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Thread: A Mathematical Interpretation of Evolution

  1. #1 A Mathematical Interpretation of Evolution 
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    In this thread I will provide mathematical models for aspects of evolution and provide an explanation as to how the rate of evolution results in the extinction of a population (and potentially species).

    Firstly, I only support this thread on the basis of the following axioms:
    1. A population can never have organisms fully adapted to their environment (i.e., an organism will never have every trait favorable for its survival).
    2. A population can never have organisms fully unadapted to their environment (i.e., an organism will never have every trait unfavorable for its survival).
    3. Populations evolve, over time, providing the organisms in that environment favorable features for survival.

    Now, for the first axiom, we will model evolution as such, for a stable environment (where F(t) is a population with organisms fully adapted to their environment, f(t) is a population of organisms in a stable environemnt, and t is time):



    If populations continually adapt to their environment, over time, than in a stable environment populations will become specialized to that environment.

    The best model that I've thought of at the moment for what we've come to so far is a Euclidean circle, with the circumference and center being discontinuous (over an interval and at a point, respectively), and varying radii of the circle representing environments. All points that exist on the circle are points that a population can attain, where the closer a population comes to the circumference of the circle, the closer it comes to fully adapting to its environment.

    Species' that are are closer to the center of the radius are what we often label as "generalist species." Similarly, those that are closer to the circumference of a circle are considered "specialist species." The advantage of being a generalist species, or closer to the center of the evolutionary circle, is being able to thrive in a wide variety of environments in addition to taking less risk of dealing with environmental change (this will be discussed more in depth later). The advantage of being a specialist species, or closer to the circumference of the circle, is that such species are more successful at surviving in such an environment. For example, in hyper-carnivores, a specialist species will very often have higher hunting percentage success rates than a generalist species. They will also be better suited for their method of locomotion and communication than a generalist.

    I'm sure there may be some sort of exceedingly sophisticated method of classifying generalist and specialist species' in terms of their anatomy and physiology, however such a method would be overly tiring and require bio-mechanical data for which I simply do not have. Therefore, the point of this thread is not to classify such species as such, but to provide a general outline into the theory of population dynamics.

    I will now discuss the disadvantages of being a specialist species in comparison to a generalist. If a population is in a stable environment for a long period of time, than it will become such a specialist. The advantages of being such a species have been discussed. However, if the environment changes rapidly, that is, the radius on which the population once lied on moves, it could result in extinction. That is, the organisms in that population now have the disadvantages of being a specialist (such as needing larger foraging ranges), while not having the advantages. If the environment changes enough, this causes the rate of death of the organisms to exceed the rate of survival. Depending on how much greater the rate of death is than the rate of reproduction, the population will go extinct or survive. If the population of organisms were specialized to their environment, but not the extent that they couldn't evolve fast enough to change the death-reproductive rate of the new environment, than they will survive. On the other hand, extremely specialized species will go extinct if such changes result (due to rate of death being far greater than the rate of reproduction). An example of an occurrence is the mass-extinction that occurred in the Pleistocene Ice Age, where extremely specialized animals (e.g., Smilodon, the famous "saber-toothed tiger" [which belonged to a different subfamily of felids altogether], or the dire-wolf [which was the dominant canid of the era]).

    We will model this as such (for all D > R):





    Where is the original size of the population, and R and D are reproduction and death, respectively, and can be modelled in terms of t, time, and m is a variable that will exist whenever the rate of death exceeds the rate of reproduction (varying depending on the magnitude of D - R).

    Now, for a population such that R > D, the model is the following:



    Where K is the carrying capacity of the environment for that population, and n is a variable that exists whenever the rate of reproduction exceeds the rate of death (varying depending on the magnitude of R - D).

    EDIT: fixing syntax.


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  3. #2  
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    Ellatha, this is a very bold undertaking, and I applaud you for the attempt. However, there are so many misconceptions of a biological nature about your premises that I am not convinced any good will come of your model on this particular sub-forum.

    If you would like me to move this to Biology I shall, and there discuss with you where I think your misconceptions arise (believe it or not, I am highly trained in population and evolutionary genetics).

    But I will not move this thread without your consent. What do you say?

    Anyone else have a thought?


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  4. #3  
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    I have posted this thread in the biology subforum as well, as it concerns both subjects. Please do post more of your opinion on the model and where you believe the misconceptions lie; as no one else has replied yet, I would be rather grateful.

    As you are also a member of this forum, rather than simply a moderator of the mathematics section, it is your choice in which forum you would like to discuss the issue.
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  5. #4  
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    Quote Originally Posted by Ellatha
    I have posted this thread in the biology subforum as well, as it concerns both subjects. Please do post more of your opinion on the model and where you believe the misconceptions lie; as no one else has replied yet, I would be rather grateful.

    As you are also a member of this forum, rather than simply a moderator of the mathematics section, it is your choice in which forum you would like to discuss the issue.
    You would be a lot better off taking Guitarist's advice and discussing this in the Biology forum.

    Youappear to be addressing population dynamics rather than evolution, and even so you might want to brush up on that subject before discussing it with the biologists.

    http://www.sosmath.com/diffeq/first/...opulation.html
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  6. #5  
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    I've been patiently awaiting a reply; will I be receiving one?
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  7. #6  
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    So sorry. I have mega-busy lately, and may not get on-line constructively for a few days. I'll see what I can do
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    I'd have to agree with the doc on this, it seems you're modeling more species specific population dynamics, not the changes and mutations that a species undergo to mutate. I do agree that your model is pretty representative of a population of a species, and I would offer though that over time, with the size of the population D would change in accordance to the environment in which the species lives until D=R, at which point there is either a stagnation or equilibrium for that species in it's current environment.

    However, I really don't see any correlation to evolution aside from your initial axioms.
    Wise men speak because they have something to say; Fools, because they have to say something.
    -Plato

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  9. #8  
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    Quote Originally Posted by Guitarist
    So sorry. I have mega-busy lately, and may not get on-line constructively for a few days. I'll see what I can do
    That's fine; if you are busy than don't even worry about it.

    EDIT:

    Arcane,

    Evolution plays a role into my model because the position of a population on any given radius of the Euclidean circle varies over time due to evolution. On the other hand, if the environment changes, than the population would shift onto another radius accordingly.
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    right. So how do you account for this movement? just saying "it moves" doesn't tell us anything about the mechanisms through which your model addresses evolution nor does it quantify evolution in any meaningful way. The model you presented, euclidean circle aside, only addresses population dynamics. How do you gather where a species will be on the circle at any given point in time?
    Wise men speak because they have something to say; Fools, because they have to say something.
    -Plato

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  11. #10  
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    Quote Originally Posted by Arcane_Mathematician
    right. So how do you account for this movement? just saying "it moves" doesn't tell us anything about the mechanisms through which your model addresses evolution nor does it quantify evolution in any meaningful way. The model you presented, euclidean circle aside, only addresses population dynamics. How do you gather where a species will be on the circle at any given point in time?
    Individual radii on the circle represent individual environments. If the environment that the population lives in changes (either from biotic or abiotic factors [i.e., the population moves to a different environment, or the current environment changes]) the population moves to a different radius. The movement of the population on the given radius tends towards the circumference in a stable environment (the outermost point on the radius represents a population with organisms fully adapted to their environment, i.e., a limit). Towards the center of the radius represents generalist species, while the centroid of the circle represents a population with no favorable features for their environment (this is also a limit).
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