In order to model random density-dependence in population dynamics, we construct the random analogue of the well-known logistic process in the branching process' framework. This density-dependence corresponds to intraspecific competition pressure, which is ubiquitous in ecology, and translates mathematically into a quadratic death rate. The logistic branching process, or LB-process, can thus be seen as (the mass of) a fragmentation process (corresponding to the branching mechanism) combined with constant coagulation rate (the death rate is proportional to the number of possible coalescing pairs). In the continuous state-space setting, the LB-process is a time-changed (in Lamperti's fashion) Ornstein-Uhlenbeck type process. We obtain similar results for both constructions: when natural deaths do not occur, the LB-process converges to a specified distribution; otherwise, it goes extinct a.s. In the latter case, we provide the expectation and the Laplace transform of the absorption time, as a functional of the solution of a Riccati differential equation. We also show that the quadratic regulatory term allows the LB-process to start at infinity, despite the fact that births occur infinitely often as the initial state goes to \infty. This result can be viewed as an extension of the pure-death process starting from infinity associated to Kingman's coalescent, when some independent fragmentation is added.
We give a new proof for a Ray-Knight representation of Feller's branching diffusion with logistic growth in terms of the local times of a reflected Brownian motion $H$ with a drift that is affine linear in the local time accumulated by $H$ at its current level. In \cite{LPW}, such a representation was obtained by an approximation through Harris paths that code the genealogies of particle systems. The present proof is purely in terms of stochastic analysis, and is inspired by previous work of Norris, Rogers and Williams \cite{NRW}.
We present a model for growth in a multi-species population. We consider two types evolving as a logistic branching process with mutation, where one of the types has a selective advantage, and are interested in the regime in which the carrying capacity of the system goes to $\infty$. We first study the genealogy of the population up until it almost reaches carrying capacity through a coupling with an independent branching process. We then focus on the phase in which the population has reached carrying capacity. After recovering a Gillespie--Wright--Fisher SDE in the infinite carrying capacity …
In this paper, we study the extinction time of logistic branching processes which are perturbed by an independent random environment driven by a Brownian motion. Our arguments use a Lamperti-type representation which is interesting on its own right and provides a one to one correspondence between the latter family of processes and the family of Feller diffusions which are perturbed by an independent spectrally positive Lévy process. When the independent random perturbation (of the Feller diffusion) is driven by a subordinator then the logistic branching processes in a Brownian environment conv…
We obtain a representation of Feller's branching diffusion with logistic growth in terms of the local times of a reflected Brownian motion $H$ with a drift that is affine linear in the local time accumulated by $H$ at its current level. As in the classical Ray-Knight representation, the excursions of $H$ are the exploration paths of the trees of descendants of the ancestors at time $t=0$, and the local time of $H$ at height $t$ measures the population size at time $t$ (see e.g. \cite{LG4}). We cope with the dependence in the reproduction by introducing a pecking order of individuals: an indivi…
We consider a class of density-dependent branching processes which generalises exponential, logistic and Gompertz growth. A population begins with a single individual, grows exponentially initially, and then growth may slow down as the population size moves towards a carrying capacity. At a time while the population is still growing superlinearly, a fixed number of individuals are sampled and their coalescent tree is drawn. Taking the sampling time and carrying capacity simultaneously to infinity, we prove convergence of the coalescent tree to a limiting tree which is in a sense universal over…
We consider a birth and death process in which death is due to both `natural death' and to competition between individuals, modelled as a quadratic function of population size. The resulting `logistic branching process' has been proposed as a model for numbers of individuals in populations competing for some resource, or for numbers of species. However, because of the quadratic death rate, even if the intrinsic growth rate is positive, the population will, with probability one, die out in finite time. There is considerable interest in understanding the process conditioned on non-extinction. In…
We establish an explicit rate of convergence for some systems of mean-field interacting diffusions with logistic binary branching towards the solutions of nonlinear evolution equations with non-local self-diffusion and logistic mass growth, shown to describe their large population limits in arXiv:1303.3939. The proof relies on a novel coupling argument for binary branching diffusions based on optimal transport, which allows us to sharply mimic the trajectory of the interacting binary branching population by certain system of independent particles with suitably distributed random space-time bir…
