Still do today. Must’ve worked, like mathematicians say.

Further to: Salamander rule 1: Stroke ain’t broke, so don’t fix it – for 270 million years (and don’t the salamanders just *look* like the type?), it turns out that, according to From *ScienceDaily*:

Animals have used the same technique to search for food that’s in short supply for at least 50 million years, a University of Southampton-led study suggests.

Researchers analysed fossilised sea urchin trails from northern Spain and found the tracks reflect a search pattern still used by a huge range of creatures today.

But this is the first example of extinct animals using such a strategy.

The findings could explain why so many modern animals use the technique, and suggest the pattern may have an even more ancient origin.

Actually, the findings don’t necessarily explain why modern animals use the technique. They don’t do the math either. It could be natural selection acting on random mutation (Darwinian evolution), it could be unrelated animals arriving at the same successful technique (convergent evolution), or for all we know, it could be horizontal gene transfer of genes that govern the initiation of such a search, or epigenetics (strategy arrived at in lifetime of individual gets encoded in the genes). All these processes sometimes occur.

Creatures including sharks, honeybees, albatrosses and penguins all search for food according to a mathematical pattern of movement called a Lévy walk — a random search strategy made up of many small steps combined with a few longer steps. Although a Lévy walk is random, it’s the most efficient way to find food when it’s scarce.

Even though a wide range of modern creatures search for food according to this pattern, scientists had no idea how the pattern came about, until now.

Professor Sims and colleagues from the University of Southampton, NERC’s National Oceanography Centre, Rothamsted Research, VU University Amsterdam and the Natural History Museum analysed the fossilised Eocene-era tracks that were made by sea urchins that lived on the deep sea floor around 50 million years ago. The long trails are preserved in rocky cliffs in a region called Zumaia in northern Spain.

“The patterns are striking, because they indicate optimal Lévy walk searches likely have a very ancient origin and may arise from simple behaviours observed in much older fossil trails from the Silurian period, around 440 million years ago,” he adds.

Now that will be interesting, if demonstrated.

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Even though a wide range of modern creatures search for food according to this pattern, scientists had no idea how the pattern came about, until now.And they still don’t know HOW.

OldArmy thanks for saying what I was also just about to post. All they know surely, is that animals half a billion years ago employed the same strategy – yet the tag line with that finding is they know how the pattern came about. What an absurd statement?!

And ID is rebuked for far less irrational statements…

In fact, we should challenge them to play the same game they challenge IDists:

“It’s not a valid answer until you have a mechanism.”

So really, they just know ancient animals used this pattern. But that is no mechanism. So this is not real science as it does not provide a mechanism. So we should reject the notion and findings.

Sound familiar?

It’s been about 10 years since I stumbled over Levy-random walk. Paul Levy was a French mathematician in the 1930’s, who was asking some questions about statistics. Here’s a brief explanation of the maths:

Let us suppose that you can walk with 10 different stride lengths–a short one when you are on ice, a medium one for grass, and a really long one when you walk on concrete. Now let us suppose that you randomize your stride, short steps followed by long or medium steps. The probability for talking a step of length L, is the PDF, the probability density function. Okay, one more assumption. Let us assume that you take a step at some arbitrary compass direction–first north, then sw followed by east and etc. How fast do you travel?

This is a restatement of the famous “drunk staggering around lamppost” illustration for diffusive travel. Now it is well known that since the drunk is equally likely to go east as west, his average displacement is zero => =0, and it looks like he’s not getting anywhere. But he is! His average squared displacement, =/= 0 is not zero. So we can define a “diffusion speed” as being the “time rate of change of the average squared displacement”, or,

D == d/dt

This math is found in every textbook, and describes the “diffusion” speed at which perfume molecules diffuse through a room. But Paul Levy asked another question. What happens when the drunk takes big steps and little steps, what happens to this diffusion speed if the PDF is not little baby steps all the time?

(Hang in there, we have one more mathematical definition to go.)

