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).

AAE

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.

Homoclinic

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.

 

Reference:

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.

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