The Anthropogenic Allee Effect: the importance of doing the math

In 2006, Franck Courchamp, and colleagues, proposed a fundamental idea in conservation called the “anthropogenic Allee effect.” It is named after the classic “Allee effect” in ecology, where populations above a certain threshold size persist and below this size go extinct* (due to the inability to locate mates for example). However, even if we assume populations grow fastest when there are few individuals (the opposite of an ecological Allee effect), changes in human behaviour can drive small populations extinct. This can occur when humans are willing to pay more for products derived from rare species.

Take a hypothetical harvested fish population that obeys the following assumptions

  1. Fishing effort increases if the price consumers are willing to pay for fish is higher than the cost required to extract the fish from the ocean
  2. Harvesters decrease effort when cost is higher than price
  3. Fishers and fish behave like particles of gas randomly bumping into each other in space
  4. The price people are willing to pay for fish stays the same through time

As fishers remove fish from this population, the population size eventually gets small enough that individuals are too expensive to locate and harvest. This leads to a stable equilibrium population size, where below it harvest is too costly and above it harvest is profitable (see fig 1A, blue line is cost per unit harvest, red line is price per unit sold).


Now if we modify assumption (4) and make price per unit harvest higher when the species is less abundant we can create a second equilibrium (price and cost intersect again at low population sizes). Now, harvest is profitable when (1) the species is abundant (because cost of harvest is low) and (2) when the species is rare (because consumers are willing to pay a high price for harvested individuals). Therefore, species with initial population sizes below the unstable equilibrium in Fig. (b) will be harvested to extinction. Initial population sizes above this equilibrium will lead to sustainabe harvest and eventually the population will approach the stable equilibrium on the right.

So is this classic argument correct? It turns out, not exactly. This is a one dimensional argument for a two dimensional model (of both fish and fishers), and while it appears intuitively correct, it is a mistake to ignore harvest effort explicitly. Today I posted a preprint on ArXiv (edit: now out in J. Theor. Biol.), which demonstrates that when you actually do the math, the classic anthropogenic Allee effect models can generate a rich set of previously undiscovered dynamics. Even abundant populations can be driven to extinction, as long as there is a small minimum price people are willing to pay when the population is very abundant.   For example, in one scenario, initial population sizes and harvest effort in the small shaded area (in Fig. 2) cycle, but persist, while populations outside the shaded area go extinct. Note that large populations to the right of the grey area are destined to extinction.**

Figure 2. More complicated population dynamics are possible than Fig. 1 suggests. Traditional theory would say all population sizes to the right of the first black circle will persist, but actually a large percentage of such initial population sizes can lead to extinction.


Jeremy Fox, has a nice list of good and bad reasons for choosing a research project. One of the good ones is

Develop the mathematical version of some verbal idea or hypothesis. Ecology is chock-a-block with influential ideas that haven’t been much developed mathematically. Often, when you try to do the math, you’ll discover key implicit assumptions that weren’t previously recognized, or else you’ll discover that the assumptions don’t actually imply the conclusions they are thought to imply. At worst, you’ll at least make the idea much more precise, and so much more testable. Now, if only someone had had a project idea along these lines back in 1979 or so…

Graphical arguments, based on models, to gain intuition can lead to great ideas, but it is eventually important to follow that up with some math [and/or simulation]. In this case, we have revealed a potential mechanism for populations deterministically crossing an Allee threshold, which would be impossible to intuit just from looking at the model. It’s is hard to tell whether the idea presented here is what drives some harvested populations to extinction (price abundance relationships are difficult to come by), but it seems like a promising mechanism to test, one that I hope will lead to interesting discussions.

*This is actually called a “Strong Allee Effect.” There are also non-threshold Allee effects where population growth rate is reduced at low densities, but is not negative.

**This figure is for a population with linear growth (in the absence of harvest). The green-red dotted loop is what we call in dynamical systems theory, a “homoclinic orbit.” It is broken if we add density dependent growth, but the dynamics in that case are similar. The grey area still exists in the density dependent case (although it isn’t a closed oval), and inside the grey area, populations spiral into the equilibrium.



Holden, MH, and Eve McDonald-Madden. (2017). High prices for rare species can drive large populations extinct: the anthropogenic Allee effect revisited. J Theor Biol. 429, 170-180.


The REAL risk of dying from shark attacks vs. car accidents: the importance of basic fractions

It is summer time here in Australia and hence I find myself at the beach quite a bit. So naturally I want to talk about gruesomely dying in the jaws of a shark. Biologists often claim that the risk of dying from a shark attack is so inconsequentially low that any rational person would ignore it, in comparison to the many risks we take doing mundane activities like driving or taking selfies. Often the statistics quoted go something like this

Number of shark attack deaths pear year: 1

Number of car accident deaths per year: 38,300*

This indeed says that deaths from shark attacks are incredibly rare, but it says absolutely nothing about the relative risk of dying from a shark vs. a car. The numbers are meaningless without an appropriate denominator (that pesky number at the bottom of a fraction). The denominator here is “years”, as the statistic is “deaths per year”, but is that the correct choice for identifying the risk of death when choosing between activities? I don’t think so. There are many people who never venture into the ocean, and of those who do, most visit only a few times per year. In comparison, the average person in the US drives nearly every day. In other words, how many times do people really have the opportunity to encounter a shark?

