ALLEE EFFECTS: POPULATION GROWTH,CRITICAL DENSITY,AND THE CHANCE OF EXTINCTION

This paper develops mathematical models to describe the growth, critical density, and extinction probability in sparse populations experiencing Allee effects. An Allee effect (or depensation) is a situation at low population densities where the per-individual growth rate is an increasing function of population density. A potentially important mechanism causing Allee effects is a shortage of mating encounters in sparse populations. Stochastic models are proposed for predicting the probability of encounter or the frequency of encounter as a function of population density. A negative exponential function is derived as such an encounter function under very general biological assumptions, including random, regular, or aggregated spatial patterns. A rectangular hyperbola function, heretofore used in ecology as the functional response of predator feeding rate to prey density, arises from the negative exponential function when encounter probabilities are assumed heterogeneous among individuals. These encounter functions produce Allee effects when incorporated into population growth models as birth rates. Three types of population models with encounter-limited birth rates are compared: (1) deterministic differential equations, (2) stochastic discrete birth-death processes, and (3) stochastic continuous diffusion processes. The phenomenon of a critical density, a major consequence of Allee effects, manifests itself differently in the different types of models. The critical density is a lower unstable equilibrium in the deterministic differential equation models. For the stochastic discrete birth-death processes considered here, the critical density is an inflection point in the probability of extinction plotted as a function of initial population density. In the continuous diffusion processes, the critical density becomes a local minimum (antimode) in the stationary probability distribution for population density. For both types of stochastic models, a critical density appears as an inflection point in the probability of attaining a small population density (extinction) before attaining a large one. Multiplicative (“environmental”) stochastic noise amplifies Allee effects. Harvesting also amplifies those effects. Though Allee effects are difficult to detect or measure in natural populations, their presence would seriously impact exploitation, management, and preservation of biological resources.     

Invasion and fixation of microbial dormancy traits under competitive pressure

Microbial dormancy is an evolutionary trait that has emerged independently at various positions across the tree of life. It describes the ability of a microorganism to switch to a metabolically inactive state that can withstand unfavourable conditions. However, maintaining such a trait requires additional resources that could otherwise be used to increase e.g. reproductive rates. In this paper, we aim for gaining a basic understanding under which conditions maintaining a seed bank of dormant individuals provides a “fitness advantage” when facing resource limitations and competition for resources among individuals (in an otherwise stable environment). In particular, we wish to understand when an individual with a “dormancy trait” can invade a resident population lacking this trait despite having a lower reproduction rate than the residents. To this end, we follow a stochastic individual-based approach employing birth-and-death processes, where dormancy is triggered by competitive pressure for resources. In the large-population limit, we identify a necessary and sufficient condition under which a complete invasion of mutants has a positive probability. Further, we explicitly determine the limiting probability of invasion and the asymptotic time to fixation of mutants in the case of a successful invasion. In the proofs, we observe the three classical phases of invasion dynamics in the guise of Coron et al. (2017, 2019).     

Co-evolutionary dynamics and Bayesian interaction games

Recently there has been a growing interest in evolutionary models of play with endogenous interaction structure. We call such processes co-evolutionary dynamics of networks and play. We study a co-evolutionary process of networks and play in settings where players have diverse preferences. In the class of potential games we provide a closed-form solution for the unique invariant distribution of this process. Based on this result we derive various asymptotic statistics generated by the co-evolutionary process. We give a complete characterization of the random graph model, and stochastically stable states in the small noise limit. Thereby we can select among action profiles and networks which appear jointly with non-vanishing frequency in the limit of small noise in the population. We further study stochastic stability in the limit of large player populations.     

Non‐linear systems in plankton population dynamics and nitrogen cycles

In this paper we propose a new perspective of population dynamics of plankton, by considering some effects of global ecological cycles, in which a mixed population of plankton is embedded. The propagation of plankton is extremely influenced by various material cycles, such as Nitrogen cycles. Taking this global effect into consideration, we will construct a mathematical model of non‐linear system. Our model is a non‐linear, non‐equilibrium system based on a stochastic model realizing population dynamics of a mixed population of two species of plankton which is placed in a global nitrogen cycle. We show, in this article, that our model gives a new mathematical foundation of phenomena such as water blooms and the predominance of one type of plankton against the other. We calculate the probability of the occurrence of the water bloom of a mixed population and that is where one type of plankton predominates. We show, as a characteristic feature of our model, that the function of predominance has some discontinuity and that there exists a threshold point among the initial values, with respect to the type of plankton that predominates the other. In other words, there is a sort of phase transition in dynamic changes of plankton population, as a result of global ecological cycles. Copyright © 2000 John Wiley & Sons, Ltd.     

