Species abundance distributions in neutral models with immigration or mutation and general lifetimes

We consider a general, neutral, dynamical model of biodiversity. Individuals have i.i.d. lifetime durations, which are not necessarily exponentially distributed, and each individual gives birth independently at constant rate λ. Thus, the population size is a homogeneous, binary Crump–Mode–Jagers process (which is not necessarily a Markov process). We assume that types are clonally inherited. We consider two classes of speciation models in this setting. In the immigration model, new individuals of an entirely new species singly enter the population at constant rate μ (e.g., from the mainland into the island). In the mutation model, each individual independently experiences point mutations in its germ line, at constant rate θ. We are interested in the species abundance distribution, i.e., in the numbers, denoted I _n (k) in the immigration model and A _n (k) in the mutation model, of species represented by k individuals, k = 1, 2, . . . , n, when there are n individuals in the total population. In the immigration model, we prove that the numbers (I _t (k); k ≥ 1) of species represented by k individuals at time t, are independent Poisson variables with parameters as in Fisher’s log-series. When conditioning on the total size of the population to equal n, this results in species abundance distributions given by Ewens’ sampling formula. In particular, I _n (k) converges as n → ∞ to a Poisson r.v. with mean γ/k, where γ : = μ/λ. In the mutation model, as n → ∞, we obtain the almost sure convergence of n ⁻¹ A _n (k) to a nonrandom explicit constant. In the case of a critical, linear birth–death process, this constant is given by Fisher’s log-series, namely n ⁻¹ A _n (k) converges to α ^k /k, where α : = λ/(λ + θ). In both models, the abundances of the most abundant species are briefly discussed.     

Population Dynamics of <Emphasis Type="Italic">Agave marmorata</Emphasis> Roezl. under Two Contrasting Management Systems in Central Mexico

Population Dynamics of Agave marmorata Roezl. under Two Contrasting Management Systems in Central Mexico. This paper evaluates the impacts of traditional management on the population dynamics of Agave marmorata, a multipurpose, useful species that is dominant in the Zapotitlán Salinas Valley, Puebla, Mexico. During 2002–2003 and 2003–2004, we constructed matrix models for two populations—one unmanaged, the other subject to plant extraction and cutting of flowering stalks. We also conducted prospective (elasticity) and retrospective (life table response experiments) analyses. Overall, the unmanaged population had higher finite rates of increase (λ) than the managed one. This variation in λ was the result of a decrease in the individual growth and fecundity in the managed population. Survival and growth were the demographic processes with the highest contribution to λ in the unmanaged population, while survival was the most important in the managed one. Our results suggest that management and plant extraction practices could be having negative effects on the population dynamics of this plant species. Recommendations are provided in order to promote positive effects on its vital rates and regeneration capacity.     

Demography of a dormancy-prone geophyte: influence of spatial scale on interpretation of dynamics

Both temporal and spatial scales are important in the evaluation of population dynamics, but the latter often receives less attention in demographic analyses. We used a 5-year demographic data set for a long-lived geophyte, Calochortus lyallii, to investigate the pattern and components of spatial variation at two scales (population and microsite). We found that neither the projected population structure nor asymptotic population growth rate (λ) varied greatly across either scale, although the underlying contributors to the variation in λ, V(λ), did differ between scales. Life table response experiment analyses showed that V(λ) among populations came primarily from variation in seedling survival and progression of non-reproductive plants, whereas V(λ) among microsites was primarily due to the variable fertility of large adults. Prolonged dormancy was important in reducing V(λ) among quadrats at both the scales, partly countering fluctuations in other transitions such as recruitment. This result represents some of the first evidence that underground “bulb banks” could function to offset the effects of a spatially heterogeneous environment in a manner analogous to seed banks. Future work is needed to isolate the specific, sometimes idiosyncratic, life history phenomena acting to modulate plant population dynamics in a spatial context.     

Interaction of maturation delay and nonlinear birth in population and epidemic models

 A population with birth rate function B(N) N and linear death rate for the adult stage is assumed to have a maturation delay T>0. Thus the growth equation N′(t)=B(N(t−T)) N(t−T) e⁻ d ₁ T−dN(t) governs the adult population, with the death rate in previous life stages d ₁≧0. Standard assumptions are made on B(N) so that a unique equilibrium N _e exists. When B(N) N is not monotone, the delay T can qualitatively change the dynamics. For some fixed values of the parameters with d ₁>0, as T increases the equilibrium N _e can switch from being stable to unstable (with numerically observed periodic solutions) and then back to stable. When disease that does not cause death is introduced into the population, a threshold parameter R ₀ is identified. When R ₀<1, the disease dies out; when R ₀>1, the disease remains endemic, either tending to an equilibrium value or oscillating about this value. Numerical simulations indicate that oscillations can also be induced by disease related death in a model with maturation delay. Received: 2 November 1998 / Revised version: 26 February 1999     

