Talk:Lotka–Volterra equation

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Deleted a link to a computer game clan - Some guy.

Help! Can anyone give me the equations for Volterra-Lotka equations for three species.

What I'm talking about is

Fox eats Rabbit

Rabbit eats Carrot

Carrot eats sunlight (infinite supply of sunlight)

Sure, I can give lots of them. Let C, R, F denote mass of Carrots, number of Rabbits, and number of Foxes. For each of C, R, and F I will give several equations, increading in realisticness. All parameters are >= 0.

Carrot equation:

C1. C' = a - bR. This assumes the number of carrots is fixed ( they do not get killed, or propagate), a is the total growth due to sunlight, and b is how much is eaten by a rabbit.

C2. C' = aC - bR. This assumes a = p - d is the difference of the propagation rate and the 'old age' death rate for carrots. This implies unlimited exponential growth in the abscence of rabbits.

C3. C' = a ( 1 - C/maxC ) C - bR. This is called a 'logistic' equation, it assumes a 'maximum mass of carrots' called MaxC. This is appropriate in the likely event that you are dealing with a finite area for your ecology experiment. This ODE gives an 'S shaped curve', rapidly growing from 0, and slowly approaching 1. This is the best equation to use, unless you are looking for an easy equation. This ignores the effect that if carrots get low, the rabbits will have a hard time finding them. if you want to deal with that effect, see the F' equations below for ideas. In many situations, the foxes keep the rabbits from eating too many carrots.

Rabbit equations. Most Lotka-Volterra models use an absurd simple Rabbit - fox interaction term of R*F, which would indicate each fox eats a fixed proportion of all the rabbits. The only good thing about that is it leads to the simplest nonlinear term.

R1. R' = cR - d RF. This is the standard lame equation ( ignoring for the moment rabbit food).

R2. R' = R [ -e + f(C/R) ] / [ 1 + g(C/R) ] = R [ -eR + fC ] / [ R + gC ]. This is a pretty good equation for the rabbit food aspect, ignoring the foxes for the moment. Here -e indicates how fast rabbits starve if the ratio C/R goes to zero, and f/g indicates how fast rabbit population grows when there are plenty of carrots to go around. For a fixed f/g, you can fiddle with f and g to better fit whatever data you have.

R3. R' = R [ -eR + fC ] / [ R + gC ] - hF. Here we assume the foxes have no trouble finding rabbits, and each one eats h rabbits per unit time.

R4. R' = R [ -eR + fC ] / [ R + gC ] - hRF / [ kF + R ]. Here each fox eats h rabbits if there are plenty to go around, and 0 as the R/F ratio goes to zero. k is a free parameter to help fit whatever curve you thing is best.

F1 F' = F [ - m + pR ]. Traditional lame equation.

F2. F' = F [ -mF + pR ] / [ F + nR ]. Here -m is the death rate due to starvation when there are no rabbits around. p/n is the birth - (old age) death rate for well fed rabbits. p and n can be adjusted to fit whatever. For example, if you have a guess for what the R/F ratio is that keeps the Fox population constant, choose p so that m/p = stable(R/F).

equation summary: A good balance between reality and simplicity is

C' = a ( 1 - C/maxC ) C - bR R' = R [ -eR + fC ] / [ R + gC ] - hRF / [ kF + R ] F' = F [ -mF + pR ] / [ F + nR ]

Before working with this set of equations, simplify by 'scaling away parameters'. You can scale four paramiters (C, R, F, and time) which will allow you to eliminate 4 of the 11 parameters. The simplist is to measure C in units of maxC, so that maxC becomes 1. There are many ways to simplify the remaining equations. For example, letting replacing R with gR and F with (g/k)F removes the parameters in the denominators of the R' equation. Scaling time by a eliminates the a. This leads to ( the parameters will now have different values ):

C' = C ( 1 - C ) - b R R' = R [ -eR + fC ] / [ R + C ] - hRF / [ F + R ] F' = F [ -mF + pR ] / [ F + nR ]

Thus you have a three dimensional ODE with seven parameters.

The first step in the analysis is to find the (nontrivial) critical points, which are values of (C,R,F) where all right hand sides are zero.

This will require

R = C(1-C)/b, F = pR/m, [-eR + fC ] / [ R+C ] = hF / [F+R ].

subbing the second eqn into the third:

-eR + fC = [ hp / (m+p) ] [ R + C ].

subbing in the first equation:

(m+p) [ (bf-e)C + eC^2 ] = hp [ ( 1+b)C - C^2 ].

Dividing thru by C:

(m+p) [ (bf-e) + eC ] = hp [ ( 1+b) - C ]

which leads to

C = [ ( e - bf ) ( m + p ) + ( 1 + b ) hp ] / [ ( m + p ) e + hp ].

From this you can find R, and then F.

