Levy Walks in the Ocean

A paper published this week in Nature about Levy Walks among marine predators.

Reference info, title, and abstract follow:

Letter

Nature 451, 1098-1102 (28 February 2008) | doi:10.1038/nature06518; Received 17 October 2007; Accepted 29 November 2007

Scaling laws of marine predator search behaviour

David W. Sims, Emily J. Southall, Nicolas E. Humphries, Graeme C. Hays, Corey J. A. Bradshaw, Jonathan W. Pitchford, Alex James, Mohammed Z. Ahmed, Andrew S. Brierley, Mark A. Hindell, David Morritt, Michael K. Musyl, David Righton, Emily L. C. Shepard, Victoria J. Wearmouth, Rory P. Wilson, Matthew J. Witt & Julian D. Metcalfe

Many free-ranging predators have to make foraging decisions with little, if any, knowledge of present resource distribution and availability¹. The optimal search strategy they should use to maximize encounter rates with prey in heterogeneous natural environments remains a largely unresolved issue in ecology^{1, 2, 3}. Lévy walks⁴ are specialized random walks giving rise to fractal movement trajectories that may represent an optimal solution for searching complex landscapes⁵. However, the adaptive significance of this putative strategy in response to natural prey distributions remains untested^{6, 7}. Here we analyse over a million movement displacements recorded from animal-attached electronic tags to show that diverse marine predators—sharks, bony fishes, sea turtles and penguins—exhibit Lévy-walk-like behaviour close to a theoretical optimum². Prey density distributions also display Lévy-like fractal patterns, suggesting response movements by predators to prey distributions. Simulations show that predators have higher encounter rates when adopting Lévy-type foraging in natural-like prey fields compared with purely random landscapes. This is consistent with the hypothesis that observed search patterns are adapted to observed statistical patterns of the landscape. This may explain why Lévy-like behaviour seems to be widespread among diverse organisms³, from microbes⁸ to humans⁹, as a 'rule' that evolved in response to patchy resource distributions.

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So there has been a lot of attention to this Levy Walk stuff in recent years. Do animal movements form power law distributions in terms of the length of each 'flight' they take? If you track an animal as it moves, it goes in a straight line for a while and then turns. These straight line lengths between turns are called paths or flights. A Levy Flight distribution is one where the histogram of these flights has a really long tail toward long flights and the probability of finding a flight of any given length is a power law distribution with an exponent between about 1 and 3 (around 2 being typical). This means there are a lot of small lengths and a few really long ones and that as length increases by some factor the probability decreases by a constant factor (the exponent). Some folks think the importance of these Levy Flights is way overblown and some people think its a big deal. If its a big deal its because we are learning something fundamental about how foraging behavior is organized and presumably how that organization reflects something about the underlying ecology or prey distribution. Are these things adaptive? To get a visual, imagine taking 'flights' while foraging that were all the same length. You might deplete all the resources in your immediate surroundings efficiently but what happens when you have to move a long way? So lots of small steps and a few long ones might be the way to go to efficiently search in your foraging habitat - and that is more or less what the Levy Flight is about.

Here they argue that Levy Flights are optimal foraging strategies that reflect the underlying distribution of prey. Hopefully we'll know more about the mechanistic links between path length distributions and prey distributions in the future. Also note that the book on Foraging that I have been blogging about does not cover this new work on Levy Flights even though this would certainly be a hot topic to some. Of course I don't blame the editors of the book as you can't fit in everything.

Anyway, its an interesting paper that is worth checking out.

Best,
Oskar

Reproducing in Cities: recent paper by R. Mace

A recent perspective by Ruth Mace published in Science (subscription required) gives a perspective on the demographic transition - the transition to reduced fertility rates among wealthy nations and economic classes. The argument is based on the costs of childrearing that seem to increase with urbanization and the lowered rates of infant and child mortality that can accompany city life in contexts where sufficient health care and sanitation are available.

Here's the paper's abstract:

"Reproducing in cities has always been costly, leading to lower fertility (that is, lower birth rates) in urban than in rural areas. Historically, although cities provided job opportunities, initially residents incurred the penalty of higher infant mortality, but as mortality rates fell at the end of the 19th century, European birth rates began to plummet. Fertility decline in Africa only started recently and has been dramatic in some cities. Here it is argued that both historical and evolutionary demographers are interpreting fertility declines across the globe in terms of the relative costs of child rearing, which increase to allow children to out compete their peers. Now largely free from the fear of early death, postindustrial societies may create an environment that generates runaway parental investment, which will continue to drive fertility ever lower."


The paper provides an interesting perspective on fertility in urban life and of course on the demographic transition in general. Mace essentially argues that the demographic transition is the result of the quantity quality tradeoff.

