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20

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 ...


15

A quick back-of-the-envelope answer to the number of generations that have passed since the estimated human-chimp split would be to divide the the split, approximately 7 million years ago (Langergraber et al. 2012), by the human generation time. The human generation time can be tricky to estimate, but 20 years is often used. However, the average number is ...


10

No, I don't think auto-regulation explain much in the population sizes of predators. Group selection may explain such auto-reagulation but I don't think it is of any considerable importance for this discussion. You'll be interested in various model of prey-predator or of consumer-ressource interactions. as @shigeta said, what the following says is: predators ...


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. ...


8

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

These equations describe how the haplotype frequencies will change over time due to a combination of recombination and natural selection. Before I proceed, I need to change your four $\delta X_i$ formulas above. Lewontin and Kojima (1960) writes the equations as: $$\Delta X_i = \frac{X_i(w_i - \bar w) \pm Drw_{14}}{\bar w}$$ where the minus sign is used ...


5

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 ...


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 ...


4

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 ...


4

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 - ...


4

Just need to solve the equation. p1 = X11 + X12; q1 = X11 + X21; 1 = X11 + X12 + X21 + X22. D = X11 - (X11 + X12) * (X11 + X21) D = X11 - (X11X11 + X11X21 + X11X12 + X12X21) D = X11 - X11X11 - X11X21 - X11X12 - X12X21 D = X11 * (1 - X11) - X11X21 - X11X12 - X12X21 D = X11 * (X11 + X12 + X21 + X22 - X11) - X11X21 - X11X12 - X12X21 D = X11 * (X12 + X21 + ...


4

Probably the best source to start would be Ilkka Hanksi's work, you can find a full list here: http://www.helsinki.fi/science/metapop/People/IlkkaPub2.htm. The seminal work would be "Ecology, Genetics and Evolution of Metapopulations" It gives a strong mathematical treatment


4

There is one book that will perfectly suits your needs: A biologist's guide to Mathematical Modeling in Ecology and Evolution, by Sally Otto It is a very good book that is very easy to understand and in the meantime goes pretty far (It ends with the use of diffusion equation in Evolutionary Biology). I highly recommend it. It covers: How to create a ...


4

This book "A primer of conservation genetics" would suit quite well I think. In particular chapter five deals with "Genetics and Extinction" and is preceded by a lot of population genetics based theory. A beginner might also combine it with "A primer of ecological genetics" (Hartl & Conner) but you seem to have enough Pop gen knowledge to not need it! ...


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

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

it is impossible to know the exact number so here is my gross ballpark estimate of an upper bound - i.e. the maximum number of organisms that could have lived on earth in the extreme best case scenario. in practice it is probably much less, but this is to get an idea of what kind of numbers we are dealing with. The earth's volume is about 1.08321 * 10^12 ...


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

According to Hartl & Clark on population genetics: "Population genetics deals with Mendel's laws and other genetic principles as they apply to entire populations of organisms.... also includes the study of the various forces that result in evolutionary changes in species through time." According to Conner & Hartl also on population ...


3

If you're interested in learning about the mathematics of population genetics, Population Genetics and Microevolutionary Theory by Alan Templeton is an absolutely amazing resource. If you check out the index, here are entries under population structure: assortative mating, admixture, linkage disequilibrium, coarse-grained spatial heterogeneity, gene flow, ...


3

Fishers Geometric Model (FGM) is a theoretical prediction about the adaptation process in traits. There are a number of things to establish before attempting comprehend FGM. Firstly, shifts in an adaptive landscape, in natural scenarios, are generally quite small. Because populations have been evolving for such a long time and the small shifts in adaptive ...


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 ...



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