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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  
    Universal Mind John Galt's Avatar
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    You do not appear to have defined what you mean by fully adapted and fully unadapted. Without such definitions your subsequent treatment appears unfounded.

    I do not understand what you mean by a discontinuous point.

    Since a generalist species can handle several environments you should be representing it as an area, so that it can embrace several radii.


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  4. #3  
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    Quote Originally Posted by Ophiolite
    You do not appear to have defined what you mean by fully adapted and fully unadapted. Without such definitions your subsequent treatment appears unfounded.
    The statements in the parenthesis right after the words adapted and unadapted define them respectively.

    Quote Originally Posted by Ophiolite
    I do not understand what you mean by a discontinuous point.
    Consider a dotted line at the circumference of the circle and an unshaded point at the center. The remainder of the circle is shaded in and are points that an are attainable for the population to lie on.

    Quote Originally Posted by Ophiolite
    Since a generalist species can handle several environments you should be representing it as an area, so that it can embrace several radii.
    No. The point on which the population lies on represents which environment it lives in; and at any given point in time an organism can only live in one environment. A generalist can be differentiated from specialist species because of its close proximity to the center of the circle.
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  5. #4  
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    Quote Originally Posted by Ellatha
    The statements in the parenthesis right after the words adapted and unadapted define them respectively..
    The definitions are wholly inadequate. Please review and revise.

    Quote Originally Posted by Ellatha
    Quote Originally Posted by Ophiolite
    Since a generalist species can handle several environments you should be representing it as an area, so that it can embrace several radii.
    No. The point on which the population lies on represents which environment it lives in; and at any given point in time an organism can only live in one environment. A generalist can be differentiated from specialist species because of its close proximity to the center of the circle.
    An organism can only live in one environment, but a species can live in multiple environments. Since you are addressing species, which means you are addressing populations, you need to consider multiple environments populated by any species.
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  6. #5  
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    Quote Originally Posted by Ophiolite
    The definitions are wholly inadequate. Please review and revise.
    Your lack of understanding is not my problem; the definitions as they stand are fine.

    Quote Originally Posted by Ellatha
    [An organism can only live in one environment, but a species can live in multiple environments. Since you are addressing species, which means you are addressing populations, you need to consider multiple environments populated by any species.
    And if you bothered to read the thread you would notice that the model concerns populations rather than species, and in the model any given population is given a single environment.
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  7. #6  
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    Quote Originally Posted by Ellatha
    Quote Originally Posted by Ophiolite
    The definitions are wholly inadequate. Please review and revise.
    Your lack of understanding is not my problem; the definitions as they stand are fine.
    It is bizarre that you would come on a discussion forum, offer a hypothesis for consideration, then choose to reject criticisms directed at the hypothesis.

    Perhaps you are unfamiliar with the scientific method. Even though this is a place for informal discussion it is appropriate that we follow the rigours of the scientific method. It is your responsibility to get your concept across clearly to your audience. It is also appropriate that you respond to criticisms of your hypothesis. Apparently you are incapable of doing so, or too petulant to do so. That is your loss. Your hypothesis had some possibilities, but if you reject questioning and positive criticism then you will have no opportunity to advance it.

    Please note that I had read your opening post a couple of times with interest, but delayed replying until I saw your appeal in the duplicate thread in mathematics(?) for a response. I took time to read your post once more in order to offer commentary. Were you expecting people to fall over themselves in awe at what you had produced? Surely not. So I have given you some bona fide criticisms of your effort. I have invested my time to do so. Your reaction will hardly encourage others to comment on your work.
    Quote Originally Posted by Ellatha
    Quote Originally Posted by Ophiolite
    An organism can only live in one environment, but a species can live in multiple environments. Since you are addressing species, which means you are addressing populations, you need to consider multiple environments populated by any species.
    And if you bothered to read the thread you would notice that the model concerns populations rather than species, and in the model any given population is given a single environment.
    Then you need to express this more clearly by avoiding the ambiguity that arises when your make statements like "species that are closer to the cente of the circle are what we label as generalist species." This clearly implies that you are dealing with species, not isolated populations of species.
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  8. #7  
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    I don't reject criticisms directed at my thesis. In fact, I welcome them when they are geniune. However, more than anything, this seems to be your intent:

    "an internet user who sends inflammatory or provocative messages designed to elicit negative responses or start a flame-war. (As a fisherman trolls for an unsuspecting fish.) : Don't answer those silly messages. Some troll is just looking for an argument."

    http://dictionary.reference.com/browse/troll

    I will not be baited so easily, and have therefore decided to ignore you (in order to to avoid a flame war and therefore avoid any repercussions from moderators and administrators alike).
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  9. #8  
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    My criticisms were directed at your thesis.

