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19

The conservation biology literature has a great deal of information, particularly with reference to developing species survival plans (e.g., Traill et al. [2007] report a minimum effective population size of ~4,000 will give a 99% persistence probability of 40 generations). Because the question specifically mentions human populations, I'll focus my answer ...


10

Population dynamics occupies a whole subset of mathematical biology. Perhaps the most pragmatic uses for modelling population dynamics come from the fields of epidemiology for modelling disease infection and transmission through a population (one such article), or ecology modelling things like forestation, fishing dynamics, predator-prey relationships (an ...


8

Leonardo's already given you an excellent answer, but I thought I'd add my perspective. I'm a mathematical epidemiologist, so I'd at least like to believe these types of models are useful. For me, there are a number of things population dynamics models are especially useful for: Highlighting data requirements. Yes, models need data, as you've mentioned. ...


7

No, I don't think there is any kind of auto-regulation. You'll be interested in various model of prey-predator or of consumer-ressource interactions. For example the Lotka-Volterra equations describe the population dynamics of two co-existing species where one is the prey and the other is a predator. Let's first define some variables… $x$ : Number of ...


7

I think it does make sense - with a population density for finland that is so low, the disease with such a low beta cannot communicate to enough people to propagate. The number of people who have this disease will be fewer each week. I think this makes sense because at 16 / km^2, you can expect that practically nobody will ever see each other. This is ...


7

Two previous answers listed many applications of population dynamics models. I want to add that they are also important for conservation of endangered species. For example classical stage-class model (Crouse et al 1987, free copy) indicate that the most effective way to protect sea turtles is reducing mortality of large juveniles. Moreover, you don't have ...


5

Here is my full derivation to the book example you gave, hopefully it'll help you clear up what went wrong: You need to remember that after there is selection acting on the population, you no longer have a total of 1 after selection. Think of selection as "killing" individuals, which means the total is now 1 minus what has been "selected out". s*y is what ...


5

Mark-recapture is the most frequently used method for small mammals. It's best when combined with uncertainty estimates and population dynamics models (e.g. projection matrix). The fluctuations of small rodent populations have long fascinated scientists, and various models have been developed. Logistic regression models can be used to estimate likelihood ...


5

I guess you meant the population size stability. It is considered that the biosystems will increase their capacity of adaptation when evolving in very fluctuating environments. I believe the population stability is embedded in the adaptability of individuals. There is a measurement about it, evolvability, when the environment changes, the faster the ...


4

The chaotic behaviour you are referring to (at least the one described in your link in the comments) is a property of the discrete version of the logistic equation, where you get chaotic dynamics at growth rates above ~3.55 (see the logistic map). The behaviour of this equation has been described in a classic paper by Robert May (1976). As you increase ...


4

Classification of equilibrium points is done on the basis of the eigenvalues. If the two eigenvalues have no real parts, it is a hyperbolic fixed point and represents undamped oscillation. If both have a negative real part, it is a stable fixed point. If any of the eigenvalues has an imaginary part then it represents damped oscillations (in that case the ...


3

Your calculations are the following. Assuming non-overlapping generations, the number of ancestors you have in the last $t$ generation is given by: $$\sum_{i=1}^t 2^t$$ This sounds correct. But there are some very strong assumptions: Generations are non-overlapping. A more realistic model would need to consider $t$ as a continuous variable a give a ...


3

As you have phrased it, the question can be understood in two ways. The population size at t+1 is 350, after births, deaths and migration have taken place. The population size is 350 after deaths, but you also have to take births and migration into account to calculate population size at t+1 For alternative 1 the growth rate is: $pgr = ...


3

This is derived from studying how heterozygosity changes over time. The standard equation for change in heterozygosity ($H$) with constant population size ($N$) is: $H_t = \left(1 - \frac{1}{2N}\right)^tH_0$ When $N$ varies between generations you use the product of this formula: $H_t = \left(1 - \frac{1}{2N_0}\right)\left(1 - ...


3

I don't believe you can produce a general function for this. It will depend on the exact gene and organism you are considering. From a molecular point of view, the vast majority of recessive mutations result from a change producing either a non-functional protein product or a truncated product that is cleaned up by the cell. We can reasonably assume that ...


