines the functional and numerical “responses” for each species by multiplying them, converts them to percentages, and plots them against prey density. Each shows a peaked curve, which in Blarina only reflects the functional response, since it showed no numerical response.

Results of varying subsidiary components:

Figure 5 is an aesthetically pleasing 3-D graph, showing that one deer mouse does not eat as many cocooned saw flies when they are buried deeper in the sand as when they are buried shallower. Figure 6 shows that one deer mouse decreases its saw fly consumption less when dog biscuits (unpalatable) are available than when sunflower seeds (palatable) are available.

Discussion:

Figure 7 is a theoretical model showing regulation of prey by predators. A horizontal line marks the ranges in percent predation where the prey birth rate = prey death rate. He states that regulation happens “when there is a rise in percent predation over some range of prey densities and an effective birth-rate that can be matched at some density by mortality from predators.” In his rambling discussion, Holling considers various models that have been proposed for oscillations of animal populations. He mentions Nicholson and Bailey’s prediction that oscillations in host (prey) numbers will increase in amplitude, and suggests that small mammal predation (in Holling’s system) can damp oscillations in prey population through the functional and numerical responses.

Holling compares his results to other systems explaining predator-prey interactions, with emphasis on Errington’s concept of compensatory predation. He then postulates four major types of predation in Figure 8, based on combining four different functional response curves with three types of numerical response.

Questions:

1. What do you think of Holling’s use of the term “anthropocentric”? Do you think he is successful in making his focus more objective?

2. What is the main point of this paper? What has Holling actually said, other than that small mammals eat more food when more food is available until they are full, and also sometimes congregate where there is more food?

3. Is statistical analysis really completely absent from this paper?

4. Holling describes various oscillating populations at some length in his discussion, but none of his graphs show oscillations. Why?

The Strategy of Ecosystem Development: An understanding of ecological succession provides a basis for resolving man’s conflict with nature. – E.P. Odum (1969)

In this article, Odum suggests the application of successional theory to ecological system studies in order to better understand the development of ecosystems. According to the author, until this article was written in 1969, the majority of the understanding of the how ecosystems developed relied upon descriptive data and “highly theoretical assumptions” with very little attempt to experimentally test hypotheses. Odum identifies the lack of experimental work in ecosystem ecology is the misinterpretation of the definition of succession. Stressing the idea that succession consists of a complex interaction of processes. In applying a successional model to ecosystem studies, he uses a tabular model to outline “components and stages of development at the ecosystem level as a means of emphasizing those aspects of ecological succession that can be accepted on the basis of present knowledge, those that require more study, and those that have special relevance to human ecology” (596[262]).

Definition of Succession

Odum defines succession according to three parameters:

(1)     It is an orderly process of community development that is reasonably directional and, therefore predictable.

(2)     It results from modification of the physical environment by the community; that is, succession is community-controlled even though the physical environment determines the pattern, the rate of change, and often sets limits as to how far development can go.

(3)     It culminates in a stabilized ecosystem in which maximum biomass (or high information content) and symbiotic function between organisms are maintained per unit of available energy.

(596[262])

In Table 1, Odum outlines 24 attributes of ecological systems grouped into six major structural and functional characteristics of ecosystems.

Bioenergetics of Ecosystem Development

Bioenergetics of the ecosystem are defined by attributes 1-5 of Table 1. These attributes refer to changes within the ecosystem itself. The theory in this is that as the ratio (P/R) of the rate of primary production or total (gross) photosynthesis (P) and the rate of community respiration (R) approaches 1, succession occurs. “energy fixed tends to be balanced by the energy cost of maintenance…in the mature or ‘climax’ ecosystem’ (597[263]).

Components of Succession in a Laboratory Microcosm and a Forest

Odum then discusses the above bioenergetics theory using laboratory aquatic microsystems from ponds cultured by Beyers (1963[6]). During the initial stages of the experiment, “the daytime production (P) exceeds nighttime respiration (R), so that biomass (B) accumulates in the system” (597[263]). After the initial bloom, both rates decline eventually becoming nearly equal with the B/P ratio increasing as the microenvironment approaches steady state.