So below I calculate a more meaningful statistic, the probability of death per instance of exposure (or at least a very rough estimate). Doing so, we can determine the distance one would have to drive in order to obtain the same chance of dying as someone going to the beach and dying from a shark attack. It starts with the numbers below

exposure source
Beach visits / year in USA 110 million (1.1 x 108) National Oceanic Atmospheric Administration
Miles driven / year in USA 3.1 trillion (3.1 x 1012) US Department of Transportation


The risk of dying from a shark attack in a given beach visit is therefore roughly 1 in 110 million and the probability of dying per mile driven is approximately 38,300 in 3.1 trillion (or roughly 1 in 81 million). What does this mean? These numbers are quite close, the risk of death from driving 0.74 miles (or 1.2 km) is about as high as dying from a shark during a beach visit.*

Now you can look at these numbers and think, the risk of dying from a shark attack is so low … it is equivalent to less than a mile (or a little over one km) of driving. Alternatively, you can look at these numbers and say wow … the statistic, “1 death from a shark attack vs. 38,300 deaths from car accidents” really makes the risk of dying from sharks sound a lot more inconsequential than the calculations above. Which camp you find yourself in might depend on how much you drive or visit the ocean without using a car. I’m gladly happy to visit the beach and take such a small risk, completely ignoring the chance of being eaten by a shark, but perhaps the risk isn’t as inconsequential as I once thought. Whatever your thoughts, the reminder here is that it is important to think about the appropriate denominator when talking statistics (there is almost always some assumed denominator, whether we realize it or not … absolute numbers are often misleading).

Photo of great white on surface with open jaws revealing meal.

This oldtime photo of a great white shark is provided by googlesite user TheBrockenInaGlory


*These calculations required some assumptions. First we assumed the numbers from the above sources were true. We also assumed that everyone at the beach goes in the water, which likely isn’t true – the risk of dying due to a shark attack might be more like the risk of driving one or two miles if for example only half of beach goes ever go past ankle-deep in the ocean. We also assumed that shark attacks and auto-accidents occur at a fixed rate for all individuals. This is of course untrue, by driving safely or taking safety precausions in the ocean you can reduce your risk of dying in either situation. We are merely looking at averages here.


Conservation needs to embrace more efficient peer review

Conservation is a crisis discipline. Species are going extinct at an unprecedented rate and therefore scientists and policy makers must act quickly to save them. The peer-review process is useful for quality control, but unfortunately a barrier for quickly disseminating information needed to make the best conservation decisions.

One challenge is that papers are often submitted and rejected from several journals, sometimes over the course of multiple years, before finally getting published. As a paper continues to get rejected, reformatted, and re-reviewed, conservation scientists (authors, reviewers and editors) each waste dozens of hours that could be allocated towards new conservation projects. In addition, policy makers must wait to get the latest credible information.

Solution: peer-review should be done for multiple journals in parallel. Imagine sending out your paper for peer-review and getting back detailed feedback along with a list of journals for which your paper is a good fit. Moreover, the service in charge of this centralized peer-review process contacts the appropriate journals and asks them whether they want the paper to be submitted. After you correct your manuscript, you send it to the interested journal, alongside a response to reviewer comments. If the journal rejects the paper, it is immediately sent to the next journal down your list. No more excessive reformatting, or unnecessary re-reviews, just a more efficient peer review process.

Now what if I told you this system already exists! A non-profit called Axios Review does exactly this, but shockingly the journals which have signed up are mostly pure biology journals, such as Ecology LettersEcology, and American Naturalist. With the exception of Frontiers in Ecology and the Environment, most of the big name conservation journals, such as, Conservation Letters, Conservation Biology, Biological Conservation and Journal of Applied Ecology, are surprisingly not lining up to be officially associated with this type of service. Axios boasts some impressive statistics: once submitted to an interested journal, 85% of their papers get accepted, and half of these accepted papers are not sent for additional review by the journal. On average, a paper going through Axios gets accepted after 1.8 rounds of review (the norm is closer to 5).

I should note that I have yet to use the service myself (partially because of the lack of conservation oriented official target journals), so this blog post is not meant as an endorsement of Axios specifically. Many ecologists already endorse the service, which is likely cost effective for authors.* I am a bit frustrated that conservation seems slow to join the party. If not Axios, we need to think how else we can reform peer-review.

Time is the most important resource in conservation! The peer-review process should reflect this.


*Cost Efficacy for Authors: The fee of 250 USD per article** is no more than the cost of one day of postdoc labour in many developed countries. So the service is likely cost effective, given that authors spend more than one day reformatting and re-submitting a paper.

**Edit: the fee is 250 USD when bought individually, and as low as 200 USD when bought in bulk