Extinction in two-species competitions

What is the time-to-extinction in a two-species competitive system? It seems like this is a crucially important theoretical question. After all, the so-called "problem of coexistence" arises precisely because competition theory (at least in the absence of so called coexistence mechanisms) can't account for large observed diversity. But, I wonder, perhaps coexistence itself isn't a stable property, but is a consequence of long waiting times in populations regulated by relatively weak competition and buffeted (or possibly, at small population sizes, buffered) by demographic stochasticity? To me, the natural next step is to look at extinction times in various models. But, turning to the theoretical literature I found relatively little of use, particularly if one is looking for relatively simple analytical solutions to inform intuition. (Please reply with links to any models I've missed.) Turning to the standard generic texts that might be of use (e.g., Gardner. 1985. Handbook of Stochastic Methods) does not induce optimism.

Much as I've wrestled with this question, I've relatively little to show for it. One small contribution is the following exercise that arose in the context of conversations with Peter Adler. Comments, improvements, and alternatives are most welcome. Link to preprint.

Societal learning in Hong Kong during the 2003 outbreak of SARS

Introduction. Recently, I proposed a model for the effects of improvement in intervention effectiveness during a disease outbreak (Drake et al. 2006). In that paper, we used data on the interval between the onset of clinical symptoms and hospitalization to estimate the learning rate during the 2003 outbreak of SARS in Singapore. We also tested (and rejected) the hypothesis that there was any substantial relaxation in the learning rate during the first eight weeks of the outbreak. Here I report results from a similar analysis for the outbreak in the same year in Hong Kong.

Methods. The average interval between symptom onset and isolation by week were obtained by digitizing Figure 2b in Anderson et al. (2004). The approach to analysis was the same: the removal rate was obtained by taking the reciprocal of the symptom-onset to admission interval. Models with and without relaxation were fit to these data using week as the independent variable. Goodness-of-fit is measured with AIC. (The likelihood function results from stipulating that errors are normally distributed with homogeneous variance—a reasonable assumption when the data are removal rate but not symptom-onset to admission interval. That is, I treated this as a simple nonlinear regression problem.)

Results. There was strong evidence for societal learning. If there was relaxation in the learning rate it was negligible. Parameter estimates and confidence intervals for both Singapore and Hong Kong are show in Table 1. The reciprocal of the base removal rate parameter (b) is an independent estimate of the duration of the infectious period in the absence of intervention by hospitalization. Accordingly, for Singapore we obtained an estimate of 8.3 d (95% confidence interval: [5.8, 14.3]). As we previously noted (Drake et al. 2006), this is consistent with measures of viral shedding, obtained by Peiris et al. (2003) using quantitative reverse transcriptase on sequential nasopharyngeal aspirates/throat and nose swabs (NPA/TNS). Compare also (WHO 2003). For Hong Kong, the estimated average period of infectiousness is 6.5 d (95% confidence interval: [4.8, 10.0]).

Discussion. The comparison between Singapore and Hong Kong can be used to address two questions. First, did the different responses adopted in Hong Kong and Singapore lead to overall differences in removal rate (and, ultimately, control). Second, if parameters like a0 and a1 are relatively well conserved across outbreaks, then at the outset of future outbreaks prior information about these rates could be extremely useful for forecasting the size and duration of outbreaks. As the 95% confidence intervals for a0 for both cities are overlapping there is no evidence for any difference between these two locations in the effectiveness of removal. Therefore, to answer the first question, we have no evidence here for a difference in the overall effectiveness of response. Concerning the second question, we have the first evidence that there is some conservation of these quantities (at least across these two societies). It remains to be seen if further information would in fact reveal differences; if the responses of these cities although relatively similar would differ from responses in other parts of the world; and, finally, if societal learning is conserved across different strains of infections and/or different infectious agents altogether.

References.

Anderson, R.M., C. Fraser, A.C. Ghani, C.A. Donnelly, S. Riley, N.M. Ferguson, G.M. Leung, T.H. Lam, and A.J. Hedley. 2004. Epidemiology, transmission dynamics and control of SARS: the 2002-2003 epidemic. Philosophical Transactions of the Royal Society of London, Series B 359: 1091-1105.