Obviously the parameters ( b,e,f,h,m,p ) will have to be such that C, R, and F are > 0. The parameter n does not affect the critical point. Interesting.

From this point on you will probably need computer assistance.

Next you classify this critical point. Find the three eigenvalues of the frechet derivative (this is a three by three matrix of partial derivatives) evaluated at the CP. If you are looking for a cyclical population situation, you probably need a complex conjugate pair with positive real part, or maybe a pure imaginary complex pair, and the third eigenvalue a positive real.

Finding the regions in (b,e,f,h,m,p) space where these two events happen would be nice, but likely involve a lot of UGLY work, maybe Monte Carlo simulation. (BTW be sure to use a GOOD random number generator. Contact me for advice. It might be a good time to find some realistic value ranges for b, e, f, h, m, n, and p.

Ralph Kelsey -- 08:08, 5 Nov 2004 (UTC)

R Kelsey's contribution is much better than the article itself; both in terms of depth, as well as in clarifying the physical significance of each term. What a shame R Kelsey is not accessible, as he deserves to be encouraged to adapt his contribution to the main article.

I also regret that R Kelsey did not leave a reference. His last paragraph points out a way: I can surely anticipate the outcome, but would be glad to have a book as my companion enroute.

BTW my initial reaction is to be skeptical of the independence of the parameter n (RK writes takes note that this is 'interesting'. I have not done the work, but I can't suppress suspicion that this may mask a computational fault.)

--Philopedia 20:00, 4 January 2007 (UTC)

R Kelsey's contribution is a great example of a population model with three species. But it does not belong in this article. The Lotka-Volterra model has only a predator and a prey, per definition (otherwise it's not a Lokta-Volterra model). :-) See Population modeling for a description of more general population models. Unfortunately, the Wikipedia coverage of population models seems to be a bit of a mess at the moment. It is incomplete, often unclear and fragmented over at least three or four articles that should all be merged. Any theoretical biologists around with too much free time on their hands? The Wikipedia:WikiProject_Ecology certainly sounds like a good initiative.

--Lvzon 23:39, 22 October 2007 (UTC)

Are these really "the" Lotka-Volterra equations?[edit]

The Lotka-Volterra equations are not necessarily the ones given in the article now, which form the Lotka-Volterra predator-prey model. There is also the well known Lotka-Volterra competition model. They have in common that the per capita growth rates are linear functions of population densities or sizes. Which of these two models deserves the name Lotka-Volterra sec is unclear to me, and if nobody justifies the choice for the predator-prey model, I think it should be changed.

On the other hand, maybe mathematicians refer to the predator-prey model with "the Lotka-Volterra equations", while ecologists call it the Lotka-Volterra predator-prey model.

Also, the name of the article is a singular, while there are clearly two equations. OpenScience (talk) 12:13, 6 February 2010 (UTC)

Discrete version[edit]

Is there a discrete version? I would expect it to be able to produce chaotic results.--Henrygb 16:35, 17 Mar 2005 (UTC)

I know the competetive version of the equations produce chaos in 4d and up systems. I've also run across papers indicating that higher dimensional systems will also produce period doubling and limit cycles. Vortmax 22:50, 2 December 2006 (UTC)

Analytic solution[edit]

There is only approximate solution revealed in this article. But what about strong analytic solution? I know (thanks for Maple) that it is expressed in terms of LambertW function. But it may be interesting if this solution would be referred in article not in so vaguely manner.

Volterra-Lotka equations.[edit]

I would like to know if the solutions of Volterra-Lotka equations are unique and global.

Reply: Yes, they are: the vector field is C^1.

Little problem with second EP[edit]

In the analysis of the second EP, the terminology and tools used are good for linear systems, not for nonlinear systems such as the LV system. ~~

That's correct and I noticed it too. All the complex eigen values reveal is that the fixed point is periodic, but does not reveal if it is a center or a stable spiral.

You have to eliminate the time dependance by dividing the dy/dt equation by dx/dt equation, which by the chain rule gives a single seperable equation for dy/dx. When you seperate this and integrate you are left with an equation in x and y which you can relate to a third variable (magnitude) giving level sets of the origional equations. Since f(x) and f(y) approach 0 as x&y approach infinity, the level curve is closed. This translates into the closed orbits. Vortmax 22:46, 2 December 2006 (UTC)Matt

Confusing; needs explication[edit]

The article has the text: In this equation, δxy represents the growth of the predator population.

This is misleading in (at least) two respects.

Most simply, it seems to be saying that the Delta-x-y term represents the totality of change in y. This is clearly in error, since there is a second term as well. However, this problem is certainly just a matter of carelessness. Presumably the writer meant that the Delta-x-y term represents the rate of births of predators.