"Industrialization and urban living enable new professions to emerge, some of which are only available to those invested with considerable capital or training." .... "Whether perceived relative costs are equivalent to actual costs is a moot point; but, in essence, historical and evolutionary demographers are converging on similar explanations for demographic change. The cost of raising a child includes enabling it to compete with its peers—for marriage partners, for jobs, or for the means to support a family—and if that competition increases costs, then basic evolutionary ecology predicts that optimal fertility will decline (16). Education introduced a new mechanism through which children could compete for future employment opportunities. School also adds pressure on parents to present adequately fed and dressed offspring for public scrutiny."

Figure 2, pasted in below, represents the basic schematic for the argument. Mace says that in many instances wealth and fertility are often positively correlated within these 'homogenous subpopulations'. While this may indeed be true (I think that it is and would argue in favor of that point), the data supporting it are really not that good and the source Mace cites as proving
that wealth and family size are usually positively correlated really provides only weak support and is based on little data.

Fig. 2. Schematic diagram of how different levels of parental investment per child can generate positive relationships between fertility and wealth in subpopulations (where each of the diagonals represents a different subpopulation within a larger society) but a negative relationship between wealth and fertility over a large heterogeneous population. Levels of parental investment per child may be highest in urban areas.

Mace is skeptical the opinion that these trends in fertility decline simply represent shifts in cultural values that have rapidly swept over the globe and I think for good reason. Clearly the pattern is consistent in very different cultural contexts (Europe, Americas, and Africa have all gone through similar demographic transitions following broadly similar correlations between economic diversification and urbanization). Instead, there is good evidence for a general evolutionary mechanism at work. So the general trend of having "few higher quality offspring" started as initially advantageous, accelerated with lower levels of child mortality and the increased importance of inherited wealth and status, and then became subject to runaway selection.

"Transfers of resources from parents to offspring are key to understanding human life-history evolution (23). In wealth-owning societies, siblings compete with each other for their parents' material and intellectual resources (24–26). If parental investment is a key influence on children's future success, and the ability to invest effectively in children is heritable (and cultural traits such as wealth and status are usually highly heritable), then it is possible that runaway cultural selection has occurred in preferred levels of investment in each child (27), driving the quantity/quality trade-off further in the direction of offspring quality. Hence, I argue that the emergence of postindustrial life, now largely free from the fear of early mortality, seems to have generated conditions under which a runaway process of ever-escalating levels of investment in our children continues to drive fertility ever lower."

The link to 'cultural selection' could be a bit more clear in order to differentiate it from the widely spreading cultural shift idea of some cultural demographers which she criticizes earlier in the paper. Readers who are not already on Mace's side of the argument may not see the difference, and its not entirely clear. However, I am on Mace's side of the argument and think that evolutionary mechanisms do indeed underly fertility decisions of this sort. The runaway selection idea is one that deserves more attention. I'd like to see some modeling or analytical work showing exactly how it could lead to below replacement fertility. Perhaps some of the references in the paper accomplish this. On the whole, its good to see this sort of approach being applied in human demography. Being a UNM graduate student though, I couldn't help but notice that some of the highly relevant work done by some faculty and/0r grad students here was not cited or included in the paper (e.g., Moses and Brown 2003; Kaplan 1996), but that is clearly only a gripe driven by hometown bias.



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Kaplan H. (1996) A Theory of Fertility and Parental Investment in Traditional and Modern Human Societies. Yearbook of Physical Anthropology, 39, 91 - 135

Moses M.E. & Brown J.H. (2003) Allometry of human fertility and energy use. Ecology Letters, 6, 295-300

Some Mac Links

A paper has recently been published in Plos one about the peopling of the new world that uses a large sample of mtDNA from Native Americans to assess population history. They argue for a three stage model that essentially supports the idea that the new world was colonized by northeast Asian groups who crossed Beringia and an ice-free corridor through northwest North America. They do argue that the origin of Clovis culture may go back to 16,000 years or more, which is a bit older than generally thought.
Blogs on this paper can be found here (Anthropology.net) and here (Dienekes).

Also of potential interest is a recent proposal to start an institute of human origins at the University of California San Diego. The news article explains that:
"The Center for Academic Research and Training in Anthropogeny will be “trans-disciplinary,” said Varki, who will be co-director with Gage and Schoeninger. (Anthropogeny is the study of human evolution.)

“It will be more than multidisciplinary,” he said. “CARTA will transcend disciplines, bringing together biologists, social scientists, neuroscientists, chemists, medical specialists – anybody who can bring insight into the question of where we come from.”

Said Schoeninger: “The center will allow us to move well beyond the bounds of any given field of study. Looking at the biological and cognitive links between humans and other primates or other animals – and doing so not only with the breadth afforded by different disciplines, but also with the depth offered by an evolutionary perspective – will give us a richer picture of the past and of today.”"

