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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 y factor if y is small. Similarly, we ought to model a cap on the birth rate in case of very abundant prey.
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biology (in part by replacing previous text and figure). I also have replaced the words "baboon" and "cheetah" with "rabbit" and "fox". This is in the first place to keep to one example: the baboons didn't add anything (there is nothing baboon specific in the example), and now the predator ties in with the atto-fox. Also because the cheetah is not a main predator of the baboon. I hope this improves the quality of the article. --
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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.
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Invasive Spices, I am ok with removal. (I'm not convinced the other sources we have on this, at the beginning of the section, are so great either.) Ami-de-la-Terre, it is often the case that people in one field re-invent ideas that have been known in other fields for a long time, thereby losing the
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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
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Hi,you provide two references, for the exact solutions. The first reference has a broken link, and the second one links to an article by 'Martin J. Gander' that gives an analytical solution without derivation , and cites: A. Steiner and M. Arrigoni, \Die Losung gewisser Rauber-Beute-Systeme", Studia
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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
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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
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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 y? However, this would seem to be only appropriate
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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
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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,
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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
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I can't believe this title said "Lotka–Volterra equation", with no final "s". I changed it. This is not about a type of equation of which any particular instance is a "Lotka–Volterra equation", nor about just one equation. In either of those situations, the singular title would be required under
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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
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for a description of more general population models. Unfortunately, the
Knowledge 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
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R2. R' = R / = R / . 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
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I agree that
Leconte is a strange choice for a source here. There are several contradictory factors: Highly regarded journal from a legitimate Association; entirely incorrect journal subject area; physics is math; recentism; utter lack of cites. I think we should ignore the other factors and rm on
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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
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I reorganised the first section to bring various important parts mentioned downstream together, and made the text more relevant to a non-mathematical audience. I have added a sub-section on the biological relevance of the model by providing some examples how the behaviour of the model relates to
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I am the contributor of the analytical solution contribution. Obviously, the cited source is recent (2022) which explains the lack of cites. About 'entirely incorrect journal subject area', I disagree, the Lotka-Volterra model may have been developed first in the ecology field, but has becomes a
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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
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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
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F2. F' = F / . 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
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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.
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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.
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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.
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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.
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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
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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.
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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 ):
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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.
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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.
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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.)
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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 :
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Did you understand the remarks on the atto-fox problem? Do you wish to scale all numbers by millions? What are you actually proposing, specifically? This is not a critic's choice "bring me another stone" forum.
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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.
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From "Ages of
Discord: A Structural-demographic Analysis of American History" by Peter Turchin (ISBN 0996139540, 9780996139540), There is an equation similar to the equations that predict social instability.
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equations have separate pages. I'd say a single page for all, or otherwise simply a retitling of this page to specific "Predator-prey" (and perhaps a clearer cross-reference in the intro) would resolve.
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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.
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R4. R' = R / - hRF / . 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.
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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.
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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.
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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,
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I thought it would be a good idea to update the "baboon/cheetah" example to use svg figures. I can make them and propose the changes, but wanted to float the idea here first. --
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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
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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.
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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.
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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.
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1547:, I just think that it would be helpful for some people to know that there is actually an analytical solution, if you find another source, it would be good too.
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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
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Proceed to propose what you think and propose, specifically, beyond passive whining. Did you read the
Wikimedia site figure's specs? Do you agree with them?
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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. ~~
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upon the motive force of the plasma? (Simply imagining what the correspondence might be. I tried reading some of the papers on this and failed.)
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Agreed; that's what I came to this page to talk about. It really demands some sort of comment about how population going to 0 ever recovers.
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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.
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the basis of 0 cites. It may in fact come back around to us as an appropriate source, having been cited by others in future but not today.
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can be "solved" (= made more realistic) by adding a little random noise to the population variables, and setting negative values to zero.
1602:, Well it is more like one large-scale component of the turbulence (a sheared flow) preys on the smaller scale turbulence which drives it.
