Preamble II. Should we include age and sex structure?
Alan Berryman
Friday, 17 April 2009 15:50 UTC
“The orbit of any one planet depends on the combined motion of all the planets, not to mention the action of all these on each other. But to consider simultaneously all these causes of motion and to define these motions by exact laws allowing of convenient calculation exceeds, unless I am mistaken, the force of the entire human intellect.”
Isaac Newton.
“Theory without practice is fantasy, practice without theory is chaos.”
Abridged from a forgotten author.
We need to get one thing straight. Our theory must be applicable to real field data, which are usually counts of organisms taken at discrete points in time over a sampling universe of given dimension. Therefore, it must be simple and direct. Long convoluted arguments will trap us in endless circular debates.
A fairly large group of scientists think that age structure is critical in defining and quantifying population changes (see Coulson et al. 2008). Others think that it is impractical or unimportant in the development of a general theory (e.g., Berryman and Lima 2006). This problem needs to be resolved before we can discuss the fundamental questions of theoretical population ecology. Herewith I present my own point of view:
I generally do not think that age (or sex) structure of a population should be considered in the general theory of dynamics. I have several reasons for this, some theoretical and some practical. First, it is often impossible to collect data that includes all the age classes of a population, or of several co-acting populations. This is especially true of small and numerous organisms with complex interactions (insects, microbes, etc.). However, many entomologists working on agricultural insects (where the crop is completely removed annually) count stages of development (e.g., eggs, larvae, pupae, adults) while workers on mammals tend to break the life cycle into easily identified stages of development (e.g., one year old and the rest). The fact is that the consideration of internal population structure greatly complicates the theory. Large numbers of variables and parameters accumulate as population structure expands to more and more species with more and more age classes. This leads us into the “curse of dimension”. Furthermore, the omission of population age structure may not seriously hamper the theory since it has minor quantitative influence on the temporal dynamics of natural populations (see Coulson et al. 2008). Thus, in order to accommodate a parsimonious and tractable theory of multi-species dynamics in relatively natural ecosystems, I believe that it is necessary to omit these often un-measurable complications. In the words of Lotka (1925), father of mathematical ecology, “the age-distribution appearing merely as an adventitious element complicating the relation, without being essential to the fundamental characterization of the species.”
Berryman, A. and M. Lima. 2006. Deciphering the effect of climate on animal populations: Diagnostic analysis provides new interpretation of Soay sheep dynamics. The American Naturalist 186: 784-795.
Coulson, T., T. H. G. Ezard, F. Pelletier, G. Tavacchia, N. C. Stenseth, D. Z. Childs, J. G. Pilkington,J. M. Pemberton, L. E. B. Kruuk, T. H. Clutton-Brock, and M. J. Crawley. 2008. Estimating the functional form for the density dependence from life history data. Ecology 89: 1661-1674.
Lotka, A. J. (1925). Elements of physical biology. Baltimore: Williams and Wilkins (reprinted as Elements of mathematical biology. New York: Dover, 1956).
Updated 30 May 2009 15:51 UTC
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Replies
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I beg to differ. Coming from a fisheries background it seems to me inconceivable that a general theory can ignore age and sex structure. The different age (and, commonly’ sex) classes generally eat different foods, live in different places, move in different ways and certainly experience different density-dependences, qualitatively different. We need also to take account of Alan Longhurst’s recent conclusion that the teleosts (bony fishes) have evolved to depend significantly on cannibalism in which the parents feed on the young as the latter grow up. Has that happened elsewhere in evolution? Sidney Holt
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I understand Sidney’s preoccupation with age and sex in harvested fish populations, where age and sex may be critically influenced by management activities. That is why I suggested, as a starting point, that we consider only unmanaged or only occasionally (rarely) managed species. This is just a starting place. We can attempt to apply the complications later. OK?