Drake, J.M., S.K. Chew, S. Ma. 2006. Societal learning in epidemics: intervention effectiveness during the 2003 SARS outbreak in Singapore. PLoS ONE 1(1): e20. doi:10.1371/journal.pone.0000020 [pdf] [web version]

Peiris, J.S.M. 2003. Clinical progression and viral load in a community outbreak of coronavirus-associated SARS pneumonia: a prospective study. Lancet 361:1767-1772.

WHO. 2005. Summary of probable SARS cases with onset of illness from 1 November 2002 to 31 July 2003. http://www.who.int/csr/sars/country/table2004_04_21/en/index.html. Accessed 27 December, 2005.

This work is licensed under a Creative Commons Attribution-ShareAlike 2.5 License.

Societal learning in in the 2003 SARS outbreak in Singapore

Standard models for disease dynamics mostly assume that parameters affecting disease transmission and spread are constant during the course of an outbreak. Although seasonal outbreaks of some infectious diseases might be driven by internal nonlinear dynamics, recent studies on seasonal outbreaks in wildlife (Altizer et al. 2006) and the age-old observation that some illnesses follow a clear seasonal pattern (Dowell 2001) suggest that there is more to the story than this. Even measles, the paradigm example of nonlinear disease dynamics, exhibits an element of seasonal forcing (Grenfell et al. 2002).

The temporal forcing addressed in the above studies is relatively slow, generally on the order of months or years. For emerging diseases, however, one suspects that disease parameters will change rapidly. And not due to the changing of seasons, but rather to a change in society--our increased effectiveness at controlling the disease as we learn about it.

In a new paper in the open access journal PLoS One, we explored this idea using SARS as a model system. First, we considered a variation of a model developed by David Kendall in the 1940's, the "non-homogeneous birth-death process", which we interpreted as a stochastic version of of the familar S-I epidemic. We studied two special cases of this model. Both assumed that the effect of "societal learning" was on the rate at which symptomatic infectious individuals were removed from the population, but they differed in how this process was represented. In the first model, we supposed that removal rate increased according to a constant, but unknown, factor. In the second model we supposed that this constant factor obtained initially, but with diminishing returns over time as control options are used up and further accelerating removal of infectious persons becomes increasingly difficult. This latter process we called "relaxation".

Interestingly, unlike most realistic disease models, which are highly computational, this model is analytically tractable. When we investigated the resulting formulas for the distribution of the duration of outbreak and the expected epidemic size, we found that the effect of relaxation was determined by its interaction with the basic societal learning rate. However, in most cases, where the basic learning rate is relatively high to begin with, the effect of relaxation was swamped by the basic learning effect. In sum, what the model suggested was common sense--tackling a disease outbreak early on is one of the most important factors determining control. What might be a little surprising is the relatively large size some outbreaks will achieve, even when very effective controls are implemented very early.

To take this idea to the next step, we fit our basic and relaxation models to weekly data on SARS in Singapore. SARS had already been shown to exhibit this kind of external forcing (Lipsitch et al. 2003), but it was not clear if a really simple model, like the one we developed, could do a very good job at recapturing the observed outbreak size. Two results emerged from this analysis. First, the simple model from Kendall did a pretty good job of predicting the final outbreak size. The actual outbreak of 238 cases was pretty near the expectation based on the model (278 cases) and well within the interval representing 95% of simulation runs (56 cases at the low end and 611 cases at the high end). Second, perhaps remarkably, the statistical analysis produced no evidence for relaxation over the 8 week duration of the epidemic studied here.

One lesson this analysis reinforces is that the magnitude of response to an outbreak at the very beginning can be a crucial factor determining its ultimate size. There may also be a technical quantitative lesson here, however. Forecasting epidemic size at the start of an outbreak is a notoriously difficult thing to do, mostly because it's so hard to know how effective the response will be. If estimates of the learning rate, such as obtained here, turn out to be relatively transportable between populations, then these could be incorporated into forecasts from the very beginning. Of course, ultimately the parameters determining infectious contact rates depend on the society in which the disease emerges and will differ among populations. What is necessary for forecasting is only that learning rate be relatively well conserved--not exactly. Whether this is so is an outstanding empirical question.