More troubling is question, why the birth rate is proportional to meetings between predator and prey. Wouldn't it be more logical to say that births among predators are proportional to meetings between male and female predators, i.e. proportional to y2? However, this would seem to be only appropriate for modelling in cases where predator-predator encounters are generally infrequent (in comparison with the period of gestation). For instance, the predators may be an endangered species. If predator-predator encounters are very frequent, we could ignore this term.

Or maybe the x-y term is meant to show the dependency on food supply. (This would make sense, else why introduce the relationships predator and prey). In this case, the xy proportionality of the term would seem to say that predators on average have some characteristic number of progeny and that this characteristic number is proportional to the amount of available food. (But since we are talking of an average birth rate for the entire predator population, shouldn't that be the amount of available food per predator, ie y/x?) However, this explanation only appears (approximately) valid for moderate ranges of populations: As I've already mentioned we ought to include a y2 factor if y is small. Similarly, we ought to model a cap on the birth rate in case of very abundant prey.

In any case, I think I've succeeded in making my point that the applicability for these equations needs to be addressed carefully and thorough if intelligent newcomers to the subject (I include myself) are to be able to learn from the article.

--Philopedia 19:32, 4 January 2007 (UTC)

The point (and also perhaps the problem) of the Lotka-Volterra model is that it is a highly simplified model. It is nice for showing that a predator-prey relationship can produce oscillating behaviour, but other than that it doesn't have much biological significance. Even less if the nullclines of the ODEs are perpendicular, as shown in many textbooks (mathematically elegant perhaps, but biologically unlikely). I think it needs to be mentioned in the article that this model is meant as a paradigm system, not a realistic model.

You are right that the delta-x-y-term shows dependency on food supply for the predators, whereby delta is the conversion-efficiency of prey into predators. This, is a simple mass-action term, like in chemical kinetics. It is the simplest way of stating that the birth-rate of predators depends on the prey density (because higher densities increase the probability of a predator catching a prey). For a more realistic model this should indeed include a saturation term.

Incidentally, attentive readers may also have noticed that the population numbers in some of the graphs fall below 1. Actually, they can become infinitely small in this model, and still no-one will go extinct. This is a general problem with these types of population models. In computer simulations it can be "solved" (= made more realistic) by adding a little random noise to the population variables, and setting negative values to zero.

--Levien 23:13, 22 October 2007 (UTC)

Elliptical diagram[edit]

What do the numbers in the key mean? What do the different solutions represent? CentralTendencyLV.jpg

Repton infinity 06:22, 27 March 2007 (UTC)

And what does it mean for the population of baboons to be between zero and one? (talk) 06:25, 28 March 2008 (UTC)

This is kind of a terrible example (talk) 02:29, 11 March 2009 (UTC)

It is only slightly misleading: it insinuates ("a serious problem with this as a biological model") that the L−V model somehow dictates an "extremely small" minimum value for the prey, x, when that is merely a choice of the parameters made. The minimum number x is the solution of the standard transcendental equation -\delta x +\gamma \ln x =V+\alpha \ln (e\beta/\alpha), where V depends on the initial conditions. A "safe" minimum may be achievable for suitable I.C.s and parameters——the solution for minimum x being a —LambertW. Since the oscillations are about the fixed point of 60 baboons and 25 cheetahs, with the maximum V*, a sufficiently large V will entail a minimum number of baboon as close to 60 as desired. Cuzkatzimhut (talk) 11:05, 17 April 2015 (UTC)

Example: baboon (prey) and cheetah (predator)[edit]

The initial conditions are 80 baboons and 40 cheetah, but what are the parameters α, β, γ and δ ? -- 10:59, 13 August 2007 (UTC)

Lol, so true. Also, if you look at the curves, none actually HIT 80 baboons, 40 cheetah ;) (talk) 02:30, 11 March 2009 (UTC)

What about the black one? (talk) 08:31, 30 April 2010 (UTC)

Proposal of Equation and citations[edit]

A. Lotka originally proposed this equation in 1910 in chemistry and extended it in 1920 to biological interactions, most definitely not as late as 1925(cited two different ways). This article could do with at lot more citations and not just some vague ref list at the end. Kae1is (talk) 00:42, 9 March 2010 (UTC)

Exact solutions[edit]

In case anyone wants to know this: The Lotka-Volterra equations are analytically solvable. Here are some papers: [1][2]. At some point someone should put these information into the article --Svebert (talk) 20:41, 24 October 2011 (UTC)

Wikipedia talk:Articles for creation/Atto-fox problem[edit]

An opinion from Wikiproject Mathematics suggests that information from the above article be merged into this article. I don't know how to do that without losing attribution, so I am leaving a note here in the hopes that an interested editor will take on this task. —Anne Delong (talk) 23:18, 11 November 2013 (UTC)

The Atto-Fox material is now in the history of this redirect: Atto-fox problem. —Anne Delong (talk) 02:53, 26 June 2014 (UTC)