2. The first principle, exponential growth
Alan Berryman
Monday, 04 May 2009 00:46 UTC
All populations grow (or decline) exponentially (or logarithmically) unless affected by other forces. This is a universal phenomenon which can be expressed in discrete time in the following way N(t) = N(t-1)G, where N(t) is population density at time t and G is the per-capita growth rate over one unit of time. Now we can determine the rate at which a population grows by subtracting the initial population size N(t-1) from both sides of this equation; i.e., N(t) – N(t-1) = N(t-1)G – N(t-1) = N(t-1)[G – 1]. From this we see that the growth rate of the population is determined by its initial density times the per-capita growth rate minus one. Now if we set this to natural logarithms, so that ln N(t) = ln N(t-1) + ln G and ln N(t) – ln N(t-1) = R where R = ln G, we see that the population growth rate is now independent of population density. This provides us with a uniform expression that allows elimination the growth process from any discussion of the feedback regulating population size. I’m not sure that this is the best description of this trivial yet vital process. What we have done, in fact, is to separate out the constant (exponential growth) from the variable growth process, or to separate the known from the unknown part of population growth. The unknown part is how R changes as a function of the densities of populations and other components of the environment within which the species lives. (If there is a better way of describing this please let me know?)
Let us therefore define the first principle of population dynamics as follows; all populations grow at constant logarithmic rates unless affected by other forces in their environment. It is what we have called the ‘‘null model’’ (Berryman and Turchin 2001). Given this fact, then it is immediately apparent that the problem of explaining and predicting the dynamics of any particular population boils down to defining how R changes with time. This is what I call the R-function, or R = f(P,H,E), where P represents a set of biotic factors (populations of different species, including the one in question), H their genetic properties, and E a set of environmental (abiotic) factors. This is basically a restatement of Lotka’s (1925) “fundamental equations” for the dynamics of living systems, except that the point of reference is now the logarithmic per-capita growth rate. This idea is quite similar to Newton’s first law (see Montroll 1978, Ginzburg 1986) since we are able to ignore, or rather put aside, the obvious fact of uniform geometric growth and focus, instead, on the forces that affect the rate of change of individuals. This all makes sense because the forces of the environment actually affect the birth and death rates of individual organisms, as summarized by changes in R.
Berryman, A. A. and Turchin, P. 2001. Identifying the density-dependent structure underlying ecological time series. Oikos 92: 265-270.
Ginzburg, L. R. 1986. The theory of population dynamics: back to first principles. J. Theor. Biol. 122: 385-399.
Lotka, A. J. 1925. Elements of physical biology, Williams and Wilkins. Reprinted as Elements of mathematical biology, Dover (1956).
Montroll, E. W. 1978. Social dynamics and the quantifying of social forces. Proc. Nat. Acad. Sci. USA. 75: 4633-4637.
Updated 30 May 2009 15:38 UTC
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The first principle, hyper-exponential growth
I agree with Berryman, and hopefully most population biologists, that we should define the first principle of population dynamics to reflect the expected growth of individuals in natural populations when extrinsic factors are constant (which they are not).
For this case, however, the expected growth is not exponential (geometrical), but hyper-exponential. The exponential law is true ONLY when there are no heritable differences between the individuals in the population; but it is one of the most basic principles of natural populations that they have individuals with different heritage, let it be due to both different genotypes and/or different epigenetic or social inheritance components. With heritable difference between individuals, the faster replicating individuals will growth faster than the slower replicating individuals, the population will evolve by natural selection, and the overall growth is hyper-exponential instead of exponential.
Hyper-exponential growth is based on Fisher’s Fundamental Theorem of Natural Selection (dr / dt = v) that states that the rate of increase in the exponential growth rate® is equal to its additive heritable variance (v). Hence, the growth rate develops as r_t = r_0 + v t, with the rate of change in population (n) being dn / dt = (r_0 + v t) n_t and, thus, n_t = n_0 e^(r_0 t + .5 v t^2). This equation reduces to exponential growth, n_t=n_0 e^(r t), when v is zero.
Someone may argue that evolutionary changes are minor in relation to laws of population growth, that they occur at a slower time scale and that they should be omitted from the first principle. However, from real studies of natural populations we already know that evolutionary changes can have a profound effect on population dynamics on the short time scale of population dynamics (e.g., Sinervo, 2000). This implies that evolutionary effects cannot, at least not as a basic principle, be ignored in our definition of population dynamics. The question that remains in relation to the first principle is then whether it should include evolutionary effects or not. I argue that it should for the reasons given below.
