# Lotka-Volterra confusion



## Skeletra (Aug 30, 2012)

Waaah. Anyone familiar with the Lotka-Volterra equation?
Looks something like this:
















"Here, ?12 represents the effect species 2 has on the population of species 1 and ?21 represents the effect species 1 has on the population of species 2. These values do not have to be equal. Because this is the competitive version of the model, all interactions must be harmful (competition) and therefore all ?-values are positive"
From this Wikipedia article:
https://en.m.wikipedia.org/wiki/Competitive_Lotka?Volterra_equations

Wat?

Yeah, it's homework.. I'm giving a presentation after new year and writing a "paper" on it. I just need to understand the fundamental concept so I can get started on this.. I failed math by the way... so... great start. Yep. Totally going to ace this D:!


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## novalax (Jun 7, 2013)

I dont know what you know or what you need help with exactly but:

r= the intrinsic rate of growth, or how much the species will grow without competition

x= the number of species 

a= competition coefficient, or the effect of inter-specific competition 

K= carry rate, or the effect of competition within the species.

for the subscript on a, the first number is the species being affected.

(k-x)/k = coefficient for intra specific competition, or how much the species fights with itself. should be between 0 and 1.

obviously the whole thing together is the rate of growth of a population with intra-secific competition.

that's almost all I know about it, but if you have any other questions feel free to ask. although, be forewarned, I may not be able to help


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## Skeletra (Aug 30, 2012)

novalax said:


> I dont know what you know or what you need help with exactly but:
> 
> r= the intrinsic rate of growth, or how much the species will grow without competition
> 
> ...


And what is d? Why are there little numbers there? Why does dx divided by dt suddenly become r2x2?
I don't understand how to solve the equation, and how it fits into a system.. It just seems so random.
Houf. I feel so hopelessly stupid. I do appreciate the effort though.


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## novalax (Jun 7, 2013)

Skeletra said:


> And what is d? Why are there little numbers there? Why does dx divided by dt suddenly become r2x2?
> I don't understand how to solve the equation, and how it fits into a system.. It just seems so random.
> Houf. I feel so hopelessly stupid. I do appreciate the effort though.


d is the derivative. it is saying per one unit time (t), but how much does the population change (x). everything after the equals sign is the function that tells you that. the little numbers, called subscripts, delineate different variables. for example x1 and x2 are both referring to population, however x1 only refers to the population of species 1 and x2 refers only to species 2. as stated before, the subscript on alpha (the thing that looks like an "a") tells you what species is praying on the other. the first subscript is the prey population and the second subscript is the predator population. to find the point where the population is not changing, set the equation equal to 0. that means either x2 is zero or that numerator (k1-n1-an2) is zero. to find when the numerator is zero, set it equal to zero and solve for n1. in doing so you will have found the isocline of zero population growth for species 1.

you are neither hopeless nor stupid. and it does seem random at first, but thats why you ask questions and learn. in due time, you will understand it. Just remember to take the time you need, dont rush yourself because you think others are faster.

if you have any other questions, ill do my best.


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## Skeletra (Aug 30, 2012)

novalax said:


> d is the derivative. it is saying per one unit time (t), but how much does the population change (x). everything after the equals sign is the function that tells you that. the little numbers, called subscripts, delineate different variables. for example x1 and x2 are both referring to population, however x1 only refers to the population of species 1 and x2 refers only to species 2. as stated before, the subscript on alpha (the thing that looks like an "a") tells you what species is praying on the other. the first subscript is the prey population and the second subscript is the predator population.* to find the point where the population is not changing, set the equation equal to 0. that means either x2 is zero or that numerator (k1-n1-an2) is zero. to find when the numerator is zero, set it equal to zero and solve for n1. in doing so you will have found the isocline of zero population growth for species 1.*
> 
> you are neither hopeless nor stupid. and it does seem random at first, but thats why you ask questions and learn. in due time, you will understand it. Just remember to take the time you need, dont rush yourself because you think others are faster.
> 
> if you have any other questions, ill do my best.


Ooo, it's starting to make sense  I think.
So small subscript numbers are just to mark them, while only superscript numbers are to multiply with?
And the 21 is actually how many of animal 2 are _affected by_ 1, and then I multiply that with x1 (which is quantity of animal 1) and add it to x2 (quantity of animal 2) and then divide by K (carry rate).. Wait, how do I find carry rate to begin with? What does it even mean? How well the species carry trough, like how many of them survive? Why should it be between 1 and 0?
You kind of lost me at the bolded part. Particularly that red word. Feel like that's all about the K, right?


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## novalax (Jun 7, 2013)

Skeletra said:


> Ooo, it's starting to make sense  I think.
> So small subscript numbers are just to mark them, while only superscript numbers are to multiply with?
> And the 21 is actually how many of animal 2 are _affected by_ 1, and then I multiply that with x1 (which is quantity of animal 1) and add it to x2 (quantity of animal 2) and then divide by K (carry rate).. Wait, how do I find carry rate to begin with? What does it even mean? How well the species carry trough, like how many of them survive? Why should it be between 1 and 0?
> You kind of lost me at the bolded part. Particularly that red word. Feel like that's all about the K, right?


Good! I'm glad it starting to make some sense.

correct, the subscripts are just there for organization while the superscripts represent exponentiation. Yup, the subscripts on alpha represent the species being affected, while the second number is the species doing the effects. Your close, you multiple the competition coefficient (alpha) by the population that is doing the effecting. so a12 is multiplied with x2 and a21 is multiplied with x1. then simplify the expression k1-x1-(a12*x2) or k2-x2-(a21*x1) depending on which species you are analyzing. then divide your answer by either k1 or k2 (again depending on which k you used in the numerator) then multiply by either (r1*n1) or (r2*n2). the carry rate is given to you (i think). The carry rate is the maximum population size an environment can sustain. (k-x)/k (which models the competition within a species) should be between 1 and 0 because it is a factor that can only limit the population. when it is equal to 1, the environment can support all of the population. less than one means it only supports part of the population. 0 means it supports none of the population. and it can't be negative as that would make no sense.

I did get kind of convoluted toward the end. I was saying that to find at what points the species population wont grow, you either set x1 (or x2) equal to zero, which means that there is no population so it can't grow, or you set the expression (k1-n1-a12*n2) equal to zero. when you do that, you solve for n1, so you have n1=k1-a12*n2. In population dynamics, Isoclines are the set of population sizes at which there is no growth with a competing species.

I hope I cleared some stuff up, but keep firing away if you have any other questions.


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## Skeletra (Aug 30, 2012)

I think I'm getting it. At least enough to get started on it .

Thanks a lot! This has been really helpful


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## novalax (Jun 7, 2013)

Skeletra said:


> I think I'm getting it. At least enough to get started on it .
> 
> Thanks a lot! This has been really helpful


Glad I could help!


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