Lotka-Volterra equations

Russ Hobbie and I don’t study population dynamics much in the 4th edition of Intermediate Physics for Medicine and Biology. To me, it’s more of a mathematical biology topic rather than an application of physics to biology. However, we do discuss one well-known model for population dynamics, the Lotka-Volterra equations, in a Homework Problem in Chapter 2.

Problem 34 Consider a classic predator-prey problem. Let the number rabbits be R and the number of foxes be F. The rabbits eat grass, which is plentiful. The foxes eat only rabbits. The number of rabbits and foxes can be modeled by the Lotka-Volterra equations

dR/dt = a R – b R F

dF/dt = - c F + d R F .

(a) Describe the physical meaning of each term on the right-hand side of each equation. What does each of the constants a, b, c, and d denote?

(b) Solve for the steady-state values of R and F.

These differential equations are difficult to solve because they are nonlinear (see Chapter 10). Typically, R and F oscillate about the steady-state solutions found in part (b). For more information, see Murray (2001).

There are two steady-state solutions. One is the trivial R = F = 0. The most interesting aspect of this solution is that it is not stable. If R and F are both small, the nonlinear terms in the Lotka-Volterra equations are negligible, and the number of foxes falls exponentially (they is no prey to eat) but the number of rabbits rises exponentially (there is no predator to eat them).

The other steady-state solution is (spoiler alert!) R = c/d and F = a/b. We claim in the problem that these equations are difficult to solve, and that is true in general, at least when searching for analytical solutions. However, if we focus on small deviations from this steady-state, we can solve the equations. Let  

R = c/d + r 

F = a/b + f ,

where r and f are small (much less than the steady state solutions). Plug these into the original differential equations, and ignore any terms containing r times f (these “doubly small” terms are negligible). The new equations for r and f are

dr/dt = - b (c/d) f 

df/dt =   d (a/b) r .

Now let’s use my favorite technique for solving differential equations: guess and check. I will guess

r = A sin(ωt)

f = B cos(ωt) .

If we plug these expressions into the differential equations, we get a solution only if ω² = ac. In that case, B = -(d/b) √(a/c) A. You can’t get A in this way; it depends on the initial conditions.

A plot of the solution shows two oscillating populations, with the rabbits lagging 90 degrees behind the foxes. In words, suppose you start with foxes at their equilibrium value, but a surplus of rabbits above their equilibrium. In this case, there are lots of rabbits for the foxes to eat, so the foxes gorge themselves and their population grows. However, as the number of foxes rises, the number of rabbits starts to fall (they are being ravaged by all those foxes). After a while, the number of rabbits declines back to its equilibrium value, but by then the number of foxes has surged above its steady-state value. Foxes continue to devour rabbits, reducing the rabbit population below equilibrium. Now there are too many foxes competing for too few rabbits, so the fox population starts to shrink as some inevitably go hungry. During this difficult time, both populations are plummeting as a large but decreasing number of ravenous foxes hunt the rare and frightened rabbits. When the foxes finally fall back to their equilibrium value there is a shortage of rabbits, so the foxes continue to starve and their number keeps falling. With less foxes, the rabbits breed like…um…rabbits and begin to make a comeback. Once they climb to their equilibrium value, there are still relatively few foxes, so the rabbits prosper all the more. With the rabbit population surging, there is plenty of food for the foxes, and the fox population begins to increase. During these happy days, both populations thrive. Eventually, the foxes return to their equilibrium value, but by this time the rabbits are plentiful. But this is just where we started, so the process repeats, over and over again. I needed a lot of words to explain about those foxes and rabbits. I think you can begin to see the virtue of a succinct mathematical analysis, rather than a verbose nonmathematical description.

For larger oscillations, the nonlinear nature of the model becomes important. The populations still oscillate, but not sinusoidally. For some parameters, one population may rise slowly and then suddenly drop precipitously, only to gradually rise again. You can see some of those results here and here.

The Lotka-Volterra model is rather elementary. For instance, there is no damping; the oscillations never decay away but instead continue forever. Moreover, the oscillations do not approach some fixed amplitude (a limit cycle behavior). Instead, the amplitude depends entirely on the initial conditions. Many more realistic models have a threshold, above which oscillations occur but below which the systems returns to its steady state.