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Anonymous
I see that there has been some debate on this site on general rules to do with population dynamics. The ‘general’ principles that Alan Berryman describes do not provide a fundamental theory of population dynamics, even if such a theory exists. What Berryman does is provide a lucid summary of one perspective on population dynamics. It has many appealing features, but is one limiting case of a much broader class of models.
If we start by making the assumption that dispersal in and out of a population is dynamically unimportant*, and that the time unit chosen is less than the minimum age at first reproduction then the population growth rate over a discrete time step is always equal to the sum of mean survival and mean reproduction. Defining population as N, time as t, mean survival as S and mean recruitment as R then N(t+1) = [S + R]N(t). This is a tautology, a mathematical identity: it will always be true. An understanding of population dynamics can consequently be obtained by understanding factors that generate variation in S and R. If a function describing variation in S, and a function describing variation in R can be identified it is straightforward to sum these functions to create a population model. Such models can be analysed in a variety of ways. Two questions arise – what factors influence S and R? And what is the functional form of the association between each factor and S and R? If there are going to be general rules in population dynamics there will be general patterns in the answers to these questions.
We can group factors potentially associated with S and R into classes. There are population level factors and individual-level factors. Population level factors include population density, inter-specific processes (including predation and disease) and sources of environmental stochasticity (including climate). Individual-level attributes include age, sex, home range, phenotypic traits and genotype. Survival and reproduction can be influenced by interactions between individual- and population-level factors – for example, survival rates of different groups of individuals may be affected in different ways by a change in population size.In general we do not know all the factors that can influence survival or reproduction. We also do not know whether population-level, individual-level or interactions between them influence survival and recruitment to the greatest extent. As far as I am aware we do not even have an expectation of the properties of the distribution of effect sizes. However, there is a large literature emerging on the identification of factors associated with survival and reproduction.
Can we construct a general framework to understand population dynamics based on functions describing variation in S and R? The answer to this is ‘yes’ but parameterisation of any such framework will be context specific, which means the equations which will emerge to describe and predict population size are likely to differ between populations in different settings and between species. The dynamics of some populations will be strongly influenced by the distribution of individual attributes – especially, for example, those populations subject to selective harvesting. Other populations will be most strongly influenced by population size, others by density-independent environmental variation, and yet others by inter-specific processes.
Not only may factors associated with survival and reproduction vary across populations, but when the same factor influences survival and reproduction in different populations the functional form describing the association between a factor and survival and reproduction may also be different. Once again there is a large statistical literature on the identification of appropriate functional forms.
Finally different factors will need mathematical treatments when incorporated into models. Where individual-level attributes influence survival and reproduction their distributions will change with time. Such individual attributes need to be tracked if they are incorporated into models. Tracking age- or stage-structure is routinely done in matrix models. Quantitative traits can be tracked in stage-structured models, including integral projection models which are solved by approximating the models as high dimensional matrices.
I believe that most biologists will accept that different factors influence survival and reproduction across populations, their settings and between species. I also believe that most biologists appreciate that population dynamics are determined by fluctuations in the birth and death rates. So I do not believe that I am stating anything that is contentious. These arguments also underlie several recent publications that I am happy to direct interested readers to.
Of course, one can combine birth and death rates into a single parameter – the population growth rate and construct models around this. One can also describe a functional form of the way the population growth rate changes with a change in density using r and K. Models constructed like this are simply a special case of the more general form described above where S and R vary with multiple factors. To my mind it is perverse to choose and champion just one factor associated with S and R, or to declare general principles and generate one or two special case models which are the claimed to be general while arguing that other models in the general framework, perhaps ones that include individual attributes like age, are in some way inadequate.
A standard retort to this argument is that models should be simple. Indeed they should be as simple as possible to answer whatever question is behind the motivation for building the model. Different questions require different models, and different populations may require different sets of factors to gain equivalent model goodness of fit and predictive power. Dogged adherence to a specific class of model within the framework I describe, while preaching claims of general principles with limited support fails to advance the field. I believe a more fruitful exercise would be the identification of factors that influence survival and reproduction across a range of systems, before constructing, analysing, simplifying and comparing models constructed from these insights.