Mathematical models for cellular systems the von Foerster equation. Part I

P. B. M. Walker (1954) and H. C. Longuet-Higgins (quoted by Walker), as well as O. Scherbaum and G. Rasch (1957), made the first attempts towards a mathematical study of the age distribution in a cellular population. It was H. Von Foerster (1959), however, who derived the complete differential equation for the age density function,n(t, a). His equation is obtained from an analysis of the infinitesimal changes occurring during a time elementdt in a group of cells with ages betweena anda+da. The behavior of the population is determined by a quantity λ which we call the loss function. In this paper a rigorous discussion of the Von Foerster equation is presented, and a solution is given for the special case when λ depends, ont (time) anda (age) but not on other variables (such asn itself). It is also shown that the age density,n(t, a), is completely known only if the birth rate,α(t), and the initial age distribution, β(a), are given as boundary conditions. In Section II the steady state solution and some plausible forms of intrinsic loss functions (depending ona only) are discussed in view of later applications. This work was performed under the auspices of the U.S. Atomic Energy Commission.     

Herbivory and population dynamics of invasive and native <Emphasis Type="Italic">Lespedeza</Emphasis>

Some exotic plants are able to invade habitats and attain higher fitness than native species, even when the native species are closely related. One explanation for successful plant invasion is that exotic invasive plant species receive less herbivory or other enemy damage than native species, and this allows them to achieve rapid population growth. Despite many studies comparing herbivory and fitness of native and invasive congeners, none have quantified population growth rates. Here, we examined the contribution of herbivory to the population dynamics of the invasive species, Lespedeza cuneata, and its native congener, L. virginica, using an herbivory reduction experiment. We found that invasive L. cuneata experienced less herbivory than L. virginica. Further, in ambient conditions, the population growth rate of L. cuneata (λ = 20.4) was dramatically larger than L. virginica (λ = 1.7). Reducing herbivory significantly increased fitness of only the largest L. virginica plants, and this resulted in a small but significant increase in its population growth rate. Elasticity analysis showed that the growth rate of these species is most sensitive to changes in the seed production of small plants, a vital rate that is relatively unaffected by herbivory. In all, these species show dramatic differences in their population growth rates, and only 2% of that difference can be explained by their differences in herbivory incidence. Our results demonstrate that to understand the importance of consumers in explaining the relative success of invasive and native species, studies must determine how consumer effects on fitness components translate into population-level consequences. Electronic supplementary material  The online version of this article (doi:) contains supplementary material, which is available to authorized users.     

Demographic differences between two sympatric lilies (<Emphasis Type="Italic">Calochortus</Emphasis>) with contrasting distributions,as revealed by matrix analysis

Related species of similar morphology can differ greatly in distribution and abundance. Elucidating reasons for such differences can contribute to an understanding of intrinsic limiting factors and the causes of rarity. We studied sympatric populations of two terrestrial lilies with contrasting distributions: Calochortus lyallii, which is geographically restricted but locally abundant, and C. macrocarpus, which is widespread but locally sparse. Marked plants of each species were monitored for 5 years in British Columbia, Canada. Matrix projection models were used to estimate annual and stochastic population growth rates (λ and λ_s) and to compare demographic traits. Annual λ-values ranged from 0.89 to 1.04 in C. lyallii and from 0.89 to 1.01 in C. macrocarpus. Stochastic projections yielded a long-term growth rate near 1 for C. lyallii, but indicated a decline for C. macrocarpus. Elasticity analysis indicated that over the 5-year period of the study, survival of flowering plants made a larger proportional contribution to λ in C. lyallii than in C. macrocarpus. LTRE analysis showed that temporal variation in λ was driven primarily by the dynamics of flowering individuals in C. lyallii, and by the dynamics of vegetative individuals in C. macrocarpus. Similarly, higher flowering rates in C. lyallii and greater vegetative stasis in C. macrocarpus made the largest contribution to the difference in λ between species. Thus, local persistence in these two morphologically similar species appears to be achieved via different demographic pathways. Our analyses show that extrapolations about demographic processes and population dynamics based on taxonomic relatedness, morphological similarity or habitat overlap may often not be justified. Electronic Supplementary Material The online version of this article contains supplementary material, which is available to authorised users.     

How small are small mutation rates?

We consider evolutionary game dynamics in a finite population of size N. When mutations are rare, the population is monomorphic most of the time. Occasionally a mutation arises. It can either reach fixation or go extinct. The evolutionary dynamics of the process under small mutation rates can be approximated by an embedded Markov chain on the pure states. Here we analyze how small the mutation rate should be to make the embedded Markov chain a good approximation by calculating the difference between the real stationary distribution and the approximated one. While for a coexistence game, where the best reply to any strategy is the opposite strategy, it is necessary that the mutation rate μ is less than N ^−1/2exp[−N] to ensure that the approximation is good, for all other games, it is sufficient if the mutation rate is smaller than (N ln N)⁻¹. Our results also hold for a wide class of imitation processes under arbitrary selection intensity.     