    My final observation was a genuine recommendation for how your thesis could be improved. You are using a metaphorical medium akin to that employed by Steve Kaufmann whose work on evolutionary mechanisms is both original and possibly of great importance. My recommendation could, in my opinion have offered you a substantial improvement in the efficacy of your hypothesis and in the communication of its principles.

    If you don't lose the arrogance you will find plenty of others ignoring you. Why do I call you arrogant: you seem to think the responsibility for clear communication lies with the reader, not the writer. That is powerful arrogance. You write unclearly, you offer inadequate definitions, and in an effort to point this out to you, for your advantage, you accuse me of trolling.
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  10. #9 Re: A Mathematical Interpretation of Evolution 
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    Quote Originally Posted by Ellatha
    Now, for the first axiom, we will model evolution as such, for a stable environment ...
    i'm afraid that's where your model is deficient - even what we call a "stable" environment is anything but : there are large variations in temperature and humidity from day to day, month to month, year to year and beyond; species migrate and your environment may be totally changed by the introduction of a new predator or pest; what were advantageous traits in one environment may be a drawback in another; and that is not to speak of calamities such as tsunamis, earthquakes and wildfires where adaptation to the environment may count for little
    "Reality is that which, when you stop believing in it, doesn't go away." (Philip K. Dick)
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  11. #10 Re: A Mathematical Interpretation of Evolution 
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    Quote Originally Posted by marnixR
    i'm afraid that's where your model is deficient - even what we call a "stable" environment is anything but :...
    I agree with your statement, but I disagree with your conclusion: namely, we are considering hypothetical, stable environments, but they aren't expected to be completely stable (nor do they need to be in order to possess applications towards population dynamics). There are no completely stable environments, but there are some that can be classified as stable (i.e., stable environments exist) and unstable (i.e., not all environments are stable).

    In other words, the frequency of change within an environment is what dictates whether it's classified as stable or unstable (and not that its frequency of change needs to be zero to be classified as stable).

    Feel free to search for criticisms though; I welcome them from posters that are genuinely interested.
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    as far as i'm aware, i know of 2 items that enhance the survival chances of a species : one is to be geographically widespread, the other is not to be a specialist in your requirements

    does your model predict these outcomes ?
    "Reality is that which, when you stop believing in it, doesn't go away." (Philip K. Dick)
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  13. #12  
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    Quote Originally Posted by marnixR
    as far as i'm aware, i know of 2 items that enhance the survival chances of a species : one is to be geographically widespread, the other is not to be a specialist in your requirements

    does your model predict these outcomes ?
    MarxnixR,

    Indeed, you make a good point in your latter statement. Being a specialist species is risky in that it will be difficult to adapt to new environments; however, it is not without its benefits. For example, specialist species often have higher survival rates than generalist species, in addition to higher hunting success rates in members of the order Carnivora. In other words, if an environment is fairly stable over time, specialist species will thrive to a greater extent than generalist species. On the other hand, if an environment begins to change after stability for an elongated period of time, specialist species face the risk of extinction as opposed to generalists, which wouldn't be effected too much.

    Geographic distribution may allow species as a whole to thrive better, but in this particular model we are considering populations (of course, I refrained from using the terminology "specialist populations" since the conventional "specialist species" is already in use).

    The brown bear is a good example of a species with a large geographic distribution (unfortunately, while grizzly bears [a subspecies of the brown bear] once thrived across the lower forty-eight states, they now only occupy less than two percent of their historical range) and is fairly general in its feeding habits. Namely, brown bears cannot hunt megafauna as well as more specialist carnivores such as big cats or gray wolves, nor can they rely on vegetation to the extent of herbivores such as bison or elk. On the other hand, if faced with environmental change it can cope better than the aforementioned species. In the Pleistocene, the cousin of the brown bear, Arctodus simus (the "giant short-faced bear"), was unable to adapt to climatic change and went extinct (despite being three times larger on average than its extant cousin).
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