Well now we need to calculate what an average, or means. If I add together all my test scores, and divide by the number of tests, I have an average score. But what happens when the prof puts more weight on the final, and less on the midterm? This “weighting” is exactly the same magic as the PDF, and the correct way to calculate it is:

Integrate( score * weighting) / Integrate(weighting).

That is, if the prof doubles the value of the final, it shouldn’t cause students to get a 150% of an A, it should be “averaged out” by lowering the midterm effects. In other words, the weighting has to come out 100% however it is distributed, and that “normalization” is the term in the denominator.

Now we are ready for Paul Levy’s contribution. If we want to calculate the “average square displacement” then we put that into the “Score” part of the integral, and the PDF goes into the “weighting” part of the integral like this:

D = Integrate (x^2 * PDF) / Integrate(PDF).

What kind of PDF can we use to calculate this? Well, if the stride size of our drunk is usually pretty small, but very rarely big, we can say:

PDF = (1/x)^n

so when stride (=x) is small, then it is very likely, and when stride is big, it is less likely, and the “n” is the power-law telling us just how unlikely a big stride will be. We use power-laws because it makes the maths easier and also because Nature seems to use them very frequently. There are very interesting reasons for Nature’s power laws which have to do with fractal numbers and the “range” over which objects interact, but Levy didn’t get into that because he was a mathematician.

Well what sort of PDF does perfume have? We usually think of the “stride” of a gas-perfume collision as being a Gaussian distribution centered on some step size X, where x = A exp[ -sig (x-X)^2], and simplifying the exponential function, we get:

x ~ 1/ (1 + x^2 + x^4/2 …) where we throw away the leading term. Plugging that into our function we get:

D = Integ(x^2 * 1/x^2) = constant.

In fact, any distribution that looks vaguely Gaussian or random, ends up with this same diffusion speed.

Then Levy noted a very interesting result. What if the PDF has more big strides in it than a Gaussian? What if it doesn’t diminish as (1/x^2) but only 1/x?

Well, when you integrate the average, one goes from -infinity to +infinity, and if the “n” in the PDF was infinity.

The diffusion speed is infinite! If this were perfume, then a PDF of 1/x would cause the perfume to speed through the room “super-diffusively”. It isn’t going the speed of light, but just a whole lot faster than you expected.

This “faster-than-perfume” speed is now known as “Levy-flight” and enables a sea-urchin or a fly to cover more territory than a stupid perfume molecule. That’s a no-brainer.

But there is one niggling detail left out of this science article. A PDF that goes as 1/x instead of 1/x^2 is not random, it shows purpose which mathematically defined is a long-range interaction. The fly or the sea-urchin shows purpose when it acts non-diffusively. The power-laws in Nature demonstrate that there are long-range interactions built into the laws of Physics. (Like the Coulomb electric interaction, for example.) This violates the Epicurean principle that Nature is made up of little atoms bouncing (with short-range repulsive interactions) through the void. It means that somewhere, something has a long-range, Aristotelean attraction, with purpose.

So thank Paul Levy, he provided one more proof that Darwin was wrong.

(with repaired brackets, since HTML stole my angle brackets “>”, so [x] is shorthand for an average “x”.)

It’s been about 10 years since I stumbled over Levy-random walk. Paul Levy was a French mathematician in the 1930’s, who was asking some questions about statistics. Here’s a brief explanation of the maths:

Let us suppose that you can walk with 10 different stride lengths–a short one when you are on ice, a medium one for grass, and a really long one when you walk on concrete. Now let us suppose that you randomize your stride, short steps followed by long or medium steps. The probability for talking a step of length L, is the PDF, the probability density function. Okay, one more assumption. Let us assume that you take a step at some arbitrary compass direction–first north, then sw followed by east and etc. How fast do you travel?