Altizer, S., Dobson, A., Hosseini, P., Hudson, P., Pascual, M., and Rohani, P. 2006. Seasonality and population dynamics: infectious diseases as case studies. Ecology Letters. 9:467-484.

Dowell, S.F. 2001. Seasonal variation in host susceptibility and cycles of certain infectious diseases. Emerging Infectious Diseases 7:369-374.

Grenfell, B. T., Bjornstad, O. N. & Finkenstädt, B. F. 2002. Dynamics of measles epidemics. II. Scaling noise, determinism and predictability with the time series SIR model. Ecological Monographs 72:185-202.

Lipsitch, M. 2003. Transmission dynamics and control of severe acute respiratory syndrome. Science 300:1966-1970.

This work is licensed under a Creative Commons Attribution-ShareAlike 2.5 License.

Extinction times in variable environments: experimental observations inconsistent with theory

The effect of temporal environmental variation on population fluctuations and extinction is a longstanding problem in population biology. Readers of my research will recall an experiment in which we manipulated the sequence of food resource made available to populations of water fleas (Daphnia magna) to test the hypothesis that extinction rates increase with environmental variability. Results from this experiment were previously reported in Ecology Letters and PLoS Biology and discussed on this blog.

A new paper reports some further analysis in which I sought to determine if the shape of the time-to-extinction distribution fit theoretical expectations. Basically, the distribution was extraordinarily peaked--more so than one would have expected. More interesting, though, is the following unexplained result. Using a non-parametric model I estimated the survival function, which is just a transformation of the probability density function (pdf) of time-to-extinction. Converting to the more familar pdf (Figure 4 in the paper) and plotting on a log-scale shows that at first the probability of extinction decreases as an exponential function (a striaght line on a log plot or log-log plot) but then around 40 days flattens out giving a "fat" tail to the distribution. In short, lots of populations go extinct very quickly. But, after that, the rest of the populations hang on for a long, long time.

This observation (which, if it happens in nature would be of considerable importance to population viability analysis) throws a kink in our understanding of extinction. To my knowledge, while the exponential part is predicted by theory (e.g., Ludwig. 1996. Am. Nat. 147:506-526 and Middleton & Nisbet 1997. Ecol. Apps. 7:107-117), the fat tail is entirely unexplained.

How common is this fat tail? Because extinction times have seldom been quantitatively studied it is difficult to say. However, a forthcoming paper by Duncan and Forsyth ("Modelling population persistence on islands") shows a qualitatively similar pattern (see Figure 1).

So, at the moment I'm stumped. Anybody with an idea or an explanation is welcome to weigh in. In case it makes it easier, I've made the raw data and R-code available on the Ecological Archives wesbite, while the original paper (and others) can be downloaded from my homepage. As a starting point (or maybe a red herring), I note that while the generation time is probably around 7 days, the point at which the pdf changes shape (40 d) is possibly around individual expected life span under these experimental conditions. That is, the pattern changes when the initial generation has died off. But knowing whether this or some yet undreamt explanation is correct awaits a cleverer scientist than me.

This work is licensed under a Creative Commons Attribution-ShareAlike 2.5 License.

How does environmental variation affect population dynamics?

The essay by Sæther and Engen [1] highlighting my experiment (with David Lodge) to test aspects of the stochastic theory of population growth [2] raises an intriguing question that we hadn't previously considered. We originally sought to determine if the chances of extinction and establishment, and the distribution of times to extinction and establishment, were affected by the magnitude of fluctuations in available food (environmental stochasticity) when the average amount of food was held constant in the long run. Our results showed that the fraction of populations going extinct increased and that the fraction of populations establishing (attaining high densities) decreased under higher levels of variation compared with controls, and that the average time to extinction decreased while there was no effect on time to establishment. The conventional explanation of these results (the one predicted by theory) is that times to extinction and establishment (and the chances of extinction and establishment) are directly affected by variation in vital rates over time because the long-run growth rate is the geometric mean of instantaneous growth rates, not the arithmetic mean [3].

