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
-
Replies
Jump to resultsResults
-
Just a side bar. In some comments there appears to be an assumption that hyper-exponential growth (alla Witting) occurs only because of the occurrence of natural selection. But in fact with diversity alone the population growth is not formally exponential – the sum of a number of exponentially growing sub-populations is not itelf exponential.
I think Lars’ fundamental revision of age-old assumptions is correct and of fundamental importance if this forum is to achieve anything at al. There is no point in rehashing century-old assumptions about population dynamics as an isolated discipline. Lars’ argument that at this time we have to look at population dynamics in an evolutionary, and hence genetic, context is incontrovertible.Elsewhere Alan has written that we have discussed age-structure as of importance and he has rejected that. I don’t think we have discussed that issue, we have just argued from pre-formed ‘positions’. I think that other qualities than numerical size are as important, generally in fact more important. Clear examples of that truth have arisen even in the rather desultory exchanges we have had about human populations How humans populations grow is determined at least as much by their structure as by numerical size; even Malthus understood that (like his contemporary and mathematically competent karl Marx he was writing mainly about class structure) AND JUST AS WE HAVE TO DISCUSS POPULATION DYNAMICS NOW IN A GENETIC-EVOLUTIONARY CONTEXT, WE ALSO HAVE SOMEHOW TO INTEGRATE THAT WITH THE DISCIPLINES OF PHYSIOòOGY AND OTHER FEATURES OF THE EXCHANGES OF ENERGY AND MATERIAL IN BIOLOGICAL SYSTEMS. Sorry about unintentional shouting! Sidney.
-
The hyper-exponential growth vs. exponential growth question can be resolved to a degree by referring to Sewell Wright’s concept of adaptive landscapes. In a particular environment, average fitness is a function of gene frequencies. The multivariate space of all possible gene frequencies produces a “fitness landscape”, a series of peaks and valleys in average fitness.
If the population’s gene frequencies place it on a slope below a local maximum, average fitness will indeed increase, and hyper-exponential growth would be displayed in a resource rich environment. However, the local maximum fitness is eventually attained; evolution has to work mainly with the material that is present, because big genetic improvements are rare. Biological organisms are fundamentally jury-rigged contraptions that for the most part have attained their performance limits (how is your back doing these days?). After the local peak is attained, populations will grow exponentially in resource rich environments. Occasionally, stochastic forces will perturb the population gene frequencies sufficiently to place it on the slope of another hill in the fitness landscape, whereupon a genetic hill-climb, and the potential for hyper-exponential growth, will ensue.
So, how much time is spend near local maxima? A pair of papers in the 1980s (by Newman et al. 1985 Nature, Lande 1985 Proc Nat Acad Sci USA) showed that under ordinary genetic stochasticity, the average time spent hill-climbing time will be far, far less than the average time spent near the summit(s).
So there is an operational prediction about what we would expect to see on a day-to-day basis. Most of the time, if we observe the population in (or we experimentally offer the population) rich resources, then we will observe exponential growth. Occasionally, we might see a significant genetic event and be able to see faster-than exponential growth, but once the fitness peak is attained the population will be back to exponential-as-usual.
If the environment itself changes with respect to fitness, the fitness landscape changes. “Environmental earthquakes” can raise new mountain ranges and drop new valleys in the fitness landscape. If such environmental events are occurring frequently, the fitness landscape will be constantly fluctuating like the surface of a lake on a windy day. The population would then be chasing fitness maximums for a larger fraction of time. However, remember that the “fixed environment” assumed for the fitness landscape is really an environmental regime which could admit ordinary environmental fluctuations (organisms are adapted to the usual temperature fluctuations found in their region, etc.). How often we would detect the potential for hyper-exponential growth as opposed to exponential growth would ultimately depend on how often environmental game-changers (changes that alter fitness landscapes) occur in the region inhabited by the population.
The fitness landscape (or adaptive landscape) concept suggests that hyper-exponential growth does not proceed indefinitely, but will become ordinary exponential growth in time. Furthermore, the landscape concept shows how maximum fitness itself does not just keep increasing, but can in fact decrease when the population is stochastically bumped to climbing a hill with a lower summit.
-
Hi Brian,
Nice to have you on this page. Let’s assume the genetics is constant for a moment. When the population is way below its maximum it is possible to have cooperation operating (mating for instance will be sub-optimal because of difficulty in finding mates). Then you can have hyper-exponential growth operating for a while (see my chapter 4, The Second Principle – Cooperation, in my book “Principles of population dynamics”). Now one of the few data sets showing this difference in numbers is the larch bud moth which has 100,000 fold increases in numbers during the cycle. Unfortunately the low density phase has great sampling errors due to the sparse densities (data can bee seen at http://www2.bren.ucsb.edu/~kendall/pubs/2003Ecology.pdf ). However, there is some indication of hyper-exponential growth at low densities (see figure 1 in this reference). Now it is well known that genetics of bud moth does in fact change during the cycle (dark larch larval morph and lighter pine morph). However, it is doubtful that the genetic changes have much to do with the cycle which is determined (I think) by exhaustion of the food supply and insect parasitoids. Genetics it seems to me has little to do with the causes of the cycles. Now Witting would disagree with me but he has no data to support his supposition. What do you think of this argument? Perhaps it would be best to answer on the page “Real Data – The Larch Bud Moth”? -
Gentlemen, I ask your indulgence in helping me to understand what you write.
1. The number of members in a population (occupying a unit of space, the density of population) is what Berryman calls N(t), what Witting calls nt in his comments and what he calls Nt in his papers and what I have called n(t). As I understand it population, population density and population abundance are all essentially the same thing. That is they vary together. Each is equal or proportional to the others.
