arXiv subjects
Publications and source records attributed to Rinaldo B. Schinazi.
It is well known that a binomial $(n,p)$ can be approximated by a Poisson distribution with parameter $np$. The typical approach in undergraduate probability texts is to show a convergence result for the distribution of the binomial as $n$ goes to infinity and $np$ converges to some $λ$. In this note we use instead the coupling technique to show a much more general result. Moreover, we only use elementary results from probability.
Let $N$ be a natural number. We consider a population which lives on $I_N=\{-N,-N+1,\dots,N-1,N\}$. Each individual gives birth at rate $λ$ on each of its neighboring sites and dies at rate 1. No births are allowed from the inside of $I_N$ to the outside or vice-versa. The population on the whole line (i.e. $N=+\infty$) survives with positive probability if and only if $λ>1/2$. On the other hand for any $1/2< λ\leq \sqrt 2/2$ there exists a natural number $N_c$ such that the population survives on $I_N$ for $N\geq N_c$ but dies out for $N<N_c$. There is no limit on the number of individuals per site so the population could grow at the center where the birth rates are maximum. Our result shows that it does not if the edge is too close.
Consider an $n\times n$ matrix $P$ with the following properties. All entries in $P$ are positive or $0$, the sum of each row is 1 and for all $i$ and $j$ in $\{1,\dots,n\}$ there exists a natural number $k$ such that the $(i,j)$ entry of the matrix $P^k$ is strictly positive. Then, there exists a unique row vector $v$ with only strictly positive entries, whose sum of entries is 1 and such that $vP=P$. We present a proof of this well-known result that uses only basic algebra and the Bolzano-Weierstrass Theorem.
Some species of salmon and sea turtle are famously good at finding their birth place to reproduce after having travelled vast expanses of ocean. In contrast, imagine now a species (maybe ancestral to the salmon or turtle) which has to find its birth place to reproduce but has no navigation skills and relies on chance alone. Would such an imaginary species survive? According to our (very simple) model it would survive if and only if the probability that a given individual find its birth place is strictly larger than 1/2.
We introduce the following model for the evolution of a population. At every discrete time $j\geq 0$ exactly one individual is introduced in the population and is assigned a death probability $c_j$ sampled from $C$, a fixed probability distribution. We think of $c_j$ as a genetic marker of this individual. At every time $n\geq 1$ every individual in the population dies or not independently of each other with its corresponding death probability $c_j$. We show that the population size goes to infinity if and only if $E(1/C)=\infty$. This is in sharp contrast with the model with constant $c$ and with the model in random environment (same random $c_n$ for all individuals at time $n$). Both of these models are always positive recurrent. Thus, genetic diversity does seem to help survival! We also study the point process associated with our model. We show that the limit point process has an accumulation point near 0 for the $c'$s. For certain $C$ distributions, including the uniform, the limit process properly rescaled is also shown to converge to a non-homogeneous Poisson process.
We introduce a population model to test the hypothesis that even a single migrant per generation may rescue a dying population. Let $(c_k)$ be a sequence of real numbers in $(0,1)$. Let $X_n$ be a size of the population at time $n\geq 0$. Then, $X_{n+1}=X_n - Y_{n+1}+1$, where the conditional distribution of $Y_{n+1}$ given $X_n=k$ is a binomial random variable with parameters $(k ,c(k))$. We assume that $\lim_{k\to\infty}kc(k)=ρ$ exists. If $ρ<1$ the process is transient with speed $1-ρ$ (so yes a single migrant per generation may rescue a dying population!) and if $ρ>1$ the process is positive recurrent. In the critical case $ρ=1$ the process is recurrent or transient according to how $k c(k)$ converges to $1$. When $ρ=0$ and under some regularity conditions, the support of the increments is eventually finite.
We consider a discrete time population model for which each individual alive at time $n$ survives independently of everybody else at time $n+1$ with probability $β_n$. The sequence $(β_n)$ is i.i.d. and constitutes our random environment. Moreover, at every time $n$ we add $Z_n$ individuals to the population. The sequence $(Z_n)$ is also i.i.d. We find sufficient conditions for null recurrence and transience (positive recurrence has been addressed by Neuts). We apply our results to a particular $(Z_n)$ distribution and deterministic $β$. This particular case shows a rather unusual phase transition in $β$ in the sense that the Markov chain goes from transience to null recurrence without ever reaching positive recurrence.