Mathematical Biology,
by James Murray.

Population dynamics is a large field, of which we only scratch the surface. One place to learn more is James Murray’s book that we cite at the end of the homework problem:

Murray, J. D. (2001) Mathematical Biology. New York, Springer-Verlag.

The most recent (3rd) edition of Murray’s book is actually in two volumes:

Murray, J. D. (2002) Mathematical Biology: I. An Introduction. New York, Springer-Verlag. 

Murray, J. D. (2002) Mathematical Biology: II. Spatial Models and Biomedical Applications. New York, Springer-Verlag.

Hear Murray talk about his research here. 

Listen to James Murray talk about Mathematical Biology.
https://www.youtube.com/watch?v=6Yj5Nyb_VyU

Alfred Lotka (1880–1949) was an American scientist. In 1925 he published a book, Elements of Physical Biology, that is in some ways a precursor to Intermediate Physics for Medicine and Biology, or perhaps an early version of Murray’s Mathematical Biology. You can download a copy of the book here.

Life’s Ratchet

Life’s Ratchet: How Molecular Machines
Extract Order from Chaos,
by Peter Hoffmann.

This week I finished reading Life’s Ratchet: How Molecular Machines Extract Order from Chaos, by Peter Hoffmann. This book is mostly about molecular biophysics, which Russ Hobbie and I purposely avoid in the 4th edition of Intermediate Physics for Medicine and Biology. But the workings of tiny molecular motors is closely related to thermal motion (Hoffmann calls it the “molecular storm”) and the second law of thermodynamics, topics that Russ and I do address. One fascinating topic I want to focus on is a discussion of Feynman’s ratchet.

Let us begin with Richard Feynman’s discussion in Chapter 16 of Volume 1 of The Feynman Lectures on Physics. I recall reading The Feynman Lectures the summer between graduating from the University of Kansas and starting graduate school at Vanderbilt University. All physics students should find time to read these great lectures. Feynman writes

Let us try to invent a device which will violate the Second Law of Thermodynamics, that is, a gadget which will generate work from a heat reservoir with everything at the same temperature. Let us say we have a box of gas at a certain temperature and inside there is an axle with vanes in it… Because of the bombardments of gas molecules on the vane, the vane oscillates and jiggles. All we have to do is to hook onto the other end of the axle a wheel which can turn only one way—the ratchet and pawl. Then when the shaft tries to jiggle one way, it will not turn, and when it jiggles the other, it will turn… If we just look at it, wee see, prima facie, that it seems quite possible. So we must look more closely. Indeed, if we look at the ratchet and pawl, we see a number of complications.

First, our idealized ratchet is as simple as possible, but even so, there is a pawl, and there must be a spring in the pawl. The pawl must return after coming off a tooth, so the spring is necessary…

Feynman goes on to explore this device in detail. He concludes that, as we would expect, the device does not violate the second law. He explains

It is necessary to work against the spring in order to lift the pawl to the top of a tooth. Let us call this energy ε… The chance that the system can accumulate enough energy ε to get the pawl over the top of the tooth is e^−ε/kT [T is the absolute temperature, and k is Boltzmann’s constant]. But the probability that the pawl will accidentally be up is also e^−ε/kT. So the number of times that the pawl is up the wheel can turn backwards freely is equal to the number of times that we have enough energy to turn it forward when the pawl is down. We thus get a “balance,” and the wheel will not go around.