We consider a trait-structured population subject to mutation, birth and competition of logistic type, where the number of coexisting types may fluctuate. Applying a limit of rare mutations to this population while keeping the population size finite leads to a jump process, the so-called `trait substitution sequence', where evolution proceeds by successive invasions and fixations of mutant types. The probability of fixation of a mutant is interpreted as a fitness landscape that depends on the current state of the population. It was in adaptive dynamics that this kind of model was first invente…
We study two types of stochastic processes, a mean-field spatial system of interacting Fisher-Wright diffusions with an inferior and an advantageous type with rare mutation (inferior to advantageous) and a (mean-field) spatial system of supercritical branching random walks with an additional deathrate which is quadratic in the local number of particles. The former describes a standard two-type population under selection, mutation, the latter models describe a population under scarce resources causing additional death at high local population intensity. Geographic space is modelled by $\{1, \cd…
Motivated by tumor growth models we establish conditions for the $R-$positivity of Markov processes and positive matrices. We then apply them to obtain the asymptotic behavior of the tumors sizes in the supercritical regime.
We provide explicit conditions, in terms of the transition kernel of its driving particle, for a Markov branching process to admit a scaling limit toward a self-similar growth-fragmentation with negative index. We also derive a scaling limit for the genealogical embedding considered as a compact real tree.
We study the maximal displacement and related population for a branching Brownian motion in Euclidean space in terms of the principal eigenvalue of an associated Schrödinger type operator. We first determine their growth rates on the survival event. We then establish the upper deviation for the maximal displacement under the possibility of extinction. Under the non-extinction condition, we further discuss the decay rate of the upper deviation probability and the population growth at the critical phase.
We give a proof of a result on the growth of the number of particles along chosen paths in a branching Brownian motion. The work follows the approach of classical large deviations results, in which paths in $C[0,1]$ are rescaled onto $C[0,T]$ for large $T$. The methods used are probabilistic and take advantage of modern spine techniques.
We consider a branching particle system where each particle moves as an independent Brownian motion and breeds at a rate proportional to its distance from the origin raised to the power $p$, for $p\in[0,2)$. The asymptotic behaviour of the right-most particle for this system is already known; in this article we give large deviations probabilities for particles following "difficult" paths, growth rates along "easy" paths, the total population growth rate, and we derive the optimal paths which particles must follow to achieve this growth rate.
A version of ``preferential attachment'' random graphs, corresponding to linear ``weights'' with random ``edge additions,'' which generalizes some previously considered models, is studied. This graph model is embedded in a continuous-time branching scheme and, using the branching process apparatus, several results on the graph model asymptotics are obtained, some extending previous results, such as growth rates for a typical degree and the maximal degree, behavior of the vertex where the maximal degree is attained, and a law of large numbers for the empirical distribution of degrees which show…
We give new results on the growth of the number of particles in a dyadic branching Brownian motion which follow within a fixed distance of a path $f:[0,\infty)\to \mathbb{R}$. We show that it is possible to count the number of particles without rescaling the paths. Our results reveal that the number of particles along certain paths can oscillate dramatically. The methods used are entirely probabilistic, taking advantage of the spine technique developed by, amongst others, Lyons et al, Kyprianou, and Hardy & Harris.
We propose and study a stochastic capital-labour model with logistic growth function. First, we show that the model has a unique positive global solution. Then, using the Lyapunov analysis method, we obtain conditions for the extinction of the total labour force. Furthermore, we also prove sufficient conditions for their persistence in mean. Finally, we illustrate our theoretical results through numerical simulations.
In this note we consider a branching Brownian motion (BBM) on $\mathbb{R}$ in which a particle at spatial position $y$ splits into two at rate $βy^2$, where $β>0$ is a constant. This is a critical breeding rate for BBM in the sense that the expected population size blows up in finite time while the population size remains finite, almost surely, for all time. We find an asymptotic for the almost sure rate of growth of the population.