Is this correct?
2. Thus n(t) (or N(t) or Nt) changes continuously with time. It changes because there are births and deaths and one or the other may dominate. But at any moment of time there are n members in the population of a given space. By n, here, I mean n(t); the time dependence is to be understood when not expressed explicitely. The important idea is that n(t) is a continuous function of time. At every instant of time n has a value.
Is this correct?
3. One generation later, or as Berryman writes, “one unit of time” later – after time t – the number in the population is n(t+1). But the number in the population has some value at any given time at all. So at some arbitrary time Δt later – maybe Δt is 0.1 of a time unit later, 0.1 of a generation – the number in the population is n(t+Δt).
Is this correct?
If these notions are correct I will know that I can procede. If not, I wait to be corrected.
-
1. I think this is true in general but it is sometimes very tricky. Populations are measured by counting numbers per unit of habitat (e.g., density per 100 grams of foliage, or density per square foot of bark) or numbers per unit of true space (i.e., spatial coordinates such as numbers per square mile). They are also trapped giving a relative estimate (numbers per trap). These can all be interpreted in terms of the true spatial coordinates, but it maybe difficult to get the relationship right.
2. This is also correct but is complicated by the yearly cycle. Death is truly a continuous variable but it changes dramatically in the year. In spring and summer it may be quite low while in winter it may be very high (e.g., insects). Births maybe purely periodic and not continuous. Many insects give birth at a particular time of the year (e.g., in spring) and produce nothing at other times. Births and longevity are a population phenomenon (species evolve them), death is mainly and environmental issue (except for longevity). You are correct that N has a value at each moment in time, but the value of dN is not consistent. It may be positive for a short period of time (due to births in spring) and then negative for the rest of the year. It has a structure! The year is the only truly repetitive event (e.g., warm summers and cold winters = high birth and low death rates in summer and low birth and high death rates in winter). This is true of most creatures (with the exception of humans).
3. This is also true but if an arbitrary unit of time is used the year is lost as a critical variable and all the pattern of uniformity is lost too. That is why I use a year as the critical time dimension.
I hope this is clear.
-
Berryman has made it clear, in his point 2 response, that there are limitations to the assumption that n(t) is a continuous variable. But, subject to Berryman’s informative caveats I gather from him that my three understandings are essentially correct. So let’s go on.
4. Because we want to extract rather than impose any yearly or other time dependence of n we want to allow n(t) to be a continuous variable measured in whatever time units one wishes: years, days, generations, … Following Berryman’s caveats we restrict ourselves to those cases where the population, n, is, indeed, well represented by a continuous single valued function of time. Then dn/dt exists where
limΔt→0(n(t+Δt)-n(t))/Δt = dn/dtand this derivative is also a single valued function of time. Just as n(t) is the instantaneous value of the population at time t, so, too, is this derivative the instantaneous value of population growth at time t. i.e. Both n(t) and dn/dt have specific values at each instant of time.
Can we agree on this?
5. There is something profoundly fundamental about exponential growth. In living systems growth is proportional to the population. The reason is that each member (or pair of members) once produced, begin, themselves, to reproduce so that the greater the number is, the greater the growth is. Cells divide. Every one becomes two. Then these divide. Each member of the community contributes growth so growth is proportional to the number of members in the community. The same for deaths. The rate is proportional to the number available to die. So net growth, dn/dt, is proportional to population number, n. Call the constant of proportionality, R. Then this idea, put mathematically, yields exponential growth. i.e. when R is constant, the differential equation that embodies the idea, is
dn/dt =Rnand its solution is
n(t) = n0eRtwhere n0 is simply the starting population; the number of its members at t=0. This is the archetypical equation of exponential growth. It was called geometrical growth by Malthus in 1778. He took pains to distinguish it from arithmetical growth; today called linear growth. Taking R as a constant with the same meaning as above (perhaps 2%/yr), and n0 as defined above, then linear growth follows the equation:
n(t) = n0(1+Rt)That these yield starkly different behaviors is seen in an animation showing populations evolving both linearly and exponentially in time at http://divineneutrality.org/exponential-growth/
The central role that exponential growth plays is seen in what Berryman writes in Oikos 103: 695-701 (2003).
“… geometric growth is a fundamental and self-evident property of all populations living under a certain set of conditions (unlimited resources), I prefer to think of it as the first founding principle of population dynamics (Berryman 1999) or, if you prefer Malthus’ principle”
Now supposing the birth rate declines. Maybe, instead, the death rate increases. Perhaps because the weather got cold or food became scarce. Then this is accounted for by a new exponent, R’; a smaller one. So R is no longer constant. It may vary with time. R = R(t). In describing living systems the idea is to retain the form of that fundamental exponential relationship and seek to explain events by variations in R.
Writes Berryman in that same paper: “the forces of the environment actually affect the birth and death rates of individual organisms, as summarized by changes in R.”
And he also says: “the problem of explaining and predicting the dynamics of any particular population boils down to defining how R deviates from the expectation of uniform growth (Berryman and Turchin 2001), or what I call the R-function. R = f(B,G,P) where B represents a set of biotic factors (populations of different species, including the one in question), G their genetic properties, and P a set of abiotic factors.”
The idea is that R depends parametrically on time through the time dependence of B and P and perhaps even G. So the fundamental exponential growth form is maintained but R is allowed to depend on time; R = R[B(t),G,P(t)] = R(t). The concept is that exponential growth is always taking place but at a rate that changes with time. Put mathematically:
(1/n)dn/dt = R(t)Estimating R, by matching empirical results to various conjectures about R, is the substance of a large number of papers.
Have I understood your point of view? Have I portrayed it fairly? If not, do correct me.
Results
-