3

To a good first approximation $\overline{\Delta f} = 0$. Where $\overline{\Delta f}$ is the mean change in fitness down to any point or indel mutation. The reasons for this are as follows: In the genome of higher organisms, most of the genome is non-functional ("junk") so most mutations will not have any effect regardless of the change made. A substantial ...


3

A solid limit: k must be greater than zero. Unless you're talking about some cannibalistic species or something that isn't suited to the model at all. As long as the species is productive in the new environment: k is greater than 1. The population is probably growing or again you probably won't be using an exponential growth model. As mentioned before you ...


3

This is a hard problem - estimates of total living things in an given environment are usually created by looking at the number of species and individuals found in a sample area and extrapolating. As far as estimating the number of living things in the world, there are still lots of species which are not known, making this number still unknown for the world ...


2

*It's been several years since I've worked with similar equations. The following reply is based on memory, and if anyone has a firmer grasp of the materials, please modify or answer as you see fit. I would assume λ is under ideal conditions or as an average of whatever species you're working with. Your modified lifetime reproductive output still leaves out ...


2

Great question! A lot of things affect how quickly a population or species can adapt to a new environment, including population size, mutation rate, generation time, standing genetic diversity, and selective pressure. The diversity of life encompasses practically all combinations of those variables. A bacterial population might very well contain enough ...


2

Is this the exact text from the book? The left side seems to represent the probability for "No coalescence in $k$ lines in $t$ generations (i.e. the $Pr(k)^t$ term), and at least one coalescence among those lines in generation $t+1$ (the $1-Pr(k)$ term)" which is the same event as "First coalescence event in $k$ lines is exactly in generation ...


2

An easy way to visualize the mistake in your thought experiment is to consider a bottleneck event, when the ancestral population was very small, maybe just a few individuals. This would mean that the entire current population is descending from just a few individuals. Your thought experiment is assuming that the "pyramid" of your ancentors is expanding all ...


1

[This is purely speculative] Assumptions: impact on fitness is measured by survival chance impact is because of protein coding genes Probability of a mutation at position $i$ $P(m=i\ |\ g)$ where $g$ is the genome with its annotations. Probability that activity of some protein changes by X-fold given mutation at $i^{th}$ position(s) in the genome: ...


1

Not in general -- there can be linkage disequilibrium among the loci. For instance, say that there are two di-allelic loci, $A/a$ and $B/b$, and that the frequencies of the $A$ and $B$ alleles are both $1/2$ and that they have the same effect on the trait, with no dominance. If all haplotypes in the population are either $Ab$ or $aB$ (with no $AB$ or $ab$ ...


1

One of the possible adjustments of these mathematical models is to introduce a "place to hide", making some (small) percent of the prey population not accessible (or much more difficult to access) for predators. After the number of predators decreases from starvation, prey individuals are relatively safer outside the "place to hide" and can grow over this ...


1

As you mention, the population-lag effect is responsible for this. From The Wikipedia article on TFR: http://en.wikipedia.org/wiki/Total_fertility_rate#Population-lag_effect A population that has recently dropped below replacement-level fertility will continue to grow, because the recent high fertility produced large numbers of young couples who ...


1

Hopefully this syllogism will answer your question. Given the following premises: In the absence of selection, fitness of individuals are at a theoretical maximum. If a theoretical maximum fitness is achieved then effective population size is maximum. If there is an allele that confers both increased and decreased fitness you have a genetic conflict (e.g. ...


1

I think at @fileunderwater provides a good explanation of the basics math behind it, and some good references. I would like to go more into detail about the modeling decisions and benefits of assuming weak selection and why it is done in the literature. When you are making evolutionary models, especially one in evolutionary game theory, the first place is ...


1

This is my take in this, without experience of using the Taylor series to analyse evolutionary game theory problems. As you know the Taylor series expansion of $f(x)$ at point $a$ can be written as: $f(x)= f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \frac{f^{(3)}(a)}{3!}(x-a)^3 + ... + \frac{f^{(n)}(a)}{n!}(x-a)^n + ...$ Often factors above second order are ...


1

It has been shown that predictions based on weak selection can be different from that on strong selection. In fact, for rare mutations, each strategy has stationary distribution at the mutation selection equilibrium. The rank of the component depicts the relative prevalence of the available strategies. Yet the rank of the components of this distribution can ...



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