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When you have finished reviewing my changes, you may follow the instructions on the template below to fix any issues with the URLs.
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Biophysica vol. 123 (1988) No. 2. I cannot find this last reference by searching online. Can you please provide another reference?
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paradigm for nonlinear interactions in many fields of
Physics, including in plasma turbulence in magnetically-confined plasmas.
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R3. R' = R / - hF. Here we assume the foxes have no trouble finding rabbits, and each one eats h rabbits per unit time.
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The existing figures are poor quality, better ones would be great. I don't see any reason at all to seek preapproval. --
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https://web.archive.org/web/20100531204042/http://entomology.wsu.edu/profiles/06BerrymanWeb/Berryman%2892%29Origins.pdf
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solution is a hardly superior implicit renaming, and it does not shed that much light, unless one understands
Lambert
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Absorbing the superfluous parameters into the variables as described in the text, retaining only the essential ratio
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The first paragraph of the section already said this, and already contained two references, before your edits. --
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The Lotka-Volterra equations are not necessarily the ones given in the article now, which form the Lotka-Volterra
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In case anyone wants to know this: The Lotka-Volterra equations are analytically solvable. Here are some papers:
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to delete these "External links modified" talk page sections if they want to de-clutter talk pages, but see the
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is unclear to me, and if nobody justifies the choice for the predator-prey model, I think it should be changed.
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I find it confusing that this page is "Lotka-Volterra equations" but only considers predator-prey, while the
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function. But it may be interesting if this solution would be referred in article not in so vaguely manner.
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R1. R' = cR - d RF. This is the standard lame equation ( ignoring for the moment rabbit food).
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On the other hand, maybe mathematicians refer to the predator-prey model with "the Lotka-Volterra
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before doing mass systematic removals. This message is updated dynamically through the template
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Lol, so true. Also, if you look at the curves, none actually HIT 80 baboons, 40 cheetah ;)
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for negative real arguments, this is all but useless here, and not suitable for inclusion.
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I would like to know if the solutions of
Volterra-Lotka equations are unique and global.
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Obviously the parameters ( b,e,f,h,m,p ) will have to be such that C, R, and F are : -->
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Help! Can anyone give me the equations for
Volterra-Lotka equations for three species.
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If you found an error with any archives or the URLs themselves, you can fix them with
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Also, the name of the article is a singular, while there are clearly two equations.
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The initial conditions are 80 baboons and 40 cheetah, but what are the parameters
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Proposal: Retitle as "Predator-Prey" or merge with other Lotka-Volterra equations
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What do the numbers in the key mean? What do the different solutions represent?
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And what does it mean for the population of baboons to be between zero and one?
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maybe a pure imaginary complex pair, and the third eigenvalue a positive real.
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fixed f/g, you can fiddle with f and g to better fit whatever data you have.
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intellectual history and context; this is presumably one of those cases. --
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http://entomology.wsu.edu/profiles/06BerrymanWeb/Berryman%2892%29Origins.pdf
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I disagree the numerical solutions are "only approximate". The "analytic"
1210:. At some point someone should put these information into the article --
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In this equation, δxy represents the growth of the predator population.
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Is there a discrete version? I would expect it to be able to produce
1034:{\displaystyle -\delta x+\gamma \ln x=V+\alpha \ln(e\beta /\alpha )}
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0. The parameter n does not affect the critical point. Interesting.
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equation summary: A good balance between reality and simplicity is
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will entail a minimum number of baboon as close to 25 as desired.
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Knowledge level-5 vital articles in Biology and health sciences
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From this point on you will probably need computer assistance.
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The Atto-Fox material is now in the history of this redirect:
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This example isn't that useful without the parameter values.
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Thus you have a three dimensional ODE with seven parameters.
