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Population Dynamics - the math behind the evolution of species

Author Topic:   Population Dynamics - the math behind the evolution of species
RAZD
Member (Idle past 1538 days)
Posts: 20714
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Joined: 03-14-2004

 Message 1 of 5 (856535) 07-01-2019 2:36 PM

Population Size vs Ecological Carrying Capacity
This is going to be a learning experience for me, as I am weak in this area. Feel free to help.
quote:
Population dynamics is the branch of life sciences that studies the size and age composition of populations as dynamical systems, and the biological and environmental processes driving them (such as birth and death rates, and by immigration and emigration). Example scenarios are ageing populations, population growth, or population decline.
Population dynamics has traditionally been the dominant branch of mathematical biology, which has a history of more than 210 years, although more recently the scope of mathematical biology has greatly expanded. ...
The rate at which a population increases in size if there are no density-dependent forces regulating the population is known as the intrinsic rate of increase. It is
$\color{white} \frac{dN}{dt} \frac{1} {N} = r$
where the derivative dN/dt is the rate of increase of the population, N is the population size, and r is the intrinsic rate of increase. Thus r is the maximum theoretical rate of increase of a population per individual - that is, the maximum population growth rate. The concept is commonly used in insect population biology to determine how environmental factors affect the rate at which pest populations increase. See also exponential population growth and logistic population growth.[2]
so r can be positive (population growth), negative(population decline) or 0 (static population). Making it per individual makes it easier to compare different populations.
quote:
(ibid) ... Population regulation is a density-dependent process, meaning that population growth rates are regulated by the density of a population. Consider an analogy with a thermostat. When the temperature is too hot, the thermostat turns on the air conditioning to decrease the temperature back to homeostasis. When the temperature is too cold, the thermostat turns on the heater to increase the temperature back to homeostasis. Likewise with density dependence, whether the population density is high or low, population dynamics returns the population density to homeostasis. Homeostasis is the set point, or carrying capacity, defined as K.[3]
$\color{white} \frac{dN}{dt} = rN{\Big (}1-\frac{N}{K}{\Big )}$
where ( 1 ’ N/K ) is the density dependence, N is the number in the population, K is the set point for homeostasis and the carrying capacity. In this logistic model, population growth rate is highest at 1/2 K and the population growth rate is zero around K. The optimum harvesting rate is a close rate to 1/2 K where population will grow the fastest. Above K, the population growth rate is negative. The logistic models also show density dependence, meaning the per capita population growth rates decline as the population density increases. In the wild, you can't get these patterns to emerge without simplification. Negative density dependence allows for a population that overshoots the carrying capacity to decrease back to the carrying capacity, K.[3]
According to r/K selection theory organisms may be specialised for rapid growth, or stability closer to carrying capacity.
If the population exceeds the carrying capacity of the ecology the population size will decrease, if it is below the carrying capacity the population will increase. If it is at K then the population size will be static.
quote:
(ibid) In fisheries and wildlife management, population is affected by three dynamic rate functions.
• Natality or birth rate, often recruitment, which means reaching a certain size or reproductive stage. Usually refers to the age a fish can be caught and counted in nets.
• Population growth rate, which measures the growth of individuals in size and length. More important in fisheries, where population is often measured in biomass.
• Mortality, which includes harvest mortality and natural mortality. Natural mortality includes non-human predation, disease and old age.
If N1 is the number of individuals at time 1 then
$\color{white} N_{1} = N _{0}+ B - D + I - E$
where N0 is the number of individuals at time 0, B is the number of individuals born, D the number that died, I the number that immigrated, and E the number that emigrated between time 0 and time 1.
If we measure these rates over many time intervals, we can determine how a population's density changes over time. Immigration and emigration are present, but are usually not measured.
This can be applied to any species, not just fish.
Thus we can define a breeding population of a species and determine if it is growing, declining or static, ... once we know the carrying capacity of the ecological environment it inhabits.
We can see from the numbers/math that each species population will tend to stabilize at the carrying capacity of the ecological environment.
If we see a population that has a static population, we can assume that it is at the carrying capacity of the ecological environment.
If we see a population that is fluctuating, we can assume that it is fluctuating around the carrying capacity of the ecological environment.
Enjoy
Edited by RAZD, : subtitle
Edited by RAZD, : .

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 Replies to this message: Message 2 by RAZD, posted 07-01-2019 2:43 PM RAZD has replied

RAZD
Member (Idle past 1538 days)
Posts: 20714
From: the other end of the sidewalk
Joined: 03-14-2004

 Message 2 of 5 (856536) 07-01-2019 2:43 PM Reply to: Message 1 by RAZD07-01-2019 2:36 PM

Evolutionary Game Theory
quote:
Evolutionary game theory (EGT) is the application of game theory to evolving populations in biology. It defines a framework of contests, strategies, and analytics into which Darwinian competition can be modelled. It originated in 1973 with John Maynard Smith and George R. Price's formalisation of contests, analysed as strategies, and the mathematical criteria that can be used to predict the results of competing strategies.[1]
Evolutionary game theory differs from classical game theory in focusing more on the dynamics of strategy change.[2] This is influenced by the frequency of the competing strategies in the population.[3]
more to come (will edit this once thread promoted)
Edited by RAZD, : .

