General unified theory
Tim Coulson
Sunday, 09 August 2009 04:24 UTC
Hello all,
I’ve been reading the posts on this discussion list with interest. And I have a couple of questions.
In one or two posts the phrase ‘general unified theory of population dynamics’ or similar has been used. I am not sure what this unified theory will explain. Will it allow accurate description and prediction of fluctuations in size of all populations? Or will it allow a comprehensive description of the types of dynamics that can be generated for a wide range of deterministic skeletons interacting with a range of distributions of environmental drivers? Or something else? Personally I would find it helpful to know what a general theory is expected to deliver.
As I see it, Alan has been championing – eloquently – one of several approaches to modeling population dynamics. This approach essentially contains two components – a deterministic skeleton characterizing density dependence, and environmental noise which is sometimes attributed to a specific driver. The two components can interact (non-additivity), and both components can be linear or non-linear, and exhibit lag effects. Obviously a large number of different models can be constructed within this framework. Is this approach the fundamental theory? If it is not, what is it lacking? If it is, why?
Cheers,
Tim
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Well Tim, the “general unified theory of population dynamics” (or whatever) is a bringing together of the ideas of many authors in the past into a general concept. As such it should cover all populations of all species. It is composed of the 5 principles discussed earlier on this web site. It should predict population changes fairly accurately most of the time (dependent on how much variation is explained by the density related changes). That being said, there are many other arguments that deflect it from its true meaning. There are also data sets that are marginally determined by the principles because of the changing physical and biotic environment, often caused by humans. To alleviate these problems, the best thing is to first look at populations that are not, or only superficially, affected by man.
Now the difficult question is – how do we define a population? Do we include other attributes of the population such as age, weight and energy? Well I for one think that we have to deal (at least at first) with just the number of individuals, for this simplifies the problem (which is by no means simple). Talking about biomass, energy content and flow, as Sidney Holt suggests, complicates the picture enormously and negates the force of the individual. Each individual gives rise to a given maximum number of offspring and this maximum may be reduced by conditions in the environment. Population dynamics, in my opinion, is the study of numbers! Study of energy relationships is another topic in my way of thinking. Now I am quite sure that Sidney will disagree with me, and that’s his prerogative. But we cannot use the principles (defined earlier) if we use energy and biomass as our measure of population. They need their own principles and, as far as I am aware, nobody has defined them yet. So we have no theory!
Now to the question of age structure. It is more difficult to dispense with because all populations can be broken down into age classes. The problem is that age is not an easily defined variable and it is continuous, not discrete, as most models assume. Some organisms live as little as a day (how does one define age?), while others can live hundreds of years (some tree species). Time is not really definable (rationally) and is only looked at (by us) in human terms. Mortality, however, acts independently of age (more or less). When there is not enough food, many individuals die, irrespective of age. Lots of predators kill many weakened individuals. Parasitoids infest many young larvae (or even eggs) of insects which only die after the larvae are full grown. What is the importance of age structure in these cases? How do you define the principles of an age structured process? What is the theory? As I see it you have to give up theory (at least that defined by past population ecologists) to include age, and that is not acceptable. The elimination of age structure greatly simplifies the analysis, allows one to define the principles that affect all populations (a theory), and removes a notion that is more aligned with man’s whims than with real life. This is what I feel, but I don’t know if I have explained it adequately? I await your (and others) responses.
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Alan, I’d like to pick a couple of points from your reply for closer examination.
Talking about biomass, energy content and flow, as Sidney Holt suggests, complicates the picture enormously and negates the force of the individual.
This statement makes it seem that you believe the individual occupies (perhaps) the critical role in determining population dynamics. Why, then, is it acceptable to use continuous variables in population dynamic models, rather than integer variables? Shouldn’t we develop models based more closely on (integer) number theory to satisfy this condition?
As for age-structure, in many species we have clearly defined, synchronous annual (or seasonal) breeding seasons – particularly with vertebrates in higher latitudes. Age has repeatedly been shown to have an impact on both breeding success and mortality, I guess you wouldn’t disagree with that.
Can you clarify if you’re suggesting that: as this sort of age-structure is not a general feature across all populations/species, it can’t be regarded as a part of a general theory? Or are you getting at something else?
I guess one potential way around the problem of different species having different lengths of lifetime could be approached by scaling all life-spans by the theoretical maximum life-span – in the same way that we model intrinsic growth rate as a theoretical maximum reproductive rate in the asbence of conspecific interactions, r0.