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C' = a ( 1 - C/maxC ) C - bR R' = R / - hRF / F' = F /
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for additional information. I made the following changes:
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Knowledge vital articles in Biology and health sciences
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Since the oscillations are about the fixed point of 25=
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Without mastery of the nasty properties of the Lambert
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C' = C ( 1 - C ) - b R R' = R / - hRF / F' = F /
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B-Class vital articles in Biology and health sciences
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Knowledge talk:Articles for creation/Atto-fox problem
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Applying Lotka–Volterra to Turchin's Ages of Discord
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704:{\displaystyle y=-W(-K(e^{x}/x)^{\gamma /\alpha })}
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159:importance scale
141:
140:
139:Ecology articles
137:
134:
131:
114:
109:
108:
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98:
91:
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77:
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53:
44:
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36:
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27:
21:
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1731:
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1725:
1671:
1670:
1646:
1626:
1611:Ami-de-la-Terre
1600:Invasive Spices
1589:Invasive Spices
1549:Ami-de-la-Terre
1516:Ami-de-la-Terre
1509:Invasive Spices
1506:
1496:Invasive Spices
1486:
1483:
1474:
1449:
1432:
1417:
1412:
1380:
1373:have permission
1363:
1337:this simple FaQ
1322:
1290:
1246:
1228:Ami-de-la-Terre
1203:
1201:Exact solutions
1183:
1123:
1068:
1063:baboons and 20=
1049:
1042:
964:
963:
956:
948:, when that is
945:
871:Repton infinity
860:
800:
774:
748:
743:
742:
736:
715:
680:
662:
633:
632:
603:
585:
560:
549:
548:
545:in the article,
538:
532:
525:
473:
472:
469:
443:
398:
313:Ralph Kelsey --
288:which leads to
178:
176:Old discussions
138:
135:
132:
129:
128:
110:
105:
103:
83:
54:on Knowledge's
51:
41:
23:
22:
15:
12:
11:
5:
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1577:
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1482:
1481:Leconte source
1479:
1473:
1470:
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1431:
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1407:
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1399:
1352:
1351:
1343:Added archive
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397:
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389:
388:
320:
312:
279:(m+p) = hp .
273:-eR + fC = .
250:
243:
218:
204:
183:
177:
174:
171:
170:
167:
166:
163:
162:
155:Mid-importance
151:
145:
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142:
116:
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112:Ecology portal
99:
87:
86:
84:Mid‑importance
78:
66:
65:
59:
37:
24:
14:
13:
10:
9:
6:
4:
3:
2:
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1312:
1308:
1307:Michael Hardy
1304:
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1209:
1207:
1200:
1198:
1197:
1193:
1189:
1180:
1176:
1172:
1168:
1167:98.245.92.246
1164:
1163:
1162:
1161:
1157:
1153:
1152:134.88.191.99
1148:
1147:
1144:
1140:
1136:
1132:
1128:
1120:
1116:
1112:
1108:
1104:
1103:
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1101:
1097:
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1058:
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1047:
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1005:
1002:
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993:
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987:
984:
981:
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975:
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962:
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951:
943:
942:
941:
940:
936:
932:
931:134.88.191.99
921:
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739:
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713:
693:
689:
685:
677:
673:
667:
663:
656:
653:
647:
644:
641:
638:
631:, so that
616:
612:
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586:
579:
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568:
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547:
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454:
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448:
440:
438:
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433:
429:
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407:
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402:predator-prey
395:
387:
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371:
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364:
361:
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349:
343:
342:
339:
334:
330:
327:
324:
318:
316:
315:132.235.14.20
310:
306:
302:
299:
295:
292:
289:
286:
285:(m+p) = hp
283:
280:
277:
274:
271:
268:
265:
262:
259:
256:
253:
248:
245:
241:
237:
234:
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209:
206:
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143:
126:
122:
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113:
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100:
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92:
88:
82:
79:
76:
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67:
63:
57:
49:
48:
38:
34:
29:
28:
19:
1647:
1627:
1609:
1584:
1484:
1475:
1433:
1411:
1408:
1383:source check
1362:
1356:
1353:
1326:
1323:
1302:
1298:
1291:
1247:
1204:
1184:
1149:
1143:141.53.37.95
1138:
1134:
1130:
1126:
1124:
1107:Cuzkatzimhut
1089:
1077:Cuzkatzimhut
1072:
1064:
1060:
949:
928:
912:Cuzkatzimhut
869:
861:
847:
843:
839:
835:
826:
822:
818:
814:
810:
807:
803:
801:
793:
782:
778:
775:
740:
737:
720:Cuzkatzimhut
542:
533:
514:216.66.24.58
513:
470:
455:
444:
425:
420:
416:
414:
409:
405:
401:
399:
378:Cuzkatzimhut
357:
344:
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331:
328:
322:
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182:
179:
154:
118:
62:WikiProjects
45:
1654:generalized
1650:competitive
1272:Anne Delong
1251:Anne Delong
883:209.6.22.99
428:OpenScience
406:competition
1675:Categories
1420:Report bug
1092:Ggmcnickle
829:Philopedia
449:results.--
338:Philopedia
1659:Kyle MoJo
1585:predating
1545:JayBeeEll
1489:JayBeeEll
1452:Eitanlees
1436:Eitanlees
1403:this tool
1396:this tool
417:equations
291:C = / .
