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| In the study of age-structured population growth, probably one of the most important equations is the '''Lotka–Euler equation'''. Based on the age demographic of females in the population and female births (since in many cases it is the females that are more limiting in the ability to reproduce), this equation allows for an estimation of how a population is growing. | | <br><br>Age does not make a difference. In this day and age when younger individuals date old people and with the Demi Moores of the globe having boyfriends like Ashton Kutcher, it just gives you some thing to believe about. Any type of relationship works irrespective of age, gender and social status. Every thing depends on the individuals involved, their compatibility and their willingness to reach compromise.<br><br><br><br>She is her own individual. She is not clingy, she is impartial and is not heading to abandon her lifestyle just because she experienced sex with you once and now she is preparing you marriage and life with each other. This is the one thing that older women have more than the restricted bodied 23-yr-olds you are probably used to dating and that you should keep in mind when you are out to seduce ladies. They are not clingy. That and they have a massive wealth of understanding when it comes to intercourse.<br><br>They know how to take treatment of themselves. For the most component, younger women have a tendency to be experimental and have a liberated mindset. But, it can be instead demanding, especially when they end up consuming a entire lot and do not truly be cautious of their own well being. guys would not have this issue with older women, simply because they are more responsible.<br><br>While some may believe that older women dating is like dating your personal mum, it is not always accurate! Believe it or not, numerous older women enjoy mentoring their companion. Partly because she feels that she is much more skilled as compared to the younger ladies.<br><br>Older women are, more than at any time before, typically financially independent. They can make it themselves. They don't need or even want a guy's cash. When they look for a man, they want someone lively, with a youthful attitude, who treats them with equality and respect.<br><br>Make the effort. She has been about, (once more, no pun meant!) so she understands all the lines and nifty methods that males use in order to seduce women and get them into mattress. Be much more than just another wannabe easy talker.<br><br>Let me give you one final piece of guidance. Steer clear of the proliferation of "Cougar dating" sites. The popularity of them makes them hugely costly. And why spend tons of money to find an older lady if you don't have to? Instead, find your self a large, mainstream dating neighborhood. 1 with hundreds of thousands of associates. Try clicking here and making a profile; I suggest it to all my visitors. Large dating communities are a fantastic way to find mature women because many of us are on a spending budget and favor not to spend lots of money on costly sites.<br><br>If you beloved this informative article along with you wish to receive details about [http://g3nx.com/members/gregg75i/activity/37875/ see cougar dating here math-preview.wmflabs.org] kindly check out our own web site. |
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| The field of mathematical [[demography]] was largely developed by [[Alfred J. Lotka]] in the early 20th century, building on the earlier work of [[Leonhard Euler]]. The Euler–Lotka equation, derived and discussed below, is often attributed to either of its origins: Euler, who derived a special form in 1760, or Lotka, who derived a more general continuous version. The equation in discrete time is given by
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| :<math>1 = \sum_{a = 1}^\omega \lambda^{-a}\ell(a)b(a)</math>
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| where <math>\lambda</math> is the discrete growth rate, ''ℓ''(''a''), is the fraction of individuals surviving to age ''a'' and ''b''(''a'') is the number of individuals born at time ''a''. The sum is taken over the entire life span of the organism.
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| ==Derivations==
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| ===Lotka's continuous model===
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| A.J. Lotka in 1911 developed a continuous model of population dynamics as follows. This model tracks only the females in the population.
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| Let ''B''(''t'') be the number of births per unit time. Also define the scale factor ''ℓ''(''a''), the fraction of individuals surviving to age ''a''. Finally define ''b''(''a'') to be the birth rate per capita for mothers of age ''a''.
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| All of these quantities can be viewed in the [[continuous function|continuous]] limit, producing the following [[integral (math)|integral]] expression for ''B'':
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| :<math> B(t) = \int_0^t B(t - a )\ell(a)b(a) \, da.</math>
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| The integrand gives the number of births ''a'' years in the past multiplied by fraction of those individuals still alive at time ''t'' multiplied by the reproduction rate per individual of age ''a''. We integrate over all possible ages to find the total rate of births at time ''t''. We are in effect finding the contributions of all individuals of age up to ''t''. We need not consider individuals born before the start of this analysis since we can just set the base point low enough to incorporate all of them.