A Theoretical Approach to Understanding Population Dynamics with Seasonal Developmental Durations

There is a growing body of biological investigations to understand impacts of seasonally changing environmental conditions on population dynamics in various research fields such as single population growth and disease transmission. On the other side, understanding the population dynamics subject to seasonally changing weather conditions plays a fundamental role in predicting the trends of population patterns and disease transmission risks under the scenarios of climate change. With the host–macroparasite interaction as a motivating example, we propose a synthesized approach for investigating the population dynamics subject to seasonal environmental variations from theoretical point of view, where the model development, basic reproduction ratio formulation and computation, and rigorous mathematical analysis are involved. The resultant model with periodic delay presents a novel term related to the rate of change of the developmental duration, bringing new challenges to dynamics analysis. By investigating a periodic semiflow on a suitably chosen phase space, the global dynamics of a threshold type is established: all solutions either go to zero when basic reproduction ratio is less than one, or stabilize at a positive periodic state when the reproduction ratio is greater than one. The synthesized approach developed here is applicable to broader contexts of investigating biological systems with seasonal developmental durations.     

Bogoliubov equations and functional mechanics

The functional classical mechanics based on the probability approach, where a particle is described not by a trajectory in the phase space but by a probability distribution, was recently proposed for solving the irreversibility problem, i.e., the problem of matching the time reversibility of microscopic dynamics equations and the irreversibility of macrosystem dynamics. In the framework of functional mechanics, we derive Bogoliubov-Boltzmann-type equations for finitely many particles. We show that a closed equation for a one-particle distribution function can be rigorously derived in functional mechanics without any additional assumptions required in the Bogoliubov method. We consider the possibility of using diffusion processes and the Fokker-Planck-Kolmogorov equation to describe isolated particles.     

The MMCPP/GE/c Queue

We obtain the queue length probability distribution at equilibrium for a multi-server, single queue with generalised exponential (GE) service time distribution and a Markov modulated compound Poisson arrival process (MMCPP) – i.e., a Poisson point process with bulk arrivals having geometrically distributed batch size whose parameters are modulated by a Markovian arrival phase process. This arrival process has been considered appropriate in ATM networks and the GE service times provide greater flexibility than the more conventionally assumed exponential distribution. The result is exact and is derived, for both infinite and finite capacity queues, using the method of spectral expansion applied to the two dimensional (queue length by phase of the arrival process) Markov process that describes the dynamics of the system. The Laplace transform of the interdeparture time probability density function is then obtained. The analysis therefore could provide the basis of a building block for modelling networks of switching nodes in terms of their internal arrival processes, which may be both correlated and bursty.     

Basic Reproduction Number in a Growing Population

The basic reproduction number of a fast disease epidemic on a slowly growing network may increase to a maximum then decrease to its equi- librium value while the population increases, which is not displayed by classical homogeneous mixing disease models. In this paper, we show that, by properly keeping track of the dynamics of the per capita contact rate in the population due to population dynamics, classical homogeneous mixing models show simi- lar non-monotonic dynamics in the basic reproduction number. This suggests that modeling the dynamics of the contact rate in classical disease models with population dynamics may be important to study disease dynamics in growing populations.     

Lower large deviations for supercritical branching processes in random environment

Branching processes in random environment (Z _n : n ≥ 0) are the generalization of Galton-Watson processes where in each generation the reproduction law is picked randomly in an i.i.d. manner. In the supercritical regime, the process survives with a positive probability and grows exponentially on the non-extinction event. We focus on rare events when the process takes positive values but lower than expected. More precisely, we are interested in the lower large deviations of Z, which means the asymptotic behavior of the probability {1 ≤ Z _n ≤ exp(nθ)} as n → ∞. We provide an expression for the rate of decrease of this probability under some moment assumptions, which yields the rate function. With this result we generalize the lower large deviation theorem of Bansaye and Berestycki (2009) by considering processes where ?(Z ₁ = 0 | Z ₀ = 1) > 0 and also much weaker moment assumptions.     

Small value probabilities for continuous state branching processes with immigration

We consider the small value probability of supercritical continuous state branching processes with immigration.From Pinsky(1972) it is known that under regularity condition on the branching mechanism and immigration mechanism,the normalized population size converges to a non-degenerate finite and positive limit W as t tends to infinity.We provide sharp estimate on asymptotic behavior of P(W≤ε) as ε→ 0+ by studying the Laplace transform of W.Without immigration,we also give a simpler proof for the small value probability in the non-subordinator case via the prolific backbone decomposition.     