Stage-structure (e.g. Larvae, Pupae, Adults), on the other hand, appears to be a different type of problem.
Looking forward to your responses!
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I should clarify that r0 should be considered the difference (or ratio, depending on the formulation) between density independent birth and death rates, rather than just a maximum reproductive rate. Apologies.
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Yes, I believe that individuals are the essence of a population. Individuals give birth and die to other individuals. Therefore I believe that discrete (difference) mathematics is more applicable to population studies than continuous (differential) mathematics (it is what I use). However, when age classes overlap completely (human population) then continuous equations may be reasonable approximations. Notice that many insect populations are synchronized with time (annual cycle) so that all stages are not overlapping and one stage dominates at a particular time. Continuous equations seem to me to be a severe over-approximation here.
Of course age has an impact on breeding success and mortality but we are not interested in this in population dynamics (at least I don’t think we are). What we are interested in is categorizing the effect of the environment (including predators and parasitoids) on the population. These 2 things are different! One asks; what effect does age have on survival? The other one; what effect does population density have on survival? These are completely different questions and require different answers. I’m not sure I have made this clear?
Age structure is present in all species of course, but it is difficult to classify and measure because of its extremely variability (e.g., from microbes to trees). Why should we include age structure in populations where we can see and recognize it, and leave it out of those which we cannot; that operate on a different time frame? It is also not a general population phenomenon in the true sense. For instance, in a relatively stable population, age structure is more or less constant. I quote again from Lotka (see Preamble II), “the age-distribution appearing merely as an adventitious element complicating the relation, without being essential to the fundamental characterization of the species.” It complicates the picture, indeed, and adds nothing (or very little) to the basic discussion. It may also make certain important mortality difficult or impossible to analyze; i.e., lateral perturbation mortality which has been shown to be very important and often missed in population studies; see my discussion of B. Mathematics of the third principle. What I am saying, therefore, is that age structure cannot be considered part of the general theory of population dynamics. I basically agree with Lotka on this.
Different species cannot be scaled to a “theoretical maximum life-span” because they can all operate in the same span of time; i.e., we can have trees interacting with microbes. We have to disregard life spans completely and just deal with numbers. You have to think of this as a practical necessity; e.g., sampling populations in nature, which we normally do once a year! The year is the only repeating entity that all life encounters, no matter how long-lived it is. What happens in a year can change the dynamics of a population completely.
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Thanks Alan for your reply.
I wonder if you can be a little more specific than “It should predict population changes fairly accurately most of the time (dependent on how much variation is explained by the density related changes)”. Could we ever all agree that we’d identified a general theory based on such a definition?
In your preamble defining ‘what is a population’ you argue that an appropriate definition of a population is a group of individuals of the same species who dynamics are determined by fluctuations in the birth and death rate. I don’t have any problems with this definition – you are essentially saying that
N(t+1) / N(t) = E(s(t)) + E(m(t))
where t represents time, E expectations, S individual survival and m individual recruitment (fertility and survival to the next census). E(s) is therefore mean survival. It is not as general as it could be though. Why not start from
N(t+1) / N(t) = E(s(t)) + E(m(t)) + E(i(t)) – E(e(t))
where i is individual immigration and e is emigration. This would then allow you to relax your criteria that dynamics are determined by birth and death alone. Isn’t this a more general starting point? You can then ask what factors influence the expectations of s, m, i and e?
Regarding age-structure. Age-structured models are special cases of stage-structured models. I do believe that most populations are structured in some manner, such that s, m, i and e differ by stage. Sometimes age may be the most obvious variable to structure on, sometimes not. In many invertebrates structuring on life history stage – egg, each instar, pupae and adult could be more appropriate than age. My starting point would be demographic models including stage, but if s, m, i and e did not differ by stage I’d simplify to un-structured models. And if i and e contributed little to the dynamics I’d simplify models of s and m. And if s and m were strongly correlated, and therefore presumably influenced by the same drivers (density, environment) I’d simplify to modelling lambda. And I’m more than happy to linearize by taking logs at any point if this helps with the maths (it usually does!). Now I accept that in most cases data are perhaps insufficient to parameterize such models with confidence, and data type, quality and quantity should influence your choice of model. But data availability shouldn’t blinker one’s view on a logical starting point.
The obvious question that follows is what processes influence s, m, i, and e? I am not sure we can make any general statements on this at present, although I suspect that density is often (but not always) an influential driver.
