population dynamics - What is the percentage of people living in England in 1500 AD whose lineage is still alive?

This sounds a bit random, but it stems from a lecture in statistical genetics which I attended a while ago. We were shown a population lineage graph from which it was clear that most lineages eventually go extinct. The further you go in to the past, the smaller the percentage of the individuals living at that time whose lineage is still alive today. This probably makes sense in the context of evolution.So I am curious about the speed of that process in human populations. I am aware that this speed, quite possibly, varies over time and geographi...Read more

Neanderthal and modern human population sizes

Are there any estimates of both the modern human and Neanderthal population sizes over the last, say, 100 k years? Since "purebread" Neanderthals are extinct, their population has to hit zero; that happened sometime around 30 k years ago. We have good estimates of both the Neanderthal (zero) and modern human (millions, billions) population sizes in the recent past -- what I'm wondering is what the numbers were (roughly) when modern humans and Neanderthals came into contact. How many people were there in each group? How many interbred? Is there ...Read more

How to check if a population density obeys replicator dynamics

Say we have a probability vector or population density $p = (p_1,...,p_n)$ with $p_i \geq 0$ and $\sum_i p_i =1$. Also assume we know the functions $g=(g_1,...,g_n)$ such that: $$p_i(t+1) = g_i(p(t))$$where $t$ is a discrete time index. Is there a way to check whether $g$ implements a discrete time replicator equation? I.e. whether there is a function $f=(f_1,...,f_n)$, called the fitness landscape, such that: $$p_i(t+1)= \frac{f_i(p(t))}{\langle f \rangle_{p(t)}} p_i(t).$$where $\langle f \rangle_{p(t)}:=\sum_j p_j(t) f_j(p(t))$ is the average...Read more

population dynamics - Two state exponential growth model

I want to write an expression to model exponential growth of a microbe that scholastically and irreversibly acquires a mutation at a fixed rate that slows growth.I can do it iteratively, but am curious whether a simple expression can be written.I started with:$P_{total} = P_{wt}e^{r_{wt}t} + P_{mutant}e^{r_{mutant}t} $How do I factor in the mutation rate that converts $P_{wt}$ to $P_{mutant}$?...Read more

Does a top heavy population grow, decline or stabilize?

Imagine a population that has a larger proportion of older individuals than younger ones, what could possible happen to it in the future?I am guessing that if the older individuals reproduce, the population will increase but as the older generation grow older they would eventually perish again decreasing the population. As they form the larger proportion, the net population would eventually grow but decline, but would the population then begin to stabilize or continue declining?...Read more