Although Table 1 refers to changes brought by biological processes within the ecosystem, Odum also stresses the importance of outside influence on Eutrophication. In the case of lakes, nutrients are imported to the lake from the watershed. Thus, he concludes that more oligotrophic conditions can be restored by slowing of nutrient input. This, in turn, can be used to deal with water pollution problems. However, this can only be achieved through functional studies of the larger landscape.

Food Chains and Food Webs

Changes in the food chains also occur as ecosystems develop. Low diversity in the early stages of ecosystem development maintain a simple and linear food chain. As the system matures, the chains become more and more complex creating food webs. Within these webs, the Odum found that the majority of biological energy flow follows a detritus pathway (598[264]).

Diversity and Succession

Within Table 1 (items 8-11), Odum lists four components of diversity: (8) species diversity – variety component; (9) species diversity – equability component; (10) biochemical diversity; and (11) stratification and spatial heterogeneity (pattern diversity). These kinds of diversity are suggested by the author to follow different developmental avenues that lead to stability of the system. According to the author, the diversity-stability relationship needs better understood. Is diversity necessary for the stability of the ecosystem?

Nutrient Cycling

As a system matures, its capacity to entrap nutrients becomes greater (599[265]). In this discussion, Odum cites Bormann and Likens (1967[24]) and questions whether or not the increased water yield of the stream is a good thing. He suggests that reducing the vegetation around the stream would be accompanied by a greater loss of nutrients, and thus affect those systems downstream. However, the nutrient retention of the biomass at this point (1969) still needs to be tested.

Selection Pressure: Quantity versus Quality

Citing MacArther and Wilson (1967[26]), Odum suggests that species with high rates of reproduction and growth are more likely to survive in uncrowded situations earlier in ecosystem development, while competitive species with low growth would fare better in a more diverse environment in the later stages of the system. Understanding the implications of the dynamics of populations could have potential implications for the adaptations that will occur with overcrowding that is occurring with human populations.

Overall Homeostasis

The review of ecosystems outlined by Odum was in an attempt to discuss the complex nature of the processes that interact with the system (600[266]). Understanding how the entirety of the system as a whole contributes to a better grasp on the nature of ecosystems. This leads to the question of how these systems age: as they get older do they return to a state of “vulnerability”?

Relevance of Ecosystem Development Theory to Human Ecology

Figure 1 is an interpretation of the human view of nature. According to Odum, humans look at the landscape and look for ways to achieve high production of the biomass and fail to see it as a limited resource. In other words, humans see a high P/B efficiency when in reality an ecosystem has a high B/P efficiency.

According to Odum, the human preoccupation with production has caused them to forget that the landscape is not merely a “supply depot” but a home where humans and other organisms must live (600[266]). In other words, are humans getting “too much of a good thing” when they focus on only one aspect of the landscape rather than the impacts on the environment as a whole.

Pulse Stability

There are cases when recurring changes in the ecosystem result in that system to remain in some intermediate stage between development and maturity. This is known as pulse stability. Organisms in the everglades, for example, have adapted to fluctuating periods wet (flooded) and dry periods. In this system wood storks only breed when water levels are falling but will not nest when conditions are still wet. A situation such as this only works in a system if the community is complete. Any sudden change (such as those caused by people) results in the inability of the ecosystem to adapt and reach any point of stability.

Prospects for a Detritus Agriculture

Natural conditions are also better for growing food. Current agricultural practices of selection of plants for rapid growth and edibility make them vulnerable for attacks by insects (602[268]). This, in turn, results in the use of harmful pesticides. Odum suggests that reversing the strategy and selecting plants that have adapted their own insecticides from using delayed consumption of detritus would result in a much clean and stable environment. This would also result in the utilization of natural systems rather than modification and destruction.

The Compartment Model

Odum concludes that a compartmentalization strategy of systems needs to be utilized so that “growth type, steady state, and intermediate-type ecosystems can be linked with urban and industrial areas for mutual benefit” (602[268]). While this would require laws governing zoning procedures, it would limit the impact of humans on the environment. If nothing else, it would provide the opportunity for an increased study and awareness of how much people depend on the environment.

On the use of matrices in certain population mathematics, P. H. Leslie (1945)

First, a primer on life tables.

https://en.wikipedia.org/wiki/Life_table

Introduction.