Lotka-Volterra's model and migrations: Breaking of the well-known center

This paper is devoted to the study of the effect of individual behavior on the Lotka-Volterra predation. We assume that the individuals have many activities in a day for example. Each population is subdivided into subpopulations corresponding to different activities. In order to be clear, I have chosen the case of two activities for each population. We assume that the activities change is faster than the other processes (reproduction, mortality, predation…). This means that we consider population in which the individuals change their activities many times in a day while the reproduction and the predation effects are sensible after about ten days, for example. We use the aggregation method developed in [1] to obtain the global dynamics. Indeed, we start with a micro-model governing the micro-variables, which are the subpopulation densities; the aggregation method permits us to obtain a simpler system governing the macro-variables, which are the global population densities. Furthermore, this method allows us to observe emergence of the dynamics. Indeed, the method implies that the dynamics of the micro-system is close to an invariant manifold after a short time. We show that the dynamics on this manifold is a perturbation of the well-known center of the Lotka-Volterra model. Finally, we prove that a weak change of behavior can lead to a subcritical Hopf bifurcation in the global dynamics.     

THE IMPACT OF AGE-, SPACE-AND STOCHASTIC-STRUCTURE ON THE EXTINCTION PROBABILITY OF A CHINOOK SALMON METAPOPULATION

ABSTRACT. Using a mechanistic model, based on chinook life history, incorporating environmental and demographic stochasticity, we investigate how the probability of extinction is controlled by age, space and stochastic structure. Environmental perturbations of age dependent survivorships, combined with mixing of year classes in the spawning population, can lower the probability of extinction dramatically. This is an analog of the more familiar metapopulation result where dispersal between asynchronously fluctuating populations enhances persistence. For a two-river chinook metapopulation, dispersal between rivers with asynchronous environmental perturbations also dramatically enhances persistence, and anti-synchronous population fluctuations provide an even greater persistence probability. Anti-synchronous fluctuations would most likely occur in pristine habitat with naturally high levels of heterogeneity. Fifty percent dispersal between two populations provides the greatest insurance against extinction, a rate unrealistically high for salmon. In contrast, dispersal between exactly correlated populations with large amplitude environmental perturbations does not help persistence, no matter how high the dispersal rate. This is in spite of weak asynchrony provided by demographic stochasticity. Dispersal between rivers, one degraded and the other pristine, can substantially increase the probability of metapopulation extinction. Population structure, combined with asynchronous environmental perturbations and dispersal (or age class mixing) lowers the probability of chinook extinction dramatically but is almost useless when survivorships are impaired.     

Threshold dynamics and ergodicity of an SIRS epidemic model with semi-Markov switching

This work studies the threshold dynamics and ergodicity of a stochastic SIRS epidemic model with the disease transmission rate driven by a semi-Markov process. The semi-Markov process used in this paper for describing a randomly changing environment is a very large extension of the most common Markov regime-switching process. We define a basic reproduction number for the semi-Markov regime-switching environment and show that its position with respect to 1 determines the extinction or persistence of the disease. In the case of disease persistence, we give mild sufficient conditions for ensuring the existence and absolute continuity of the invariant probability measure. Under the same conditions, we also prove the global attractivity of the Ω-limit set of the system and the convergence in total variation norm of the transition probability to the invariant measure. Compared with the existing results in the Markov regime-switching environment, the results generalized require almost no additional conditions.     

Approximating the ideal free distribution via reaction-diffusion-advection equations

We consider reaction-diffusion-advection models for spatially distributed populations that have a tendency to disperse up the gradient of fitness, where fitness is defined as a logistic local population growth rate. We show that in temporally constant but spatially varying environments such populations have equilibrium distributions that can approximate those that would be predicted by a version of the ideal free distribution incorporating population dynamics. The modeling approach shows that a dispersal mechanism based on local information about the environment and population density can approximate the ideal free distribution. The analysis suggests that such a dispersal mechanism may sometimes be advantageous because it allows populations to approximately track resource availability. The models are quasilinear parabolic equations with nonlinear boundary conditions.     