Sounds pretty cool. We need more work on figuring out what makes humans human.

Human macroecology is interested in underlying mechanisms of the growth and form of human institutions/groups/populations at multiple scales. A recent study in Science magazine covered geometric principles of the growth and form of cities that is covered by Science Daily and worth a quick read.

Cheers,

O

Quantity/quality offspring tradeoff in humans and other primates

The following article is available online in firstcite with the Proceedings of the Royal Society B.

Title: The tradeoff between number and size of offspring in humans and other primates

Abstract: Life-history theory posits a fundamental trade-off between number and size of offspring that structures the variability in parental investment across and within species. We investigate this ‘quantity–quality’ trade-off across primates and present evidence that a similar trade-off is also found across natural-fertility human societies. Restating the classic Smith–Fretwell model in terms of allometric scaling of resource supply and offspring investment predicts an inverse scaling relation between birth rate and offspring size and a −¼ power scaling between birth rate and body size. We show that these theoretically predicted relationships, in particular the inverse scaling between number and size of offspring, tend to hold across increasingly finer scales of analyses (i.e. from mammals to primates to apes to humans). The advantage of this approach is that the quantity–quality trade-off in humans is placed into a general framework of parental investment that follows directly from first principles of energetic allocation.

Authors: Robert Walker, Michael Gurven, Oskar Burger, Marcus Hamilton

I have a biased perspective but I think this is a really good paper. It combines the classic model of the quantity/quality tradeoff in life history theory developed by Smith and Fretwell with a recent model by Charnov and Ernest (citations below).
The Smith-Fretwell model basically says that the relative cost of a kid C is the total energy budget mom has to put toward making kids R divided by the number of kids she has N. so C = R/N. This also means that the number of kids then is given by N = R/C. This is a pretty straight forward model that works well. So the higher the cost of an average kid to an average mother in a species or population the fewer kids the average mom will have. We know from life history theory that mom's energy budget R is a function of her mass and that a 3/4 power allometry of body mass is a pretty reasonable estimate for this energy budget. We also know that a reasonable estimate of the cost of a kid seems to be mass at weaning, which is a linear function of mom's mass - about .3M among mammals where M is mom's mass. This means that on average mammal offspring are dependent on their moms for energy until they are about 1/3 her size (Charnov 1993 and others). Anyway, these two observations can be placed into the Smith-Fretwell model to predict another well-known allometric relationship, the -1/4 power scaling of fertility rate with body mass. This happens because R is proportional to M^3/4 and C is proportional to M^1 so we get that N can be predicted by M^3/4 divided by M^1 which gives us M^-1/4. Or we can use these same expressions to look at the relationship between the number and size of offspring, which is predicted to be an inverse relationship (N/R = 1/C ~ M^-1). Our analysis demonstrates that the theory predicts the actual empirical trends in humans and primates.

If this is kind of thing is new to you just realize that when we look for patterns across large numbers of species - like all mammals or all birds - we find these really consistent relationships where a lot of important traits seem to be largely constrained (or at least well-predicted) by the average adult mass of the species. Three of these traits used here are metabolic rate, which is taken to be energy budget, the size of the offspring when its independent from its mom (important because that's when the mom can start making new kids if she wants so its a key constraint on fertility), and these two predict another - that fertility rate is slower with larger animals than with smaller ones.

We need to keep in mind that mass at weaning is generally a good proxy for the measure we are really interested in which the energetic cost of the offspring to the mother. For primates and in humans in particular, however, this may not be a good approximation. Human mothers invest much more in their offspring as human kids are often dependent long after they are weaned.
We take this into account by looking at mass of the offspring at ages older than the typical age at weaning and find that the model works better as a result. So this basic prediction of life history theory, that a tradeoff exists between the number and size of offspring, is met with data on human groups (natural fertility, small-scale populations) and among primates. There are some other points to the paper as well and we hope people find it interesting and that it provokes further research in this area. comments welcome of course.

I'll post a stable link to the pdf of the paper as soon as I can but something is wacky with UNM's server so I can't right now. [Ok, here's a link to a pdf of the paper.] If you are on a computer that has access to the proceedings of the royal society then you should be able to get it here.

best,
Oskar



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Charnov, E.L. & Ernest, S.K.M. 2006 The offspring-size/clutch-size trade-off in mammals. Am. Nat. 167, 578–582, (doi:10.1086/501141).

Charnov, E.L. 1993 Life history invariants: some explorations of symmetry in evolutionary ecology. Oxford, UK: Oxford University Press.

Smith, C.C. & Fretwell, S.D. 1974 The optimal balance between number and size of offspring. Am. Nat. 108, 499–506, (doi:10.1086/282929).