- I am not claiming it is unimportant I am making this assumption to keep things simple. The approach I describe can easily be extended to incorporate dispersal.
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Dear Anonymous (07 Jun 2009 | 17:29),
Thanks for your considered response – however, I’d really appreciate it if you’d give yourself a much more telling user profile. This will make it much easier for us to regulate and moderate comments on this Forum – we want to avoid spam and other messages that don’t conform to the Community Guidelines as far as possible, and keep the debate open and polite/friendly.
This is not a comment about the relevance of your post, simply a request to allow easier moderation of the discussion.
Thanks,
Mike -
I think that the question si simple, no biologist can deny that several individual-level factors (phenotypic, spatial, sex, age, stage) and population-level factors (climate, density, predators) can affect survival and reproductive rates in populations. However, to me the question is, we need always all this information for understanding population dynamics?, we need to describe, identify and model each specific factor and estimate several parameters and functional forms to inlcude them in population dynamic models? Well, some of us think that not, that all these multiplicity of factors operating at individual level wash-out when population dynamic patterns are analyzed, others think that is strictly neccesary. That is why is so important to use diagnostic tools to determine the degree of complexity of a determined dynamics. However, I think that as in many areas of science, there different shools of thinking about population dynamics, time and the societal demands for understanding and prediction will increase in the next years, we will see what approach is able to generate more confident predictions and general understanding.
regards
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Mauricio Lima stated it perfectly; the theory of “population dynamics” attempts to explain the population-level consequences (changes in numbers) of various population-level processes (e.g., feedback, density, predators, climate) and does not concern itself with individual-level complications (age, stage, sex, spatial structure). Lotka said the same, as cited in my text at the beginning of this section. Many people do not agree. So be it. But they, in my opinion, miss out on the actual causal mechanisms because of the complexity inherent in their models, and the fact that only one of many “population-level” factors regulate the density of the population “in the vicinity of equilibrium” (see section E. Mathematics of the fifth principle (limiting factors)). They are faced with the “curse of dimension”, as noted by Isaac Newton and cited in my introduction at the beginning of this section. Also, as far as I am aware, lateral perturbations (Royama 1992, pp. 37), which are very common in population systems (e.g., Einum 2005, Lindstrom et al. 2006, Berryman and Lima 2006), cannot be detected in age structured models. This is a severe problem in my opinion. As I have noted on many occasions, the theory as stated by myself and others only applies to populations that are under equilibrium dynamics, or that are in the vicinity of equilibrium. Human populations, and those that they impact severely, are not. This is why people working with such populations have difficulty with the theory.
Now let us talk about data. The theory is used as a tool for understanding the causes of population changes in real life. How do we measure population changes? Well the most common type of measurement is a time series; a series of annual measurements of the numbers of individuals in a population taken over a fairly long period of time. There may be hundreds of such measurements taken (or correlates, like tree rings, measured) for a large number of years; e.g., the larch budmoth (Baltensweiler et al. 2004). These data almost never include age classes and must therefore be considered useless by the age-class modelers. I personally have analyzed hundreds of data sets and some can be seen at http://entomology.wsu.edu/Profiles/06BerrymanWeb/BerrymanOnlinePubs.htm. A couple of these data series have stages of development and have also been analyzed by age-structured models. For example,
The spruce needle-miner (Epinotia tedella).