The experimental results we reported are consistent with this theory and were therefore interpreted as providing an important piece of theory confirmation. Sæther and Engen [1] introduce an interesting alternative explanation for these results, which would also account for the failure to detect an effect of environmental variation on time to establishment. They suggest that environmental variation could have a direct effect on mean vital rates, and not just on temporal variation in vital rates. This could occur, for example, if model parameters are sensitive to extreme values in the distribution of environmental conditions, though we are not aware of any physiological (or other) theory that predicts whether the effect would be to increase or decrease the long-run growth rate. One might nevertheless expect the results to be different. This hypothesis deserves further theoretical elaboration and experimental investigation. It is conceivable that such a phenomenon might affect populations of threatened and endangered species, and many other species if, as predicted, global climate change magnifies environmental stochasticity [4].

Although we did not consider this hypothesis in our original paper, further analyses [5] of the original data now allow us to reject it, at least with respect to the original experiment. These analyses suggest that the population fluctuations observed in this experiment can be modeled with a stochastic version of the conventional logistic model of population growth. It is convenient from the perspective of the Sæther-Engen hypothesis that the two parameters of the stochastic model have biological interpretations, which pertain to the per capita rate of population increase at small population sizes (r) and the carrying capacity (K). If the Sæther-Engen hypothesis—that environmental variation directly affected the mean vital rates—is correct, then parameter estimates will differ when fit to data from each of three different experimental treatments (low, medium, and high levels of environmental variation). Using a subset of the original data, we fit the best estimates of r and K using conditional least squares. Confidence intervals for these parameters were obtained by fitting the model to one thousand datasets for each experimental treatment obtained by nonparametric bootstrap. The estimated parameters for the different treatments are indistinguishable (Table 1).

treatment	r		K
	estimate	95% CI	estimate	95% CI
low (CV=0)	1.6	(1.5, 1.7)	12.7	(11.9, 13.7)
medium (CV=1)	1.5	(1.4, 1.7)	13.5	(12.1, 15.4)
high (CV=2)	1.6	(1.4, 1.8)	12.0	(10.7, 13.9)

Table 1. Estimates of stochastic model parameters do not differ among populations experiencing three levels of environmental variation.

Although the alternative hypothesis fails to account for the experimental results observed here, it is an intriguing idea and two aspects particularly deserve future consideration. First, we need a theory to predict how mean vital rates might respond to environmental variation. Such a theory will go beyond the classical theory of storage effects [6, 7] integrating ideas from population biology and physiological ecology to develop a theory of metabolism in fluctuating environments. Consider, for example, the population dynamical consequences for organisms that consume at higher average rates in fluctuating environments to ensure that sufficient energy stores are available for metabolism during leaner periods. Because of high average consumption rates, such organisms might have higher average lifetime productivities resulting in higher (average) rates of increase. Alternatively, if environmental fluctuations are of low frequency so that energy reserves cannot carry organisms from one period of resource abundance to another, then the carrying capacity might be determined by the fatness of the tails of the distribution of available resources, rather than by the mean. Such a diminished carrying capacity could exhibit both direct and indirect effects on fertility and survivorship. These admittedly sketchy considerations suggest that the population dynamical consequences of environmental variation could be complex, and the development of such theories will require careful consideration.

Second, theoreticians and empiricists need to test the effect of spatial heterogeneity through time [e.g., 8]. Specifically, we need to know the conditions under which individual organisms can metabolically average over temporal fluctuations in resource availability and conditions under which they cannot, and to determine how fluctuations considered spatially-temporally will affect the dynamics of populations differently than simple spatial heterogeneity. Thus the theory of population dynamics in fluctuating environments continues to suggest novel and important questions for research, which require the integration of physiological and ecological considerations.