Suppose that $(Z_n)_{n\geq0}$ is a supercritical branching process in independent and identically distributed random environment. The right tail function of the scaled growth rate for $(Z_n)_{n\geq0}$ is studied. The upper bounds for $\displaystyle\mathbb{P}\left[\frac{\log Z_n}{Mn}-μ\geq x\right]$ for any $x\geq3$ are obtained, by applying an extension of the Hoeffding type inequalities.
We consider a particular Branching Random Walk in Random Environment (BRWRE) on $\sN_0$ started with one particle at the origin. Particles reproduce according to an offspring distribution (which depends on the location) and move either one step to the right (with a probability in $(0,1]$ which also depends on the location) or stay in the same place. We give criteria for local and global survival and show that global survival is equivalent to exponential growth of the moments. Further, on the event of survival the number of particles grows almost surely exponentially fast with the same growth r…
We study a spatial branching model, where the underlying motion is Brownian motion and the branching is affected by a random collection of reproduction blocking sets called "mild" obstacles. We show that the quenched local growth rate is given by the branching rate in the `free' region . When the underlying motion is an arbitrary diffusion process, we obtain a dichotomy for the local growth that is independent of the Poissonian intensity. Finally, and most importantly, we obtain the asymptotics (in probability) of the quenched (when $d\le 2$) and the annealed (arbitrary d) global growth rates,…
Consider a rooted infinite Galton-Watson tree with mean offspring number $m>1$, and a collection of i.i.d. positive random variables $ξ_e$ indexed by all the edges in the tree. We assign the resistance $m^d ξ_e$ to each edge $e$ at distance $d$ from the root. In this random electric network, we study the asymptotic behavior of the effective resistance and conductance between the root and the vertices at depth $n$. Our results generalize an existing work of Addario-Berry, Broutin and Lugosi on the binary tree to random branching networks.
We study branching random walks in random environment on the $d$-dimensional square lattice, $d \geq 1$. In this model, the environment has finite range dependence, and the population size cannot decrease. We prove limit theorems (laws of large numbers) for the set of lattice sites which are visited up to a large time as well as for the local size of the population. The limiting shape of this set is compact and convex, though the local size is given by a concave growth exponent. Also, we obtain the law of large numbers for the logarithm of the total number of particles in the process.
In Li (2011), Example 2.2, the notion of a multi-type continuous-state branching process (MCSBP) was introduced with a finite number of types, with the countably infinite case being proposed in Kyprianou and Palau (2017). One may consider such processes as a super-Markov chain on a countable state-space of types, which undertakes both local and non-local branching. In Kyprianou and Palau (2017) it was shown that, for MCSBPs, under mild conditions, there exists a lead eigenvalue which characterises the spectral radius of the linear semigroup associated to the process. Moreover, in a qualitative…
Under a first order moment condition on the immigration mechanism, we show that an appropriately scaled supercritical and irreducible multi-type continuous state and continuous time branching process with immigration (CBI process) converges almost surely. If an $x \log(x)$ moment condition on the branching mechanism does not hold, then the limit is zero. If this $x \log(x)$ moment condition holds, then we prove $L_1$ convergence as well. The projection of the limit on any left non-Perron eigenvector of the branching mean matrix is vanishing. If, in addition, a suitable extra power moment condi…
We consider a stochastic logistic growth model involving both birth and death rates in the drift and diffusion coefficients for which extinction eventually occurs almost surely. The associated complete Fokker-Planck equation describing the law of the process is established and studied. We then use its solution to build a likelihood function for the unknown model parameters, when discretely sampled data is available. The existing estimation methods need adaptation in order to deal with the extinction problem. We propose such adaptations, based on the particular form of the Fokker-Planck equatio…
We are interested in the dynamic of a structured branching population where the trait of each individual moves according to a Markov process. The rate of division of each individual is a function of its trait and when a branching event occurs, the trait of the descendants at birth depends on the trait of the mother and on the number of descendants. In this article, we explicitly describe the penalized Markov process, named auxiliary process, corresponding to the dynamic of the trait along the spine by giving its associated infinitesimal generator. We prove a Many-to-One formula and a Many-to-O…
In this paper, we formulate a stochastic logistic fish growth model driven by both white noise and non-Gaussian noise. We focus our study on the mean time to extinction, escape probability to measure the noise-induced extinction probability and the Fokker-Planck equation for fish population X(t). In the Gaussian case, these quantities satisfy local partial differential equations while in the non-Gaussian case, they satisfy nonlocal partial differential equations. Following a discussion of existence, uniqueness, and stability, we calculate numerical approximations of the solutions of those equa…
We consider a supercritical branching process and define a contact tracing mechanism on its genealogical tree. We calculate the growth rate of the post tracing process, and give conditions under which the tracing is strong enough to drive the process to extinction.