Populations were analyzed by Mikael Munster-Swendsen (1985) by age-structured life table methods developed by Varley et al. (1975). He measured (for 19 years) the density of needle-miners and their primary insect parasitoids emerging annually from the litter of a Danish spruce plantation. He also constructed detailed life tables, which included all suspected mortality factors, by sampling several times each year for 9 of those years. He found that the needle-miner population exhibited spatially synchronous 6-7 year cycles of abundance, and that the key factor affecting the population change was reduced fecundity due to unknown causes. When I started working with Mikael he was not very impressed with my analytical procedure. I only needed one measurement a year, which gave me 19 years of data, while his method of studying developmental stages only gave him 9 years. Thus, I had over twice as many years as he, but he had several samples for each of his years. Anyway, the result of my diagnostic analyses of his data showed that parasites regulated the cycles of spruce needle-miner populations (Berryman 1999, Munster-Swendsen and Berryman 2005). This puzzled Mikael because parasites could not regulate the population for they did not attack enough needle-miner larvae. He however, being a true scientist, started looking at the data again and soon found a remarkable fact: when moth larvae and parasitoid adults were very abundant, the parasites were frequently disturbed during oviposition (by needle-miner movements), and this disturbance often resulted in a needle-miner larvae being sterilized (the first part of the attack) but no parasitoid egg was laid in it (the last part of the attack) (Munster-Swendsen 1994). This turned out to be the cause of his reduced fecundity, because the moths would emerge from hibrenation but were infertile and produced no young. But what is important is not the details of the parasitism process but the outcome, the population-level process of parasitoids being responsible for spruce needle-miner cycles. Notice that this was accomplished by the simple once-a-year sampling of needle-miner and parasitoid populations rather than multiple yearly samples.Another example can be found in our work with the Soay sheep (Berryman and Lima 2006) where we produced a simple non-structured annual model to predict Soay sheep abundance. This model was then criticized by Coulson et al. (2008) who used an age-structured model. Note that the dynamics of our model was strongly influenced by a lateral perturbation which could not be reproduced in the age-distributed model.
Baltensweiler, W., Weber, U. M and Cherubini, P. 2008. Tracing the influence of larch-bud-moth insect outbreaks and weather conditions on larch tree-ring growth in Engadine (Switzerland). Oikos 117: 161-172.
Berryman, A. A. 1999. Principles of population dynamics and their application. Stanley Thornes, UK.
Berryman, A.A. and M. Lima. 2006. Deciphering the Effects of Climate on Animal Populations: Diagnostic Analysis Provides New Interpretation of Soay Sheep Dynamics. American Naturalist 784-795.
Coulson, T., Ezard, T. H. G., Pelletier, F., Tavecchia, G., Stenseth, N. C., Childs, D. Z., Pilkington, J. G., Pemberton, J. M., Kruuk, L. E. B., Clutton-Brock, T. H. and Crawley, M. J. 2008. Estimating the functional form for density dependence from life history data. Ecology 89: 1661–1674.
Einum, S. 2005. Salmonid population dynamics: stability under weak density dependence? Oikos 110: 630-633.
Lindstrom, A., Enemar, A., Andersson, G., von Proschwitz, T. and Nyholm, N. E. I. 2005. Density-dependent reproductive output in relation to a drastically varying food supply: getting the density measure right. Oikos 110: 155-163.
Munster-Swendsen, M. 1985. A simulation study of primary-, clepto- and hyperparasitism in Epinotia tedella (Cl.) (Lepidoptera: Tortricidae). J. Anim. Ecol. 54: 683-695.
Munster-Swendsen, M. 1994. Pseudoparasitism: detection and ecological significance in Epinotia tedella (Cl.)(Tortricidae). Norw. J. Agric. Sci. Suppl. 16: 329-335.
Münster-Swendsen, M. and Berryman, A. 2005. Detecting the causes of population cycles by analysis of R-functions: the spruce needleminer, Epinotia tedella, and its parasitoids in Danish spruce plantations. Oikos 108: 495-502.
Varley, G. C., Gradwell, G. R. and Hassell, M. P. 1975. Insect population ecology: an analytical approach. Blackwell.