This section just explains the way to get estimate values for the female population in any unit of time. For m number of time periods that a population's age structure can be broken up into, you would have to solve m+1 equations to get these different populations for each unit of time. This can be represented by some matrix algebra using the matrix M that they define. The top row of M is the number of daughters that are born to women in a certain age group. The Ps are just the probability of living so that they can be part of the next generation. Thus we get all of the population and how many individuals are in each age group.

Derivation of the matrix elements.

This just explains some of math things that are going on to make the matrix stuff work work. Instead of thinking of the age structures as being continuous from n to n+1, they just assume all people in the age group are n+1/2 to make it easier to run the numbers. Also, since populations should be evenly distributed between n and n+1, the average age of that population would end up being n+1/2 anyways.

Numerical Example.

This gives an example of using the matrix algebra to solve the population numbers for an imaginary population. They use numbers from the life table of a sample species and also do the matrix algebra Mn0 to get the new population structure n1. Here they look at the errors and see that their approximation is actually pretty good with the caveat that the age groups can't be too large in comparison to the life-span of the organism.

Properties of the basic matrix.

This section can be a little confusing, but what they have done is to separate the matrix M into a couple of different parts. It has zeroes in the right upper quadrant because at some point, there will be no more reproduction from those individuals once they get to a certain age. So the submatrix A shows the subpopulation in which reproduction for the organism is occurring. Now they have submatrix A, which is where all of the stuff they are interested in is happening and also it has a determinant and is non-singular and invertible, which makes mathematical operations with it non-trivial.

Transformation of the co-ordinate system.

Once again we get a little weird math stuff going on. They introduce a new vector that is the same as the old n vectors just with different notation (squiggle), xi. Then they introduce another new vector (weird n) eta. This eta is just a made up vector that satisfies the equation eta*xi is equal to the square of the length of the vector. Because it satisfies these things, they can make new vectors, phi (circle with a line) and psi (trident), which can help describe a matrix H such that H*A*H(inverse) is actually the B matrix from the previous page. This B matrix becomes important because it is another way to describe what is happening in the population. Now instead of having the births and likelihood of surviving split up like in matrix A, everything has been collapsed into a single term which is indicated by top row of the matrix.

Relation between the canonical form B and the Lxmx column.

When they are talking about this column, they are talking about life tables again. They are looking at the proportion of surviving individuals from one age group to the next as being the Px value. In reporting these values though, they look at it as if they started with one individual in the 0-1 age group and then look at the proportion surviving on to each age group. Using this process, they can find B with less work and just look at the Lxmx column from the life tables.

The stable age distribution.

Here we get introduced to another matrix algebra concept in the form of eigenvalues (the lambdas). Eigenvalues are important because they can show some characteristics of a matrix as we see, the matrix times a vector is equal to the lambda times the same vector, which makes the job of figuring out solutions of a matrix times a vector much easier. They show that there is a positive eigenvalue and that it will be larger than the others. This eigenvalue becomes the important one as it can show the change in populations represented by xi and psi.

Properties of the stable vectors.

Just some properties of the vector as well as an explanation for why they did it the way they did it.

The spectral set of operators.

Once again they are using the eigenvalues as well as finding a cofactor matrix which they can use to look at the contribution to the total population at time t per individual in a certain age group.

Reduction of B to classical canonical form.

More transformation, making B an easier matrix to use without computers.

The relation between phi and psi vectors.

Not necessarily useful at the time but having to do with solving things in the future. He even says so on page 197.

Case of repeated latent roots.

From here they look at the non-positive eigenvalues that were obtained earlier. They tried to find some of the other roots and whether or not they were important. They found that there are certain cases in which there will be two non-negative eigenvalues though one will be much larger than the other and will thus be dominant. Since it is dominant, the other is pretty trivial.

Special case of the matrix with only a single non-zero Fx element.

Here they are looking at the case where only one age class reproduces. This is an example of what they were doing before but they have actual numbers for this. They show how the matrices A and B end up looking. Because of the numbers, there is a stable population that ends up happening and oscillating between the years but always oscillates in the same pattern. From this example, they take it a little further and change up the numbers since the 6 that are born are born sometime between the 2nd and 3rd year. With this assumption, they find another stable population that tends to occur, but it is a smaller population than before, but it still has the same ratios of individuals in the specific generations and just takes longer to get there.