Hoffmann explains that a lot of molecular machines important in biology operate analogously to Feynman’s ratchet and pawl. He writes

What kind of molecular device could channel random molecular motion into oriented activity? Such a device would need to allow certain directions of motion, while rejecting others. A ratchet, that is, a wheel with asymmetric teeth blocked by a spring-loaded pawl, could do the job... Maybe nature has made molecular-size ratchets that allow favorable pushes from the molecular storm in one direction, while rejecting unfavorable pushes from the opposite direction…

For the ratchet-and-pawl machine to extract energy from the molecular storm, it has to be easy to push the pawl over one of the teeth of the ratchet. The pawl spring must be very weak to allow the ratchet to move at all. Otherwise, a few water molecules hitting the ratchet would not be strong enough to force the pawl over one of the teeth. Just like the ratchet wheel, the pawl is continuously bombarded by water molecules. Its weak spring allows the pawl to bounce up and down randomly, opening from time to time, allowing the ratchet to slip backward… Worse, because the spring is most relaxed when the pawl is at the lowest point between two teeth [the compressed spring pushes the pawl down against the ratchet], the pawl spends most of its time touching the steep edge of one of the teeth. When an unfavorable hit pushes the ratchet backward just as the pawl has opened, it does not need to go far to end up on the incline of the next tooth—rotating the ratchet backward!... The ratchet will move, bobbing back and forth, but it will not make any net headway.

How then do molecular machines work? They require in input of energy, which eventually gets dissipated into heat. Hoffmann concludes

We could, in fact, make Feynman’s ratchet work, if from time to time, we injected energy to loosen and then retighten the pawl’s spring. On loosening the spring, the wheel would rotate freely, with a slightly higher probability of rotating one way rather than the other. Tightening the pawl’s spring would push the wheel further in the direction we want. On average, the wheel would move forward and do work. In fact, it can be shown that any molecular machine that operates on an asymmetric energy landscape and incorporates and irreversible, energy-degrading step can extract useful work from the molecular storm.

This may all seem abstract, but Hoffmann brings it down to specifics. The molecular machine could be myosin moving along actin (as in muscles) or kinesin moving along a microtubule (as in separating chromosomes during mitosis). The energy source for the irreversible step is ATP. This step allows the motor to extract energy from the “molecular storm” of thermal energy that is constantly bombarding it.

I WENT TO PARIS AND I MISSED THE VERY BEST THING!

Three summers ago, my wife and I visited Paris for our 25th wedding anniversary. We carefully planned our trip so we could see all the most famous sites—the Eifel Tower, the Arc de Triomphe, the Notre Dame Cathedral, the Palace of Versailles, the Pantheon, the Louvre, and the Musee d’Orsay—but somehow WE MISSED THE MOST IMPORTANT THING! Apparently there is a giant painting in the Musee d’Art Moderne by Raoul Dufy, depicting many scientists who have contributed to the study of electricity. What more could a physicist like me ask for? I first learned about this painting in a book I am now reading, The Spark of Life: Electricity and the Human Body, by Frances Ashcroft. I’ll have more on that book in a future post. Here is what she writes about the painting:

An unusual tribute to the scientists and philosophers who contributed to the discovery of electricity hangs in Musee d’Art Moderne in Paris. A giant canvas known as “La Fee Electricite,” which measures 10 metres high and 60 metres long, it was commissioned by a Paris electricity company to decorate its Hall of Light at the 1937 world exhibition in Paris. It is the work of French Fauvist painter Raoul Dufy, better known for his wonderful colourful depictions of boats, and it took him and two assistants four months to complete. The Electricity Fairy sails through the sky at the far left of the painting above some of the world’s most famous landmarks, the Eiffel Tower, Big Ben and St Peter’s Basilica in Rome among them. Behind her follow some 110 people connected with the development of electricity, from Ancient Greece to modern times. As time and the canvas progress, the landscape changes from scenes of rural idyll to steam trains, furnaces, the trappings of the industrial revolution and finally the giant pylons that support the power lines carrying electricity to the planet.

Short of going to see the painting in Paris, the next best thing is to view it in sections at the Electricity Online website of the University of Leeds. I won’t list all the scientists depicted in it, but let me note those Russ Hobbie and I mention in the 4th edition of Intermediate Physics for Medicine and Biology (roughly in chronological order): Newton, Bernoulli, Laplace, Poisson, Gauss, Ohm, Oersted, Clausius, Clapeyron, Fourier, Savart, Fresnel, Biot, Ampere, Faraday, Gibbs, Helmholtz, Maxwell, Poincare, Moseley, Lorentz, and Pierre Curie. A few were only present in IPMB because they have a unit named after them: Pascal, Watt, Joule, Kelvin, Roentgen, Becquerel, Hertz, and Marie Curie. Galvani is shown with a frog, Faraday with a coil and galvanometer, Pierre Curie (mentioned in IPMB through the Curie temperature) is standing next to his wife Marie Curie (only mentioned in IPMB in association with her unit, and the only female scientist in the painting), and Edison is next to his light bulbs.