50:is rated
1409:Cheers.—
1288:plural!!
1054:LambertW
1052:being a
1041:, where
1333:my edit
1212:Svebert
786:Vortmax
459:Vortmax
451:Henrygb
447:chaotic
157:on the
130:Ecology
125:ecology
81:Ecology
52:B-class
1631:Pips15
1485:Hello
1295:WP:MOS
1188:Kae1is
897:Vttale
850:Levien
58:scale.
791:Matt
421:model
360:Lvzon
201:= 0.
39:This
1663:talk
1652:and
1635:talk
1615:talk
1593:talk
1567:talk
1553:talk
1535:talk
1520:talk
1500:talk
1462:talk
1440:talk
1311:talk
1299:pair
1276:talk
1270:. —
1255:talk
1232:talk
1216:talk
1192:talk
1171:talk
1156:talk
1141:? --
1137:and
1111:talk
1096:talk
1081:talk
935:talk
916:talk
901:talk
887:talk
724:talk
541:is
432:talk
382:talk
323:much
1563:JBL
1531:JBL
1458:JBL
1377:RfC
1347:to
1065:α/β
1061:γ/δ
534:γ/α
410:sec
200:-->
149:Mid
1677::
1665:)
1637:)
1617:)
1569:)
1555:)
1537:)
1522:)
1464:)
1442:)
1390:.
1385:}}
1381:{{
1313:)
1305:.
1278:)
1257:)
1234:)
1218:)
1194:)
1173:)
1158:)
1133:,
1129:,
1113:)
1098:)
1083:)
1026:α
1018:β
1009:
1006:ln
1003:α
988:
985:ln
982:γ
973:δ
970:−
937:)
918:)
903:)
889:)
848:--
827:--
768:.
726:)
694:α
686:γ
654:−
645:−
617:α
609:γ
577:−
566:−
555:−
434:)
423:.
384:)
376:.
358:--
336:--
1661:(
1633:(
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1454::
1450:@
1438:(
1422:)
1418:(
1405:.
1398:.
1309:(
1274:(
1253:(
1249:—
1230:(
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1169:(
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1139:δ
1135:γ
1131:β
1127:α
1109:(
1094:(
1079:(
1073:V
1069:V
1056:.
1050:x
1043:V
1029:)
1022:/
1015:e
1012:(
1000:+
997:V
994:=
991:x
979:+
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957:x
952:.
946:x
933:(
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899:(
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750:C
722:(
716:W
711:.
699:)
690:/
682:)
678:x
674:/
668:x
664:e
660:(
657:K
651:(
648:W
642:=
639:y
613:/
605:)
601:x
597:/
591:x
587:e
583:(
580:K
574:=
569:y
562:e
558:y
539:K
526:W
500:W
497:t
494:r
491:e
488:b
485:m
482:a
479:L
430:(
380:(
161:.
64::
20:)
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