Numerical comparison with the usual methods of computation

Finally the payoff. Now that they have built this system for solving these equations with less work than before, they can do some comparison to try and work out how well their model fits real data. And that is what they do in this section. This section is probably the most relevant to people and also can be more easily understood than the previous sections. They look at the actual numbers vs the matrix calculated numbers and they go back to the earlier point that the age groups have to be fairly fine in relation to the age of the organism to actually provide meaningful results as they take a look at what happens if you make the age gradient coarser.

Further practical applications

This section is just describing the best situation in which to use these matrices given the time period that this paper was written. It look at new invading species which are important in a lot of ecological systems. These matrices can be used to better anticipate what might happen with these kinds of invasive species or when a niche opens up to a new species. This section also talks about the limitations of these matrices and talks about needing to be able to change with time. This ability to change matrices as they progress through time is something that we can do and is an important facet of modeling biological systems.

Appendix

Check out this section if you want to learn a little bit more about where the got their numbers from in the first 14 sections of the paper. They show the derivations as well as the life tables they were using for their calculations.

Hopefully we will get a chance to talk about some matrix math on Tuesday so that when you read the paper for Thursday you can kind of see what they are doing to some of these matrices.

Forest Tree Pollen in South Swedish Peat Bog Deposits - Lennart von Post (1967[1916])

Introduction by K. Faegri and J. Iversen

The Lennart von Post paper (1967) begins with an introduction stressing the vital importance of his 1916 Lecture to the 16^th convention of Scandinavian naturalists in the progression of pollen analysis. Primarily a Quaternary geologist, von Post attempted to use pollen analysis as a method for investigating climatic change during the Quaternary period. Unfortunately, since his presentation was never published, the paper was difficult to access. Furthermore von Post’s “aristocratic arrogance” in believing his work to stand on its own merit, made the paper know more for its implications than its wording.

The Lecture (Translated by M.B. Davis and K. Faegri)

In the 25 years prior to 1916, investigations followed two approaches for determining the age of peat bog strata. The paleofloristic method used Japetus Steenstrup as a model, focusing on isolating and identifying plant remains from peat. The second approach led by Rutger Sernander and C.A. Weber, was paleophysiognomic. This method carefully examined the stratigraphy in an attempt to interpret the ontogeny of each bog. While both approaches are limited in their ability to provide dates to peat-bog sequences, von Post considered the paleophysiognomic approach the most successful.

Using the paloephysignomic approach, Sernander was able to identify a distinct, regularly occurring stratigraphic marker. Termed the Grenzhorizon, this subboreal-subatlantic contact was determined to coincide with the postglacial climatic deterioration. According to von Post, this horizon can be used as a starting point when using stratigraphic methods for dating peat.

Because previous methods of dating peat bog strata proved unreliable, von Post suggested that fossil pollen within the different stratigraphic layers could be used to illuminate the history of the vegetation. von Post realized during his investigations that the character of preserved pollen flora in peat was constant and characteristic for each sample. Therefore, this composition must reflect the average configuration of the pollen rain at the time of deposit. As such, pollen rain itself must be a general reflection of the forest at that time. Realizing this, von Post believed that reconstruction of the changes in pollen flora composition through a sediment profile should make it possible to observe the changes of the general makeup of the forest through time.

To test the usefulness of using pollen to reconstruct the content of peat strata, von Post examined the pollen flora of South Swedish peat bogs collected in a systematic investigation throughout 1915. Peat bogs were selected using a set of five principles: (1) large bog size; (2) be above the maximum limit of the marine transgression; (3) general features of sediments and developmental history had to be known; (4) the subboreal-subatlantic contact was clearly developed; and (5) the profiles must be from peat that completely preserved the pollen flora.

Identification of pollen flora was undertaken using 200x magnification, with higher and lower magnification settings employed depending on the coarseness of the pollen. The composition of pollen flora in a particular peat strata was calculated as a percentage based on the total number of grains in a series of preparations.

In his presentation, von Post discussed his results using a “generalized diagram based on the averages of the individual curves” (387). Unfortunately, the translators could not locate the original diagrams of the pollen curves. According to the footnotes, pollen percentages from each bog were plotted along the x-axis, while the y-axis represented individual pollen types.

von Post separated the eight main curves plotted on the diagram into three groups:

1.      The first group, consisting of birch and pine pollen. This group exhibited high frequencies at the beginning of post-glacial time that subsequently declined until its rebound in subatlantic time.