1. Sæther, B.-E. & Engen, S. (2004). Stochastic population theory faces reality in the laboratory. Trends Ecol. Evol., 19:351-353.
2. Drake, J.M. & Lodge, D.M. (2004). Effects of environmental variation on extinction and establishment. Ecol. Lett., 7:26-30.
3. Lewontin, R.C. & Cohen, D. (1969). On population growth in a randomly varying environment. Proc. Natl. Acad. Sci. USA, 62:1056-1060.
4. McLaughlin, J.F., Hellmann, J.J., Boggs, C.L. & Ehrlich, P.R. (2002). Climate change hastens population extinctions. Proc. Natl. Acad. Sci. USA, 99:6070-6074.
5. Drake, J.M. (2005). Density dependent demographic variation determines extinction rate of experimental populations. PLoS Biology 3:1300-1304.
6. Cáceres, C.E. (1997). Temporal variation, dormancy, and coexistence: a field test of the storage effect. Proc. Natl. Acad. Sci. USA 94:9171-9175.
7. Warner, R.R., & Chesson, P.L. (1985). Coexistence mediated by recruitment fluctuations: a field guide to the storage effect. Am. Nat. 125:769-787.
8. Pickett, S.T.A. & Cadenasso, M.L. (1995). Landscape ecology: spatial heterogeneity in ecological systems. Science 269:331-334.

This work is licensed under a Creative Commons Attribution-ShareAlike 2.5 License.

Modelling ecological niches with support vector machines

For a new application of machine learning to ecology, see our paper on Modelling ecological niches with support vector machines.

Support vector machines are a new type of machine learning algorithm. Particularly, some support vector machines are specially devised for estimating the support of a high dimensional distribution. These include the support vector data description of Tax and Duin (2004) and also the approach of Scholkopf et al. (1999). From G.E. Hutchinson's conception of the ecological niche as the set of environments supporting population persistence it follows that ecological niche identification can be conceived as a special case of support estimation and that support vector machines might therefore be effectively used for this purpose (see also Guo et al. 2005). This paper reports results from comparing three different methods for ecological niche identification with support vector machines using data on the presence and absence of 106 woody plants in the sub-Alpine region of the Canton de Vaud, Switzerland. We find that these approaches are comparble or superior to other frameworks for niche modeling currently available.

Drake, J.M., C. Randin. A. Guisan. 2006. Modelling ecological niches with support vector machines. Journal of Applied Ecology. [web version] [pdf]

The purpose of this blog

This weblog is to be a forum for disseminating and discussing results from research in ecology and environmental science, population biology, and epidemiology. It is intended to contain primarily technical content. For a short bio and brief desciption of my research interests visit my webpage at NCEAS, the National Center for Ecological Analysis and Synthesis, where I work.

Part of the justification for a publicly accessible blog is to engage other scientists in the development and analysis of ideas and interpretation of results prior to their being presented at professional conferences and submitted to journals for publication, as research often can be improved by engaging other researchers earlier in the process. Another part is to devise and explore possible explanations for anomalous or confusing results. A final justification for a publicly accessible blog is to solicit opinions from the American public (whose taxes pay for a great deal of the basic research performed in this country) and the global public (who have an interest in scientific developments and technology wherever it is produced and regardless of who pays for it). Therefore, the reader is asked: Is this research relevant? Is it useful or interesting? How might it have come to wrong conclusions or be misleading? What has beeen missed? What other questions could or should be asked? What problems in ecology or environmental health are not being adequately investigated? Depending on context, answers to these questions can probe the limits of shared scientific understanding and be of the highest importance to our self-understanding in the modern, technological world or might simply matter to my workaday life and the lives of other researchers with whom I share it. At whatever level, fellow scientists and other interested persons are invited to participate.

Some details:

1. Necessarily, the opinions expressed on this webpage are my own and do not reflect official positions of my past, current, or future employers.
2. Comments to posts on this blog are welcome, but will be moderated. Inappropriate or gratuitous comments (positive or negative) will not be posted. Any serious ideas, however unsophisticated or developed, will be posted.
3. This blog is not anonymous (you know who I am - John Drake). The primary reason being that, in my view, one's ideas only have force when one is willing to stand behind them. This means inviting constructive criticism. There are times or topics for which anonymity is appropriate. This blog is not one of them.

This work is licensed under a Creative Commons Attribution-ShareAlike 2.5 License.