In this paper we consider a branching particle system consisting of particles moving according to the Ornstein-Uhlenbeck process in R^d and undergoing a binary, supercritical branching with a constant rate λ>0. This system is known to fulfil a law of large numbers (under exponential scaling). In the paper we prove the corresponding central limit theorem. Moreover, in the second part of the paper we consider U-statistics of the system, for which, under mild assumptions, we prove a law of large numbers and a central limit theorem. The limits are expressed in terms of multiple stochastic integral…
We study the high temperature phase of a family of typed branching diffusions initially studied in [Astérisque 236 (1996) 133--154] and [Lecture Notes in Math. 1729 (2000) 239--256 Springer, Berlin]. The primary aim is to establish some almost-sure limit results for the long-term behavior of this particle system, namely the speed at which the population of particles colonizes both space and type dimensions, as well as the rate at which the population grows within this asymptotic shape. Our approach will include identification of an explicit two-phase mechanism by which particles can build up i…
For a set $A\subset C[0,\infty)$, we give new results on the growth of the number of particles in a dyadic branching Brownian motion whose paths fall within A. We show that it is possible to work without rescaling the paths. We give large deviations probabilities as well as a more sophisticated proof of a result on growth in the number of particles along certain sets of paths. Our results reveal that the number of particles can oscillate dramatically. As a byproduct of our methods we also obtain new results on the number of particles near the frontier of the model. The methods used are entirel…
We study the long time behavior of an asymmetric size-structured measure-valued growth-fragmentation branching process that models the dynamics of a population of cells taking into account physiological and morphological asymmetry at division. We show that the process exhibits a Malthusian behavior; that is that the global population size grows exponentially fast and that the trait distribution of individuals converges to some stable distribution. The proof is based on a generalization of Lyapunov function techniques for non-conservative semi-groups. We then investigate the fluctuations of the…
We study the survival probability and the growth rate for branching random walks in random environment (BRWRE). The particles perform simple symmetric random walks on the $d$-dimensional integer lattice, while at each time unit, they split into independent copies according to time-space i.i.d. offspring distributions. The BRWRE is naturally associated with the directed polymers in random environment (DPRE), for which the quantity called the free energy is well studied. We discuss the survival probability (both global and local) for BRWRE and give a criterion for its positivity in terms of the …
We propose and analyze a new class of controlled multi-type branching processes with a per-step linear resource constraint, motivated by potential applications in viral marketing and cancer treatment. We show that the optimal exponential growth rate of the population can be achieved by maintaining a fixed proportion among the species, for both deterministic and stochastic branching processes. In the special case of a two-type population and with a symmetric reward structure, the optimal proportion is obtained in closed-form. In addition to revealing structural properties of controlled branchin…
We consider excursions for a class of stochastic processes describing a population of discrete individuals experiencing density-limited growth, such that the population has a finite carrying capacity and behaves qualitatively like the classical logistic model when the carrying capacity is large. Being discrete and stochastic, however, our population nonetheless goes extinct in finite time. We present results concerning the maximum of the population prior to extinction in the large population limit, from which we obtain establishment probabilities and upper bounds for the process, as well as es…