I’m still not sure how I never knew about this magnificent painting. I guess we need to take another trip to Paris. Honey, start packing!

Resource Letter BSSMF-1: Biological Sensing of Static Magnetic Fields

In the October 2012 issue of the American Journal of Physics, physicist Leonard Finegold published “Resource Letter BSSMF-1: Biological Sensing of Static Magnetic Fields” (Volume 80, Pages 851–861). Finegold recommends that a good starting point for mastering the topic of magnetoreception is Kenneth Lohmann’s News and Views article in Nature.

35. “Magnetic-field perception: News and Views Q and A,” K. J. Lohmann, Nature, 464, 1140–1142 (2010). (E) 

I looked it up, and it does indeed provide a well-written summary of the field in a reader-friendly question-and-answer format.

In the 4th edition of Intermediate Physics for Medicine and Biology, Russ Hobbie and I discuss magnetotactic bacteria. We write that

Bacteria in the northern hemisphere have been shown to seek the north pole. Because of the tilt of the earth’s field, they burrow deeper into the environment in which they live. Similar bacteria in the southern hemisphere burrow down by seeking the south pole.

Finegold also reviews this topic. The excerpt reproduced below serves both as an up-date to IPMB and as a sample of the style of an American Journal of Physics resource letter.

Certain bacteria move in response to the earth’s magnetic field (Ref. 35), swimming along the field lines, and have been excellently reviewed (Ref. 36). The “sensing” element is magnetite (an iron oxide) or greigite (an iron sulfide) (Ref. 37). The bacteria would swim toward the boundary between oxygenated and oxygen-poor regions. Until recently, there was the comforting idea that there are two groups of bacteria with opposite sensors, depending on which of the earth’s hemispheres they reside. Alas, both groups have now been found in the same place; it appears that their polarity is correlated with the local redox potential (Ref. 38 and 39). In addition, some bacteria use only the axial property of the field (i.e., they swim both with or against the field direction), whereas others use the vector property (i.e., they swim either with or against the field direction). Details of the behavior have been elucidated by applying magnetic fields to bacteria in a spectrophotometer cuvette, with genetic analysis (Ref. 39).

35. “South-seeking magnetotactic bacteria in the Southern Hemisphere,” R. P. Blakemore, R. B. Frankel, and Ad. J. Kalmijn, Nature 286, 384–385 (1980). (A)

36. “Bacteria that synthesize nano-sized compasses to navigate using Earth’s geomagnetic field,” L. Chen, D. A. Bazylinski, and B. H. Lower, Nature Education Knowledge 1(10), 14 (2010). (I)

37. “The identification and biogeochemical interpretation of fossil magnetotactic bacteria,” R. E. Kopp and J. L. Kirschvink, Earth-Sci. Rev. 86, 42–61 (2008). (A)

38. “South-seeking magnetotactic bacteria in the northern hemisphere,” S. L. Simmons, D. A. Bazylinski, and K. J. Edwards, Science 311, 371–374 (2006). (A)

39. “Characterization of bacterial magnetotactic behaviors by using a magnetospectrophotometry assay,” C. T. Lefevre, T. Song, J. P. Yonnet, and L. F. Wu, Appl. Environ. Microbiol. 75, 3835–3841 (2009). (A)”

Magnetoreception is a field that often stirs debate. Russ and I outline one such debate in IPMB

Kirschvink (1992) proposed a model whereby a magnetosome in a field of 10⁻⁴–10⁻³ T could rotate to open a membrane channel. As an example of the debate that continues in this area, Adair (1991, 1992, 1993, 1994) argued that a magnetic interaction cannot overcome thermal noise in a 60-Hz field of 5 × 10⁻⁶ T. However, Polk (1994) argues that more biologically realistic parameters, including a large number of magnetosomes in a cell, could allow an interaction at 2 × 10⁻⁶ T.