Population models with partial migration

Populations exhibiting partial migration consist of two groups of individuals: Those that migrate between habitats, and those that remain fixed in a single habitat. We propose several discrete-time population models to investigate the coexistence of migrants and residents. The first class of models is linear, and we distinguish two scenarios. In the first, there is a single egg pool to which both populations contribute. A fraction of the eggs is destined to become migrants, and the remainder become residents. In a second model, there are two distinct egg pools to which the two types contribute, one corresponding to residents and another to migrants. The asymptotic growth or decline in these models can be phrased in terms of the value of the basic reproduction number being larger or less than one respectively. A second class of models incorporates density dependence effects. It is assumed that increased densities in the various life history stages adversely affect the success of transitioning of individuals to subsequent stages. Here too we consider models with one or two egg pools. Although these are nonlinear models, their asymptotic dynamics can still be classified in terms of the value of a locally defined basic reproduction number: If it is less than one, then the entire population goes extinct, whereas it settles at a unique fixed point consisting of a mixture of residents and migrants, when it is larger than one. Thus, the value of the basic reproduction number can be used to predict the stable coexistence or collapse of populations exhibiting partial migration.     

A solution to the distribution problems arising in the studies of a two-sex population process

Given a population of two sexes, the birth rate of one sex of which depends upon the population size of the other, it is very difficult to find an explicit expression for the probability distribution of the former. In this paper we have explicitly found the probability generating function of the joint distribution from which individual probability distributions and, in particular, moments of all orders in each case can be obtained in principle. As an example, using this probability generating function we have worked out explicitly the first and second order moments of the male and female populations and the explicit expression for the distribution of the male population in a particular case. This method can be successfully applied for the same purpose in the studies of chemical and biological processes where the synthesis or production of one species depends upon the concentration of another species.     

Dynamics of a system of three interacting populations with Allee effects and stocking

A model of three interacting populations where two populations engage in competition and two populations are in predator–prey type interaction is proposed and analysed. One of the two competing populations is subject to Allee effects and is also a pest population. The other competing population is regarded as a control agent and is the host for the predator population. There is a constant level of the external control agents released into the interaction at each generation after parasitism. We provide asymptotic dynamics of the competition subsystem and prove that a Neimark–Sacker bifurcation occurs for the host–parasitoid subsystem when the unique interior steady state loses its stability. The three interacting populations are impossible to persist for all positive initial conditions. Sufficient conditions based on the initial population size of the population with Allee effects are derived for persistence of the three populations.     

Basic Reproduction Number of Rabies Model with Stage Structure

The mathematical model is proposed to simulate the dynamics of rabies transmissions in the raccoon population where juveniles stay with their mother and become adults until they establish their own habitats. The basic reproduction number of rabies transmission is formulated and is shown to be a threshold value of disease invasion. The bifurcation direction from the disease-free equilibrium is proved to be forward when the basic reproduction number passes through unity for spatial homogenous environment. The global stability of the disease-free steady state is also studied.     

Competitive exclusion in a nonlocal reaction–diffusion–advection model of phytoplankton populations

We continue our study on the global dynamics of a nonlocal reaction–diffusion–advection system modeling the population dynamics of two competing phytoplankton species in a eutrophic environment, where both populations depend solely on light for their metabolism. In our previous work, we proved that system (1.1) is a strongly monotone dynamical system with respect to a non-standard cone related to the cumulative distribution functions, and further determined the global dynamics when the species have either identical diffusion rate or identical advection rate. In this paper, we study the trade-off of diffusion and advection and their joint influence on the outcome of competition. Two critical curves for the local stability of two semi-trivial equilibria are analyzed, and some new competitive exclusion results are obtained. Our main tools, besides the theory of monotone dynamical system, include some new monotonicity results for the principal eigenvalues of elliptic operators in one-dimensional domains.     

On a Feasible–Infeasible Two-Population (FI-2Pop) genetic algorithm for constrained optimization: Distance tracing and no free lunch

We explore data-driven methods for gaining insight into the dynamics of a two-population genetic algorithm (GA), which has been effective in tests on constrained optimization problems. We track and compare one population of feasible solutions and another population of infeasible solutions. Feasible solutions are selected and bred to improve their objective function values. Infeasible solutions are selected and bred to reduce their constraint violations. Interbreeding between populations is completely indirect, that is, only through their offspring that happen to migrate to the other population. We introduce an empirical measure of distance, and apply it between individuals and between population centroids to monitor the progress of evolution. We find that the centroids of the two populations approach each other and stabilize. This is a valuable characterization of convergence. We find the infeasible population influences, and sometimes dominates, the genetic material of the optimum solution. Since the infeasible population is not evaluated by the objective function, it is free to explore boundary regions, where the optimum is likely to be found. Roughly speaking, the No Free Lunch theorems for optimization show that all blackbox algorithms (such as Genetic Algorithms) have the same average performance over the set of all problems. As such, our algorithm would, on average, be no better than random search or any other blackbox search method. However, we provide two general theorems that give conditions that render null the No Free Lunch results for the constrained optimization problem class we study. The approach taken here thereby escapes the No Free Lunch implications, per se.