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| Let us then guess an [[Exponential function|exponential]] solution of the form ''B''(''t'') = ''Qe''<sup>''rt''</sup>. Plugging this into the integral equation gives:
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| :<math>Qe^{rt} = \int_0^t Q e^{r(t - a)}\ell(a)b(a) \, da </math>
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| or
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| :<math> 1 = \int_0^t e^{-ra}\ell(a)b(a) \, da. </math>
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| This can be rewritten in the [[discrete mathematics|discrete]] case by turning the integral into a sum producing
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| :<math>1 = \sum_{a = \alpha}^\beta e^{-ra}\ell(a)b(a)</math>
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| letting <math>\alpha</math> and <math>\beta</math> be the boundary ages for reproduction or defining the discrete growth rate ''λ = ''e''<sup>''r''</sup> we obtain the discrete time equation derived above:
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| :<math>1 = \sum_{a = 1}^\omega \lambda^{-a}\ell(a)b(a)</math>
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| where <math>\omega</math> is the maximum age, we can extend these ages since ''b''(''a'') vanishes beyond the boundaries.
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| ===From the Leslie matrix===
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| Let us write the [[Leslie matrix]] as:
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| :<math>\begin{bmatrix}
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| f_0 & f_1 & f_2 & f_3 & \ldots &f_{\omega - 1} \\
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| s_0 & 0 & 0 & 0 & \ldots & 0\\
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| 0 & s_1 & 0 & 0 & \ldots & 0\\
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| 0 & 0 & s_2 & 0 & \ldots & 0\\
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| 0 & 0 & 0 & \ddots & \ldots & 0\\
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| 0 & 0 & 0 & \ldots & s_{\omega - 2} & 0
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| \end{bmatrix}</math>
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| where <math>s_i</math> and <math>f_i</math> are survival to the next age class and per capita fecundity respectively.
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| Note that <math>s_i = \ell_{i + 1}/\ell_i</math> where ''ℓ''<sub> ''i''</sub> is the probability of surviving to age <math>i</math>, and
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| <math>f_i = s_ib_{i + 1}</math>, the number of births at age <math>i + 1</math> weighted by the probability of surviving to age <math>i+1</math>.
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| Now if we have stable growth the growth of the system is an [[eigenvalue]] of the [[matrix (math)|matrix]] since <math>\mathbf{n_{i+ 1}} = \mathbf{Ln_i} = \lambda \mathbf{n_i}</math>. Therefore we can use this relationship row by row to derive expressions for <math>n_i</math> in terms of the values in the matrix and <math>\lambda</math>.
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| Introducing notation <math>n_{i, t}</math> the population in age class <math>i</math> at time <math>t</math>, we have <math>n_{1, t+1} = \lambda n_{1, t}</math>. However also <math>n_{1, t+1} = s_0n_{0, t}</math>. This implies that
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| :<math>n_{1, t} = \frac{s_0}{\lambda}n_{0, t}. \, </math>
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| By the same argument we find that
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| :<math>n_{2, t} = \frac{s_1}{\lambda}n_{1, t} = \frac{s_0s_1}{\lambda^2}n_{0, t}. </math>
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| Continuing [[mathematical induction|inductively]] we conclude that generally
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| :<math>n_{i, t} = \frac{s_0\cdots s_{i - 1}}{\lambda^i}n_{0, t}. </math>
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| Considering the top row, we get
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| :<math>n_{0, t+ 1} = f_0n_{0, t} + \cdots + f_{\omega- 1}n_{\omega - 1, t} = \lambda n_{0, t}.</math>
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| Now we may substitute our previous work for the <math>n_{i, t}</math> terms and obtain:
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| :<math>\lambda n_{0, t} = \left(f_0 + f_1\frac{s_0}{\lambda} + \cdots + f_{\omega - 1}\frac{s_0\cdots s_{\omega - 2}}{\lambda^{\omega - 1}}\right)n_{(0, t)}. </math>
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| First substitute the definition of the per-capita fertility and divide through by the left hand side:
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| :<math>1 = \frac{s_0b_1}{\lambda} + \frac{s_0s_1b_2}{\lambda^2} + \cdots + \frac{s_0\cdots s_{\omega - 1}b_{\omega}}{\lambda^{\omega}}. </math>
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| Now we note the following simplification. Since <math>s_i = \ell_{i + 1}/\ell_i</math> we note that
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| :<math>s_0\ldots s_i = \frac{\ell_1}{\ell_0}\frac{\ell_2}{\ell_1}\cdots\frac{\ell_{i + 1}}{\ell_i} = \ell_{i + 1}. </math>
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| This sum collapses to:
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| :<math>\sum_{i = 1}^\omega \frac{\ell_ib_i}{\lambda^i} = 1, </math>
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| which is the desired result.