The key citations in the debate are

Adair, R. (1991) “Constraints on biological effects of weak extremely-low-frequency electromagnetic fields,” Phys. Rev. A, Volume 43, Pages 1039–1048.

Kirschvink, J. L. (1992) “Comment on “Constraints on biological effects of weak extremely-low-frequency electromagnetic fields,” Phys. Rev. A, Volume 46, Pages 2178–2184.

Adair, R. (1992) “Reply to “Comment on ‘Constraints on biological effects of weak extremely-low-frequency electromagnetic fields’,” Phys. Rev. A, Volume 46, Pages 2185–2187.

For those of you who like this sort of thing, here is another example from Finegold’s resource letter. The debate is about, of all things, if cows align themselves in magnetic fields!

A surprising finding is that cattle and deer seem to align themselves in an approximate north-south (geomagnetic) direction. The evidence is from world-wide satellite photographs from Google Earth, supported by ground observations of more than 10,000 animals, and is hard to rebut. The satellite photographs do not have enough resolution to show the direction (north or south) in which the animals face.

72. “Magnetic alignment in grazing and resting cattle and deer,” S. Begall, J. Cerveny, J. Neef, O. Vojtech, and H. Burda, Proc. Natl. Acad. Sci. U.S.A. 105, 13453–13455 (2008). (I)

As Usherwood asks, why on Earth should cattle and deer prefer this alignment? Possible interpretations are that the satellite photographs are made close to noon, so there may be physiological reasons (heating, cooling) for animals to align or to view predators better.

73. “Cattle and deer align north (-north-east),” J. Usherwood, J. Exp. Biol. 212, iv (2009). (E)

Partly to rule out sun compass effects, Burda et al. investigated ruminant alignment under high-voltage (and hence high-current, low-frequency) power lines and found that the geomagnetic north-south alignment was disturbed; the disturbance was correlated with the alternating fields. Such disturbance might instead be because the animals felt protected by (or preferring) the overhead lines or pylons or because of the audible (to humans at least) corona discharge. A good control for this would be to look at ruminants under power lines being repaired, carrying no current; this is difficult to do. The authors ingeniously compared the nonalignment under N-S and E-W trending power lines and found that the nonalignment followed the resultant total magnetic field. Their conclusions have been challenged (Ref. 75), and they have a lively rebuttal (Ref. 76), to which the challengers have replied (Ref. 77). Hence, the initially persuasive evidence, that cattle and deer detect magnetic fields, may need re-examination.

74. “Extremely low-frequency electromagnetic fields disrupt magnetic alignment of ruminants,” H. Burda, S. Begall, J. Cerven, J. Neef, and P. Nemec, Proc. Natl. Acad. Sci. U.S.A. 106, 5708–5713 (2009). (I)

75. “No alignment of cattle along geomagnetic field lines found,” J. Hert, L. Jelinek, L. Pekarek, and A. Pavlicek, J. Comp. Physiol., A 197, 677–682 (2011). (I)

76. “Further support for the alignment of cattle along magnetic field lines: Reply to Hert et al.,” S. Begall, H. Burda, J. Cerveny, O. Gerter, J. Neef-Weisse, and P. Nemec, J. Comp. Physiol. [A] 197, 1127–1133 (2011). (I)

77. “Authors’ Response,” J. Hert, L. Jelinek, L. Pekarek, and A. Pavlicek, J. Comp. Physiol. [A] 197(12), 1135– 1136 (2011). (I) 

Finegold also discusses magnet therapy, a topic I am extremely skeptical about, and that I have discussed before in this blog. He cites his own editorial with Flamm

“Magnet therapy,” L. Finegold and B. L. Flamm, Br. Med. J. 332, 4 (2006) (E) 

which concludes

Extraordinary claims demand extraordinary evidence. If there is any healing effect of magnets, it is apparently small since published research, both theoretical and experimental, is weighted heavily against any therapeutic benefit. Patients should be advised that magnet therapy has no proved benefits. If they insist on using a magnetic device they could be advised to buy the cheapest—this will at least alleviate the pain in their wallet.