It should be obvious that our definition of population dynamics should arise from population thinking; which essentially is to realize that natural populations are collections of individuals with different, and most often connected, heritage. By doing this we allow us not only to describe population dynamics in the traditional way where all individuals have the same heritage and the first law reduces to exponential growth, but we also allow us to describe population dynamics more generally with evolutionary changes by natural selection included [which might actually be a major, if not the major, cause for population cycles when intra-specific competition is accounted for (Witting, 2000; Ginzburg & Colyvan, 2004)].
An additional, and very important, benefit of population thinking is that population dynamics becomes inherently integrated with natural selection, which is not the case if the traditional exponential growth principle route is taken. And why is this essential? It is essential because theoretical considerations and evidence now suggest that it is the natural selection pressure that arises from population dynamics that actually shape the evolution of the major life-history traits of the organism (Witting, 2008), and that this pressure is only appropriately described if it incorporates the density dependent interactions between the individuals in the population. A better future understanding of both life history evolution and population dynamics is likely to be a joint process that requires that the two theories are fully integrated and dependent upon one another.
I think that this forum should bring the concepts of population dynamics further than just repeating the basic ideas that were laid down by others almost 100 years ago, i.e., we should bring the theory to a position where it can become a more prominent and dominant field in biology as a whole. And the most obvious way to bring population dynamic thinking further into the central heart of biology is to extend the focus to consider also the population dynamic consequences of heritable differences between individual in natural populations. The theory of life history evolution evidently don’t work unless it explicitly integrates population dynamics (see Witting, 2008), and the sword seems to cut both ways (e.g., Sinervo, 2000; Witting, 2003) so that the theory of population dynamics also don’t work probably unless it, at least in some and probably in many circumstances, integrates evolutionary changes in life histories by natural selection on genetic and epigenetic inheritance components. So why not let both theories have the same first principle?
References
Ginzburg, L. R. and Colyvan, M. 2004. Ecological orbits. How planets move and populations grow. Oxford University Press, Oxford.
Sinervo, B., Svensson, E. and Comendant, T. 2000. Density cycles and an offspring quantity and quality game driven by natural selection. Nature, 406: 985-988
Witting, L. 2000. Population cycles caused by selection by density dependent competitive interactions. Bull. Math. Biol., 62: 1109-1136
Witting, L. 2003. Reconstructing the population dynamics of eastern Pacific gray whales over the past 150 to 400 years. J. Cetacean Res. Manag., 5: 45-54
Witting, L. 2008. Inevitable evolution: back to The Origin and beyond the 20th Century paradigm of contingent evolution by historical natural selection. Biol. Rev., 83: 259-294
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The proposition of Lars Qitting is compilling and interesting, ecological and evolutionary forces are linked and must be taking into account together, at the end of the day ecology and evolution are simply manners to study, understand and predict the forces underlying the “struggle for existence”. However, I do not think that hyper-exponential growth should be the first principle. Following the arguments of Lars, if the individuals with heritable differences have differents intrinsic growth rates, the consequence is that different genotypes will have different intrinsic groth rates and those with higher r_0 will growth faster per unit of time than the others. But in all case the growth will be exponential if no forces are present.
In absence of ecological forces (ilimited food and space, no enemies)natural selection will favor genotypes with higher ro, but may be other genotypes will be favored when competition or cooperation is in action.
more later
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I was just about to draw attention to the important but insufficiently known work of Lars Witting (Danish geneticist/ evolutionist working in Greenland) when I saw that he is a member of this forum and already got his words in. His theory of hyper-exponential growth has immense implications for assessments of exploited populations of fishes and whales – his own paper on the gray/grey whale, which he cites, gave us strong evidence of that. Sidney Holt
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Mauricio is correct that the growth of the different evolutionary lineages in a population (at least for the asexual case) is exponential – and this is exactly the point: the growth of the different lineages is exponential BUT the growth of the POPULATION is hyper-exponential
-and population dynamics relate to the growth of the population, not to the growth of identical individuals! Hence, the first principle should be hyper exponential.It is also true that other genotypes are favored in different situations and this may be one of the major causes for cyclic population dynamics when intra-specific competition is taken into consideration (but this has nothing to do with the first principle).
Others may try and argue facts back to the old stand point of exponential growth by saying that with evolution the heritable variation will soon be depleted and then the population will return to exponential growth. While this argument is also essentially correct – it misses the point – for the simple reason that the purpose of the first principle law is not to describe a realistic type of growth that may occur in natural populations (because it will never occur in natural populations) – but instead to outline the most basic growth POTENTIAL of a population and here it is very important to note that it is hyper-exponential, with exponential growth being only a species case.
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sorry – last sentence should read: with exponential growth being only a special case.
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Hi everyone,
Thanks for this stimulating discussion – I’m very glad to see this Forum is proving a useful arena for promoting and discussing these ideas.