We introduce the following discrete time model. Each natural number represents an ecological niche and is assigned a fitness in $(0,1)$. All the sites are updated simultaneously at every discrete time. At any given time the environment may be normal with probability $p$ or a catastrophe may occur with probability $1-p$. If the environment is normal the fitness of each site is replaced by the maximum of its current fitness and a random number. If there is a catastrophe the fitness of each site is replaced by a random number. We compute the joint fitness distribution of any finite number of sites at any fixed time. We also show convergence of this system to a stationary distribution. This too is computed explicitly.
We introduce a model to study the impact of catastrophes on evolutionary paths. If we do not allow catastrophes the number of changes in the maximum fitness of a population grows logarithmically with respect to time. Allowing catastrophes (no matter how rare) yields a drastically different behavior. When catastrophes are possible the number of changes in the maximum fitness of the population grows linearly with time. Moreover, the evolutionary paths are a lot less predictable when catastrophes are possible. Our results can be seen as supporting the hypothesis that catastrophes speed up evolution by disrupting dominant species and creating space for new species to emerge and evolve.
We consider a model for which every site of $\mathbb{N}$ is assigned a fitness in $[0,1]$. At every discrete time all the sites are updated and each site samples a uniform on $[0,1]$, independently of everything else. At every discrete time and independently of the past the environment is good with probability $p$ or bad with probability $1-p$. The fitness of each site is then updated to the maximum or the minimum between its present fitness and the sampled uniform, according to whether the environment is good or bad. Assuming the initial fitness distribution is exchangeable over the site indexing, the empirical fitness distribution is a probability-valued Markov process. We show that this Markov process converges to an explicitly-identified stationary distribution exhibiting a self-similar structure.
We propose a Markov chain approach for the evolution of a genealogical line of genomes. Our idealized genome has $N$ sites and each site can be in state $0$ or $1$. At each time step we pick a site at random. If the site is in state $0$ we flip it to state 1 with probability $p$ or we keep it in state $0$ with probability $1-p$. If the site is in state $1$ we flip it to state 0 with probability $1-p$ or we keep it in state $1$ with probability $p$. Even when state 1 has a selective advantage (i.e. $p>1/2$) the Markov chain is quite unlikely to approach the most fit allele (i.e. all 1's). In fact, randomness (i.e. which site is picked for a possible mutation) and selection (i.e. the value of $p$) balance each other out so that the number of $1$'s in the genome converges to a Gaussian distribution centered around $Np$.
We consider a random walk with catastrophes which was introduced to model population biology. It is known that this Markov chain gets eventually absorbed at $0$ for all parameter values. Recently, it has been shown that this chain exhibits a metastable behavior in the sense that it can persist for a very long time before getting absorbed. In this paper we study this metastable phase by making the parameters converge to extreme values. We obtain four different limits that we believe shed light on the metastable phase.
We study a pure death process. At each discrete time every individual dies or not independently of each other with a constant probability. We give examples showing that in a certain limit extinction happens along a path where one and only one individual is lost at a time. We also exhibit an example for which such a path goes from infinity to 0 in a finite time. This is what we call a process implosion.
We propose a purely probabilistic model to explain the evolution path of a population maximum fitness. We show that after $n$ births in the population there are about $\ln n$ upwards jumps. This is true for any mutation probability and any fitness distribution and therefore suggests a general law for the number of upwards jumps. Simulations of our model show that a typical evolution path has first a steep rise followed by long plateaux. Moreover, independent runs show parallel paths. This is consistent with what was observed by Lenski and Travisano (1994) in their bacteria experiments.
Random population dynamics with catastrophes (events pertaining to possible elimination of a large portion of the population) has a long history in the mathematical literature. In this paper we study an ergodic model for random population dynamics with linear growth and binomial catastrophes: in a catastrophe, each individual survives with some fixed probability, independently of the rest. Through a coupling construction, we obtain sharp two-sided bounds for the rate of convergence to stationarity which are applied to show that the model exhibits a cutoff phenomenon.
Many species live in colonies that thrive for a while and then collapse. Upon collapse very few individuals survive. The survivors start new colonies at other sites that thrive until they collapse, and so on. We introduce spatial and non-spatial stochastic processes for modeling such population dynamic. Besides testing whether dispersion helps survival in a model experiencing large fluctuations, we obtain conditions for the population to get extinct or to survive.
We consider branching like models in local, global and fixed environments. We show that survival is more likely in a locally changing environment than in a fixed environment and that survival in a fixed environment is itself more likely than in a globally changing environment.
There are numerous stochastic models for cancer risk for a given tissue. Many rely on the following two hypotheses. 1. There is a fixed probability that a given cell division will eventually lead to a cancerous cell. 2. Cell divisions are nefarious or not independently of each other. We show that recent data on cancer risk and number of stem divisions is consistent with hypotheses 1 and 2.