I do agree with you regarding how a theory of the dynamics of biomass will be more complicated that a theory of the dynamics of numbers. As I see it the dynamics of biomass are determined by the product of dynamics of numbers and the dynamics of the average mass. This does not mean that such a theory is not desirable or useful – I can see many cases when such theory would be of great use.
I agree with Mike regarding time – sometimes discrete time models will be a more appropriate choice than continuous time models, sometimes not. It depends on the life history, and whether reproduction (and the timing of death) occurs predictably with time.
Cheers,
Tim
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Hi guys,
Actually, what I was getting at by mentioning number theory wasn’t related to the time variable. Rather, the discreteness of individuals, constituting the N (population size) variable.
Now I certainly don’t claim to know much (if anything) about number theory, but the present discussion (and most of ecological modelling) focuses on a type of mathematics that assumes N is a continuous variable.
For example, the Ricker function tells us that if we have a single individual at time t, with intrinsic growth rate r0 = 1, in a habitat which can carry a population size of K = 100:
N t+1 = N t exp(r(1 – N t / K)) = 2.6912
which, without rounding, is not often true of any real ‘population’. It should be an integer number. The same problem applies to most, if not all, commonly used discrete and continuous time models.
I basically agree that discrete and continuous time models will be appropriate in different cases (though many of the tricks we use to analyse these models don’t change), but what about dealing with models that only use (and predict) discrete integer values for the number of individuals in a population in a deterministic framework? Is it an unavoidable feature of any useful ecological model, or can we employ different types of maths to incorporate this?
I’ve (literally) just come across this paper] which may help with this issue, but haven’t had time yet to read it. The basic model (master equation) defines the probability of finding a given number of particles at a given time, which could easily apply to the number of individuals in a population at a given time.
1 Assaf & Meerson (2006) Spectral Theory of Metastability and Extinction in Birth-Death Systems. Physical Review Letters 97: 200602
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I understand your problem Mike. In truth you are right but in the world of reality we are sampling populations in part of space (not total space) and therefore fractions become possible. For instance, if we measure numbers per square meter of foliage on a sample of say 20 square meters of foliage we can obtain a fraction, as is often encountered in real life. If we are dealing with actual populations (e.g., totals) then we are involved in undefined space and we can still get fractions as individuals leave and enter your space. On the other hand, we can also round off. The model is, after all, an approximation! It is important to remember that I am talking about real life and sampling real populations.
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P. S. I should have said the average number on 20 square meters of foliage (i.e., sum Numbers / 20 = fractional number). The fraction comes from averaging. If you don’t do that then whole numbers are the rule. Perhaps there is a need for a new mathematics, and number theory may be useful. I would be interested in what you find out about it. That individuals are discrete rather than continuous is one of the facts of life. A population, made up of individuals, is also discrete, is it not? Maybe we should never round down, always round up? But in practice that is difficult or sometimes impossible? Well you have introduced something intriguing!
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Alan, it’s probably easier, and possibly more sensible, to represent space as a continuous variable, rather than population size. For example, let’s stick to the 20 individual insects we count on a leaf of 24.312 cm2.
It’s not clear to me why the ‘approximation’ argument is relevant here. We’re starting from a mathematical model, and can design that around whatever assumptions we want. It’s only when applying our model to a real population that we start to have trouble matching the assumptions in the model to what is going on in the real world. Space is flexible enough to allow us to maintain an integer approach to population size.
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Time, in answer to your initial topic:
I would find it helpful to know what a general theory is expected to deliver
I guess I’ve been following and participating in these discussions with the idea that a general theory is an underlying foundation that will be applicable to all (or as many as possible) population scenarios.
It can be built upon by adding specific components that apply to specific populations.
So, while I agree with you that N(t+1) = Survivors + Births – Emigrants + Immigrants is a very sensible way to think about populations, I think excluding dispersal makes this a more general formulation.
As has been mentioned, the dispersal part of this is problematic in real populations, e.g., differentiating between emigration and death can be very tricky unless all individuals are followed around all subpopulations.
It was proposed in one of the earlier topics that we can focus on local survival and reproduction, as a local population can be defined as being local because it is regulated more by local birth/death processes than dispersal processes. As movement between subpopulations becomes more important, we need to reconsider exactly what is the most important spatial scale interactions between individuals occur over.
I think what I’m trying to say is that dispersal processes can vary more easily between populations than the occurrence of births and deaths, making dispersal a less general feature.
Any thoughts?
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