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(This is a general comment – not a comment related to age structure)
I can only fully agree with our anonymous friend and continue to disagree with Alan. I also find that it is inadequate to choose and champion just one special view to declare principles which are claimed to be general while arguing that other models in the general framework are inadequate. The more fruitful exercise is the identification of factors that influence survival and reproduction across a broad range of systems, in order to provide a wider view and a better integration with other disciplines of ecology and evolution. To me this forum is no longer a forum but a continued argument for an outdated and simplistic view of population dynamics (exponential, traditional density regulation, and cycles by predator-prey interactions). It is a fundamental misunderstanding that “population level consequences” can be explained exclusively by “population level processes”. In order to understand and predict population level processes we need in many cases first to understand the underlying individual level processes (although these processes may not need to be exclusively formulated in the final population dynamic equations).
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I have to agree with Lars and for that reason have been considering withdrawing from the Forum.
My earlier suggestion about age and size structure has been misunderstood, unfortunately. My point is that it is not very useful to define “population” in purely numerical terms. If, for example, I found myself restricted by some rule that only one measure can be included in a definition I would chose biomass, not number. But that, too, would be inadequate because the dynamics such as mortality and reproductive rates are determined by the composition of the biomass as well as its size.This Forum discussion will get nowhere if it is limited to very narrow definitions. and, a priori, a few designated primary processes. Sidney Holt -
What I have seen here, and in various recent books, is how fragmented our discipline has become. There is extraordinarily little overlap in the bibliographies of various reviews and so on.And methodologically we are fragmented, too. For example Alan writes that in dealing with time series we are usually looking at counts of numbers. Not so, at least not generally. In many areas of marine biology, for example we are often looking at masses, not numbers, or indices of masses: chlorophyll densities, or ultrasonic signal strengths, for example. If we are to continue this discussion usefully we have to get away from narrow assumptions, from each of our limited experience about what it is possible, or necessary or desirable to try to observe. Sidney Holt
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Anonymous
[From anon shib, June 13, 2009]
“. . . the theory as stated by myself and others only applies to populations that are under equilibrium dynamics, or that are in the vicinity of equilibrium. Human populations, and those that they impact severely, are not.” (Berryman 10 June 2009 | 05:17 )
“. . . as in many areas of science, there different schools of thinking about population dynamics, time and the societal demands for understanding and prediction will increase in the next years, we will see what approach is able to generate more confident predictions and general understanding.” (Lima 09 June 2009 | 15:04)
“I have to agree with Lars and for that reason have been considering withdrawing from the Forum.”
. . .
“If we are to continue this discussion usefully we have to get away from narrow assumptions, from each of our limited experience about what it is possible, or necessary or desirable to try to observe.” (Holt 12 June 2009 | 17:30 )
I am not the “Anonymous” of the 07 June 2009 | 17:29 post, but another anonymous (I will henceforth refer to myself as “anon shib” to differentiate myself from that participant; I suggest that he or she do the same. The reason I prefer to remain anonymous is not to “hide” behind a “handle” but to prevent the kind of personalization of the discussion that has taken place here. The value of the Forum is not in displays but in the issues.
To that end, may I suggest that all participants adopt anonymous “handles” or pseudonyms? May I further suggest that this particular discussion confine itself to the originator’s stated purpose, and if an alternative purpose is desired that a separate discussion be started that can be similarly confined to that alternative. Let us give Berryman’s idea a chance. Let us examine it for its merit and give it a try.
To facilitate that examination, shall we state or restate the most fundamental question behind population ecology, followed by the subsets that advance that issue? Otherwise, it seems to me that the discussion will become ensnarled in pedantic yarn spun of egos, rather than “continue this discussion usefully” as Holt suggests.
May I also suggest that responses be as brief and specific as possible and confined to the point initially made? Perhaps Berryman should re-initiate the discussion with a simple declaration such as "We should not include age/sex structure for the following reasons (list and number), from which a more or less particularly relevant discussion might follow? However, I do think it would be useful to re-state the most fundamental purpose behind the discipline first, if for no other reason than to maintain a disciplined structure “tree” upon which relevant details can be assembled with as little decoration as possible.
anon shib
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