Friday, June 28, 2013

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.

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.

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.

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