I’d like to add my own viewpoint here. I certainly agree that we should do more to incorporate ecological and evolutionary dynamics in the same framework – Alan has started another topic here relating to this.
However, I feel the discussion of hyper-exponential growth leads us into a slightly separate area – the species concept. Using the example of asexual/clonal organisms with different intrinsic growth rates (r0 values) makes me wonder why you’d consider them to be the same species? As individuals within this population are not interacting (competing/mating/sharing genes etc.) and their only identifiable, heritable characteristics (leading to r0) differ, can they sensibly be considered a single species?
Intrinsic growth rate, r0, arises only at the population level, from the mean field approach. By using this value we assume the the natural variation in fitness (the difference or ratio between per-capita reproductive and survival rates) across the whole population can be suitably captured by a single ‘mean’ number. I think this is a critical assumption that does not mean we assume all individuals are identical.
To echo Mauricio’s earlier comment, the exponential growth concept assumes that there is an unlimited supply of limiting resources – i.e., no negative feedback is occurring at the individual or population levels. I wonder exactly how natural (non-neutral) selection could occur in such an idealised situation?
To classify some portion of the population in one way (r0,i = a_) and the remaining portion in another way (_r0,j = b) immediately suggests that the mean field approach is insufficient to describe this focal population.
As I mentioned above, I don’t believe the mean field approach assumes that there is no variation among individuals in a population (v = 0), rather, it assumes that the variation that is present (v > 0) leads to behaviour at the population level that can still be sufficiently described by a single value. Perhaps this model assumption is too restrictive, or too unrealistic to be of general use, and eventually we’ll have to drop it?
I’ll deliberately avoid the idea of introducing different sexes into this topic until I can start to get my head around this simpler case! I stress that I accept there is natural variation in phenotypic characters and hereditary of this variation occurring in natural populations, but I hope someone can clarify these points in the context of hyper-exponential growth!
- Also, I’ll try to put up a post with useful tips on writing mathematical formulae today. Keep your eyes peeled!
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Ahhh, the lunch break has given me a chance to consider and clarify a point in my previous comment:
I wonder exactly how natural (non-neutral) selection could occur in such an idealised situation?
I realise that the gene (and/or phenotype – r0) frequencies will change over time – in a predictable way assuming no further mutation occurs. This does complicate the issue. But, I think we could only sensibly derive a mean-field r0 estimate over a given time-period. This will change according to the amount of data available.
Hyper-exponential growth still relies on the underlying assumption of exponential growth within geno-/phenotypes, but adds structure to the population, so represents a fundamentally more complex scenario.
Which led me to another realisation – perhaps it is not sensible to try to rank or order the “principles” of population growth. Exponential growth may not be any more biologically important than positive or negative feedback mechanisms.
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As individuals within this population are not interacting (competing/mating/sharing genes etc.) and their only identifiable, heritable characteristics (leading to r0) differ, can they sensibly be considered a single species?
Why not? We’ve got plenty of species concepts to use. I don’t see that having variation in a phenotype would preclude individuals from being members of the same species.
Hyper-exponential growth still relies on the underlying assumption of exponential growth within geno-/phenotypes, but adds structure to the population, so represents a fundamentally more complex scenario.
I’ve been thinking that the exponential growth principle (and perhaps the others) should be defined in terms of individual behaviour. I guess the approach would be to define the principles in terms of how individuals behave, and how these lead to population-level consequences. I think I need to think this through a bit more, though. I’m sure there are subtleties lurking here.
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I hope you will permit me to enter your discussions on this very compelling topic even though I am not credentialled in it. Population dynamics overlaps other interests of mine.
What I would very much like to know is this: How much weight of empirical evidence supports your principle of exponential growth? Evidently exponential growth cannot continue indefinitely so the principle must refer to very constrained conditions. What are they? Seems like exponential growth can only obtain ideally when a pristine environment allows it. My computations tell me that interactions with an environment produces growth linearly or quadratically with time and the growth must oscillate. That’s why I am so interested in empirical evidence.
I happen to have posted a viewer activated motion diagram and discussion of exponential growth on my blog a few months ago. For anyone interested it’s at:
http://divineneutrality.org -
Exponential growth is a logical truism of constant population growth as shown in the web site posted in your letter (previous post). The logarithm of population size therefore changes at a constant rate. Now this is very rarely seen because the growth rate is rarely constant. But there are a few organisms in which you can see this constant logarithmic rate, at least for a short period of time (I can send references if you wish). This simple fact of exponential growth (law or principle) must be included in any model of population dynamics. However, the log-linear transformation allows one to ignore it in practicality, unless of course you want to get an estimate of the population size in the next time increment. It is easy to prove this logical principle as you did in your post. I don’t know if this is what you wanted?
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