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| ==Analysis of expression==
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| From the above analysis we see that the Euler–Lotka equation is in fact the [[characteristic polynomial]] of the Leslie matrix. We can analyze its solutions to find information about the eigenvalues of the Leslie matrix (which has implications for the stability of populations).
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| Considering the continuous expression ''f'' as a function of ''r'', we can examine its roots. We notice that at negative infinity the function grows to positive infinity and at positive infinity the function approaches 0.
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| The first [[derivative]] is clearly −''af'' and the second derivative is ''a''<sup>2</sup>''f''. This function is then decreasing, concave up and takes on all positive values. It is also continuous by construction so by the intermediate value theorem, it crosses ''r'' = 1 exactly once. Therefore there is exactly one real solution, which is therefore the dominant eigenvalue of the matrix the equilibrium growth rate.
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| This same derivation applies to the discrete case.
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| ==Relationship to replacement rate of populations==
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| If we let ''λ'' = 1 the discrete formula becomes the [[replacement rate]] of the population. | |
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| ==Bibliography==
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| * Kot, M. (2001) ''Elements of Mathematical Ecology'', Cambridge. [[Cambridge University Press]].
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| {{DEFAULTSORT:Euler-Lotka Equation}}
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| [[Category:Demography]]
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Age does not make a difference. In this day and age when younger individuals date old people and with the Demi Moores of the globe having boyfriends like Ashton Kutcher, it just gives you some thing to believe about. Any type of relationship works irrespective of age, gender and social status. Every thing depends on the individuals involved, their compatibility and their willingness to reach compromise.
She is her own individual. She is not clingy, she is impartial and is not heading to abandon her lifestyle just because she experienced sex with you once and now she is preparing you marriage and life with each other. This is the one thing that older women have more than the restricted bodied 23-yr-olds you are probably used to dating and that you should keep in mind when you are out to seduce ladies. They are not clingy. That and they have a massive wealth of understanding when it comes to intercourse.
They know how to take treatment of themselves. For the most component, younger women have a tendency to be experimental and have a liberated mindset. But, it can be instead demanding, especially when they end up consuming a entire lot and do not truly be cautious of their own well being. guys would not have this issue with older women, simply because they are more responsible.
While some may believe that older women dating is like dating your personal mum, it is not always accurate! Believe it or not, numerous older women enjoy mentoring their companion. Partly because she feels that she is much more skilled as compared to the younger ladies.
Older women are, more than at any time before, typically financially independent. They can make it themselves. They don't need or even want a guy's cash. When they look for a man, they want someone lively, with a youthful attitude, who treats them with equality and respect.
Make the effort. She has been about, (once more, no pun meant!) so she understands all the lines and nifty methods that males use in order to seduce women and get them into mattress. Be much more than just another wannabe easy talker.
Let me give you one final piece of guidance. Steer clear of the proliferation of "Cougar dating" sites. The popularity of them makes them hugely costly. And why spend tons of money to find an older lady if you don't have to? Instead, find your self a large, mainstream dating neighborhood. 1 with hundreds of thousands of associates. Try clicking here and making a profile; I suggest it to all my visitors. Large dating communities are a fantastic way to find mature women because many of us are on a spending budget and favor not to spend lots of money on costly sites.
If you beloved this informative article along with you wish to receive details about see cougar dating here math-preview.wmflabs.org kindly check out our own web site.