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 This message is a reply to: Message 1 by RAZD, posted 07-01-2019 2:36 PM RAZD has replied

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RAZD
Member (Idle past 1538 days)
Posts: 20714
From: the other end of the sidewalk
Joined: 03-14-2004

 Message 3 of 5 (856539) 07-01-2019 2:53 PM Reply to: Message 2 by RAZD07-01-2019 2:43 PM

Fisherian Runaway Selection
quote:
Fisherian runaway or runaway selection is a sexual selection mechanism proposed by the mathematical biologist Ronald Fisher in the early 20th century, to account for the evolution of exaggerated male ornamentation by persistent, directional female choice.[1][2][3] An example is the colourful and elaborate peacock plumage compared to the relatively subdued peahen plumage; the costly ornaments, notably the bird's extremely long tail, appear to be incompatible with natural selection. Fisherian runaway can be postulated to include sexually dimorphic phenotypic traits such as behaviour expressed by either sex.
The peacock tail in flight, the classic example of an ornament
assumed to be a Fisherian runaway
Peacock flying on rice field, at Karaikudi, Tamilnadu. Image
taken on 12th November 2012, by Haribabu Pasupathy.

more to come (will edit this once thread promoted)

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 Replies to this message: Message 4 by RAZD, posted 07-04-2019 3:00 PM RAZD has seen this message but not replied

RAZD
Member (Idle past 1538 days)
Posts: 20714
From: the other end of the sidewalk
Joined: 03-14-2004

 Message 4 of 5 (856996) 07-04-2019 3:00 PM Reply to: Message 3 by RAZD07-01-2019 2:53 PM

Zygosity
quote:
In population genetics, the concept of heterozygosity is commonly extended to refer to the population as a whole, i.e., the fraction of individuals in a population that are heterozygous for a particular locus. It can also refer to the fraction of loci within an individual that are heterozygous.
Typically, the observed (Ho) and expected (He) heterozygosities are compared, defined as follows for diploid individuals in a population:
Observed
$\color{white} H_o = \frac{\sum_{i=0}^{n} {(1\ {\textrm {if}}\ a_{{i1}}\neq a_{{i2}})} } {n}$
where $\color{white} n$ is the number of individuals in the population, and $\color{white} a_1, a_2$ are the alleles of individual $\color{white} i$ at the target locus.
Expected
$\color{white} H_{e}=1 - \sum_{i=1}^{m} {(f_{i})^{2} }$
where m is the number of alleles at the target locus, and $\color{white} f_{i}$ is the allele frequency of the $\color{white} i^{th}$ allele at the target locus.
Heterozygosity values of 51 worldwide human populations.[7] Sub-Saharan Africans
have the highest values in the world.
By David Lpez Herráez , Marc Bauchet , Kun Tang , Christoph Theunert,
Irina Pugach, Jing Li, Madhusudan R. Nandineni, Arnd Gross, Markus Scholz,
Mark Stoneking - Lpez Herráez D, Bauchet M, Tang K, Theunert C, Pugach I,
Li J, et al. (2009) Genetic Variation and Recent Positive Selection in Worldwide
Human Populations: Evidence from Nearly 1 Million SNPs. PLoS ONE 4(11): e7888.
doi:10.1371/journal.pone.0007888 http://journals.plos.org/plosone/article?i
d=10.1371/journal.pone.0007888, CC BY 2.5, https://commons.wikimedia.org/
w/index.php?curid=54659673

Just to be clear, the value for the observed heterozygosity function is 0 when the individual in question is homozygous, and ai1 = ai2. The function just counts up the number of individuals that are heterozygous at locus i.
quote:
A heterozygote advantage describes the case in which the heterozygous genotype has a higher relative fitness than either the homozygous dominant or homozygous recessive genotype. ...
When two populations of any sexual organism are separated and kept isolated from each other, the frequencies of deleterious mutations in the two populations will differ over time, by genetic drift. It is highly unlikely, however, that the same deleterious mutations will be common in both populations after a long period of separation. Since loss-of-function mutations tend to be recessive (given that dominant mutations of this type generally prevent the organism from reproducing and thereby passing the gene on to the next generation), the result of any cross between the two populations will be fitter than the parent.
Interesting that (non-lethal) deleterious mutations are involved here. This explains how "hybrid vigour" can occur.
quote:
Heterosis, hybrid vigor, or outbreeding enhancement, is the improved or increased function of any biological quality in a hybrid offspring. An offspring is heterotic if its traits are enhanced as a result of mixing the genetic contributions of its parents. These effects can be due to Mendelian or non-Mendelian inheritance.
The physiological vigor of an organism as manifested in its rapidity of growth, its height and general robustness, is positively correlated with the degree of dissimilarity in the gametes by whose union the organism was formed . The more numerous the differences between the uniting gametes ” at least within certain limits ” the greater on the whole is the amount of stimulation . These differences need not be Mendelian in their inheritance . To avoid the implication that all the genotypic differences which stimulate cell-division, growth and other physiological activities of an organism are Mendelian in their inheritance and also to gain brevity of expression I suggest . that the word 'heterosis' be adopted.
When a population is small or inbred, it tends to lose genetic diversity. Inbreeding depression is the loss of fitness due to loss of genetic diversity. Inbred strains tend to be homozygous for recessive alleles that are mildly harmful (or produce a trait that is undesirable from the standpoint of the breeder). Heterosis or hybrid vigor, on the other hand, is the tendency of outbred strains to exceed both inbred parents in fitness. ...
BUT the range of genetic difference between the two populations where interbreeding occurs doesn't always produce hybrid vigor (heterosis):
quote:
(ibid) Not all outcrosses result in heterosis. For example, when a hybrid inherits traits from its parents that are not fully compatible, fitness can be reduced. This is a form of outbreeding depression.
So once the differences become too great, too incompatible, then you get outbreeding depression, increasing until a point is reached where outbreeding fails to produce viable offspring ... and you have speciaton instead, the populations are reproductively isolated.
Enjoy
Edited by RAZD, : .

we are limited in our ability to understand
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 This message is a reply to: Message 3 by RAZD, posted 07-01-2019 2:53 PM RAZD has seen this message but not replied

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