31

No, I don't think auto-regulation explain much in the population sizes of predators. Group selection may explain such auto-regulation but I don't think it is of any considerable importance for this discussion. The short answer is, as @shigeta said [predators] tend to starve to death as they are too many! To have a better understanding of what @shigeta ...


26

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 ...


14

Remi.b's answer is an excellent one, and this should be taken as a supplement to it: It's possible your simulation is correct The Lotka-Volterra equations are what is known as a deterministic model, and it describes the behavior of predator-prey systems (in a somewhat simplified fashion) in large populations. Small populations are subject to what is known ...


14

Let's see! I took the most recent WHO data from here and did a quick an dirty analysis in R. Here is the histogram as well as a normal distribution with the same mean and standard deviation as the actual data: Does not look very normally distributed. In fact, the shapiro test confirms this impression: Shapiro-Wilk normality test data: df$life_expectancy ...


12

It is certainly possible to use general relationships to predict human population density or abundance. The relationship between population density/abundance and body size is an old topic in ecology that falls within the field of allometrics (how different features of organisms scale with body size). A closely related allometric relationship is Kleiber's law,...


12

You might want to look for asymmetric dispersal. Asymmetric dispersal has been found in many freshwater fishes (such as bullhead; Junker 2012), freshwater mussel (Terui et al., 2014) and in marine kelp (bull kelp; Collins et al., 2010). That being said asymmetric dispersal does not mean that dispersal goes exclusively one way. Maybe Blondel et al. (2020) ...


11

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 ...


11

This article discusses the origin of the terms. They come directly from the equation used to describe population dynamics. As Canadianer mentioned the Wiki also covers it quite well. "r" stands for "rate" {growth rate}, r strategists have a high r value and a low K value. They grow fast but most die. "K" stands for Kapazit├Ątsgrenzen which is german for ...


9

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 ...


9

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 ...


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 ...


8

You can make the continuous approximation when the population size is large. As mentioned by arboviral, there are algorithms that allow you to perform stochastic simulations with discrete variables. However, these are computationally much more intensive than integration of ODEs. Moreover, analytical solutions for the master-equations (time evolution of ...


8

I commented on this question, but the OP's response prompted me to think again. Here is the graph from the document that the OP linked to: Clearly what is confusing is that the parameter referred to as 'antibody level' rises quickly, but not as a step. In terms of the x-axis, vaguely labelled as weeks, it looks as if the level of antibody continues to rise ...


7

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 ...


7

We call it Colony Collapse Disorder (CCD). Consequences The phenomenon is observed worldwide and is pretty serious. Northern Ireland lost 50% of its beehives for example. Between 1997 and 2003, 10 millions beehives were lost. Many cultivated crops are pollinated by bees and we don't quite have an alternative today. In 2005 a study showed that the worth of ...


7

As Alan Boyd says, the relatively slow rise is due to gradual uptake of the injected antibody. If you deliver the antibodies by intravenous injection or another mode that allows rapid uptake (I use intraperitoneal injection in mice) then the antibody levels peak rapidly, less than a day and probably a couple hours, and then drop off; there is no continued ...


6

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 ...


6

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 ...


6

First of all, here is a program which simulates the evolution of the G-matrix over multiple generations, it's a few years old (they seem to have stopped developing it) and I've only played with it briefly. This could solve how to model the evolution of the G-matrix. Fisher's fundamental theorem is a great place to start off with the theory of this: The ...


6

First, Allee effects (also positive density dependence) can be modelled in several different ways, and the equation you give is one example. The terms weak and strong Allee effects are in my experience used in a couple of different ways. Most often, strong density dependence is used to denote Allee effects where the per capita population growth rate can ...


6

Here is a tree based on mitochondrial DNA variations in human populations. van Oven M, Kayser M. Hum Mutat. 2009 Feb;30(2):E386-94. Updated comprehensive phylogenetic tree of global human mitochondrial DNA variation. Looking at genetic distance between populations via mitochondria DNA, all nonafricans are descended from a founder in one mtDNA group (L3). ...


5

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 = \frac{(230+21)-(...


5

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 ...


5

You need to add Bell curves to your simulation. The most important curve to simulate is the nutritional quality of the prey though there are plenty more thing to curve like speed and virility for prey and predators both. Nature uses lots of Bell curves so they must be good for something, such as softening the harsh effects of pure exponential growth. I ...


5

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 + ...


5

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


5

I am presenting a speculative approach since nobody has mentioned about any existent models yet. Assuming that selection is based on performance in certain tasks; performance is a function of traits which in-turn is a function of genotype. Performance is a non-linear function of genotype and selection imposes a cutoff/bandpass filter on the performance ...


5

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 ...


5

Since you are asking for the biological interpretations about these parameters, it is important to realize that the model you are presenting is a non-dimensionalized version of this model: $$ \frac{dx_1}{dt} =b_1x_1\Big(1- \Big(\frac{x_1 + \beta_{12}x_2}{K_1} \Big)\Big) $$ $$ \frac{dx_2}{dt} =b_2x_2\Big(1- \Big(\frac{x_2 + \beta_{21}x_1}{K_2} \Big)\Big), $$ ...


5

I'd hardly call myself an expert on this topic by any stretch of the imagination, but you can actually come up with good approximations based on ODE-based models by rounding off to the nearest whole number (assuming that your populations are sufficiently large). The key word is "approximation" - it's not actually all that big of a deal to have to round your ...


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