2.      The curves from the second group, mixed oak-forest, alder, and hazel pollen, show a rapid increase over time as birch and pine pollen began to decrease. Their frequency diminishes following the onset of subatlantic time.

3.      The third group, made up of beech, hornbeam, and spruce, are represented sporadically at the beginning of post glacial time. These species begin to increase along with the pine and birch during the subatlantic.

At this point in the paper, von Post discusses the precautions of using pollen diagrams. He states that in the absence of pollen productivity and dispersal indices of various trees, the diagrams are not sufficient in expressing the composition of the forest communities. Rather, they should be studied for the trends they represent of changes in frequency between forest types (390).

Taking these considerations into account, von Post concludes that trees adapted for warmer climates made their way into southern Sweden after the post-glacial period began. von Post further tested this conclusion and the potential for the use of pollen sequences in chronological relation to sea-level change following deglaciation by analyzing pollen from peat and gyttja sequences of Ancylus lake (9500-8000 BP) and Litorina sea (7500-4000 BP). Although he could not prove it, he was certain that southern forest elements immediately replaced glacial period flora in the area. An important aspect of this conclusion concerns the beech-spruce pollen limit horizon.

During his analysis, von Post found that beech and spruce occurred in low percentage frequencies throughout the profile. The beech in southern Scandinavia first appeared during the Atlantic period. Spruce pollen, however, occurred in much lower frequencies. However, von Post was inclined to believe that its history was similar to that of beech pollen. While they were most likely subordinate species during the time of mixed oak-forest, they most likely took advantage of the drier climatic conditions during subboreal time and, eventually, became the dominant species: beech in the south and spruce in the north. von Post credited the differential dominance of spruce versus beech not only to climatic conditions but also human activity.

From the pollen analyses of Ancylus lake and Litorina sea, von Post concluded that:

1.      Postglacial time can be divided into a time of mixed-oak forest and a beech-spruce forest epoch.

2.      The Ancylus maximum epoch of southern Sweden are characterized by a pollen floral with dominating pine and birch, and low frequencies of alder, elm, lime or hazel (399). These flora were later to be replaced by beech-spruce forests, distinguished by higher frequencies of these pollen.

At the end of his lecture, von Post suggested that the use of pollen curves derived from analyzing sedimentary deposits could assist providing a better understanding the changes in climate during the late Quaternary period. This conclusion was made on the basis that during the presentation he showed that the variation of pollen flora in peat diagrams could be correlated with the stratigraphy of peat bogs. In all, he hoped to someday use this research to develop comparative forest maps of the post arctic period for the different regions of the world.

Questions for Discussion

1. What were the primary contributions of this study to ecology?

2. What are the strengths and weaknesses of this paper? (Consider in both a historical and modern context)

3. Is this paper suitable as a paper included in the "Methodological Advances" section of this volume?

The Intrinsic Rate of Natural Increase of an Insect Population, L. C. Birch (1948)

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

https://creativecommons.org/licenses/by-sa/2.5/legalcode

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 N_t  = N₀e^rt, where N₀ is the starting population, N_t 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), l_x (probability at birth of being alive at age x), m_x (age-specific fecundity rates), and the product l_xm_x. The net reproductive rate, R₀, or the ratio of total female births in two successive generations, is the sum of all the products l_xm_x for the population, and R₀ = N_T/N₀.

Next, he sets out to calculate the mean length of a generation. He starts with the original equation, N_T = N₀e^rt.  Substituting, R₀ = e^rT.  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?

Ecology Readings for Part Four

During the first week after fall break, we'll throw you a curve ball, and discuss the introduction to the methodological section, along with only the first three papers. On Tuesday, October 13, we'll explore Lennart von Post together; on Thursday, October 15, we'll compare musings regarding Leslie and Birch. Please remember to read the excellent introduction, written by James H. Brown.

During the second week of part 4, we'll tackle the last four papers, in the order they're presented in our book. On Tuesday, October 20, we'll trade pithy commentary on Holling, and Porter & Gates. On Thursday, October 22, we'll engage in a group dialogue adventure about Bray & Curtis, and Odum.