This is a restatement of the famous “drunk staggering around lamppost” illustration for diffusive travel. Now it is well known that since the drunk is equally likely to go east as west, his average displacement is zero => [x]=0,

and it looks like he’s not getting anywhere. But he is! His average squared displacement, [x^2] =/= 0 is not zero. So we can define a “diffusion speed” as being the “time rate of change of the average squared displacement”, or,

D == d[x^2]/dt

This math is found in every textbook, and describes the “diffusion” speed at which perfume molecules diffuse through a room. But Paul Levy asked another question. What happens when the drunk takes big steps and little steps, what happens to this diffusion speed if the PDF is not little baby steps all the time?

(Hang in there, we have one more mathematical definition to go.)

Well now we need to calculate what an average, or [] means. If I add together all my test scores, and divide by the number of tests, I have an average score. But what happens when the prof puts more weight on the final, and less on the midterm? This “weighting” is exactly the same magic as the PDF, and the correct way to calculate it is:

Integrate( score * weighting) / Integrate(weighting).

That is, if the prof doubles the value of the final, it shouldn’t cause students to get a 150% of an A, it should be “averaged out” by lowering the midterm effects. In other words, the weighting has to come out 100% however it is distributed, and that “normalization” is the term in the denominator.

Now we are ready for Paul Levy’s contribution. If we want to calculate the “average square displacement” then we put that into the “Score” part of the integral, and the PDF goes into the “weighting” part of the integral like this:

D = Integrate (x^2 * PDF) / Integrate(PDF).

What kind of PDF can we use to calculate this? Well, if the stride size of our drunk is usually pretty small, but very rarely big, we can say:

PDF = (1/x)^n

so when stride (=x) is small, then it is very likely, and when stride is big, it is less likely, and the “n” is the power-law telling us just how unlikely a big stride will be. We use power-laws because it makes the maths easier and also because Nature seems to use them very frequently. There are very interesting reasons for Nature’s power laws which have to do with fractal numbers and the “range” over which objects interact, but Levy didn’t get into that because he was a mathematician.

Well what sort of PDF does perfume have? We usually think of the “stride” of a gas-perfume collision as being a Gaussian distribution centered on some step size X, where x = A exp(-sig (x-X)^2), and simplifying the exponential function, we get:

x ~ 1/ (1 + x^2 + x^4/2 …) where we throw away the leading term. Plugging that into our function we get:

D = Integ(x^2 * 1/x^2) = constant.

In fact, any distribution that looks vaguely Gaussian or random, ends up with this same diffusion speed.

Then Levy noted a very interesting result. What if the PDF has more big strides in it than a Gaussian? What if it doesn’t diminish as (1/x^2) but only 1/x?

Well, when you integrate the average, one goes from -infinity to +infinity, and if the “n” in the PDF was infinity.

The diffusion speed is infinite! If this were perfume, then a PDF of 1/x would cause the perfume to speed through the room “super-diffusively”. It isn’t going the speed of light, but just a whole lot faster than you expected.

This “faster-than-perfume” speed is now known as “Levy-flight” and enables a sea-urchin or a fly to cover more territory than a stupid perfume molecule. That’s a no-brainer.

But there is one niggling detail left out of this science article. A PDF that goes as 1/x instead of 1/x^2 is not random, it shows purpose, which mathematically defined is a long-range interaction. The fly or the sea-urchin shows purpose when it acts non-diffusively. The power-laws in Nature demonstrate that there are long-range interactions built into the laws of Physics. (Like the Coulomb electric interaction, for example.) This violates the Epicurean principle that Nature is made up of little atoms bouncing (with short-range repulsive interactions) through the void. It means that somewhere, something has a long-range, Aristotelean attraction, with purpose.

So thank Paul Levy, he provided one more proof that Darwin was wrong.

Rob, does this relate to the law of Conservation of Information?

are you sure this Levy wasn’t drunk when he did this? could explain the french losing so quick to the Germans.

perfume? equals drunks equals Darwin wrong?? I’m a creationist but somethings not right here!

Math is useless in biology.

One has to wonder whether Levy’s seeing eye dog knew he was blind.