Chemotactic collapse for the Keller-Segel model

 This work is concerned with the system (S) {u _t =Δu − χ∇ (u∇v) for x∈Ω, t>0Γ v _t =Δv+(u−1) for x∈Ω, t>0 where Γ, χ are positive constants and Ω is a bounded and smooth open set in ℝ². On the boundary ∂Ω, we impose no-flux conditions: (N) ∂u∂n =∂v∂n =0 for x∈∂ Ω, t>0 Problem (S), (N) is a classical model to describe chemotaxis corresponding to a species of concentration u(x, t) which tends to aggregate towards high concentrations of a chemical that the species releases. When completed with suitable initial values at t=0 for u(x, t), v(x, t), the problem under consideration is known to be well posed, locally in time. By means of matched asymptotic expansions techniques, we show here that there exist radial solutions exhibiting chemotactic collapse. By this we mean that u(r, t) →Aδ(y) as t→T for some T<∞, where A is the total concentration of the species. Received 9 March 1995; received in revised form 25 December 1995     

Approximation of the Basic Reproduction Number <Emphasis Type="Italic">R</Emphasis><Subscript>0</Subscript> for Vector-Borne Diseases with a Periodic Vector Population

The main purpose of this paper is to give an approximate formula involving two terms for the basic reproduction number R ₀ of a vector-borne disease when the vector population has small seasonal fluctuations of the form p(t) = p ₀ (1+ε cos (ωt − φ)) with ε ≪ 1. The first term is similar to the case of a constant vector population p but with p replaced by the average vector population p ₀. The maximum correction due to the second term is (ε²/8)% and always tends to decrease R ₀. The basic reproduction number R ₀ is defined through the spectral radius of a linear integral operator. Four numerical methods for the computation of R ₀ are compared using as example a model for the 2005/2006 chikungunya epidemic in La Réunion. The approximate formula and the numerical methods can be used for many other epidemic models with seasonality. MSC 92D30 ⋅ 45C05 ⋅ 47A55     

Remnant population dynamics in the facultative biennial Carum carvi in fragmented semi-natural grasslands

Transition matrix models were used to examine the population dynamics in the facultative biennial Carum carvi L. in semi-natural grasslands, specifically to assess what life cycle stages are important for population development and to evaluate the effects of environmental stochasticity on population persistence and, hence, the ability to develop remnant populations. The demographic studies were conducted over a 4-year period in three moderately grazed grasslands that differed in onset and duration of grazing. Experimental seed-sowing was also conducted in disturbed and undisturbed plots in the populations. Deterministic and stochastic models yielded overall negative population growth (λ < 1) for the populations. λ was sensitive to transitions in the most frequent vegetative stage classes. Elasticity analysis indicated that a large proportion of population growth could be ascribed to the stasis of individuals in the largest vegetative stage class. Life-table response experiment (LTRE) analyses showed also that progression to larger stage classes was important in explaining the between-population variation in λ. The expected time to extinction was on the order of several decades for the study populations. Seed-sowing indicated that seedling establishment was limited by both seed and micro-site availability. The populations of C. carvi seem to be able to persist for a rather long time in moderately grazed semi-natural grasslands, even in cases where populations are destined to become extinct. The results, thus, indicate that “biennials” are able to maintain remnant populations in managed semi-natural grasslands.     

Enhanced <Emphasis Type="SmallCaps">l</Emphasis>-phenylalanine production by recombinant <Emphasis Type="Italic">Escherichia coli</Emphasis> BR-42 (pAP-B03) resistant to bacteriophage BP-1 via a two-stage feeding approach

The l-phenylalanine (l-Phe) production by Escherichia coli WSH-Z06 (pAP-B03) was frequently prevented by bacteriophage BP-1 infestation. To cope with the bacteriophage BP-1 problem for an improved l-Phe production, one bacteriophage BP-1-resistant mutant, E. coli BR-42, was obtained from 416 mutant colonies of E. coli WSH-Z06 after N-methyl-N’-nitro-N-nitrosoguanidine (NTG) mutagenesis by selection for resistance to bacteriophage BP-1. The recombinant E. coli BR-42-carrying plasmid pAP-B03 had a high capacity in l-Phe production and a remarkable tolerance to 1 × 10¹⁰ pfu (plaque-forming unit)/ml bacteriophage stock. For an enhanced l-Phe production by E. coli BR-42 (pAP-B03), the effects of different feeding strategies including pH–stat, constant rate feeding, linear decreasing rate feeding, and exponential feeding on l-Phe production were investigated; and a two-stage feeding strategy, namely exponential feeding at μ _set = 0.18 h⁻¹ in the first 20 h and a following linear varying rate feeding with F = (−0.55 × t + 18.6) ml/h, was developed to improve l-Phe production. With this two-stage feeding approach, a maximum l-Phe titer of 57.63 g/l with a high l-Phe productivity (1.15 g/l/h) was achieved, which was 15% higher than the highest level (50 g/l) reported so far according to our knowledge. The recombinant E. coli BR-42 (pAP-B03) is a potential l-Phe over-producer in substantial prevention of bacteriophage BP-1 infestation compared to its parent strain WSH-Z06 (pAP-B03).     

HIV spread in the San Francisco cohort: Scaling of the effective logistic rate for seropositivity

A simple scaling (semigroup) property is manifest in the functional form of the effective logistic rate for the increase in the HIV seropositive fraction in the San Francisco (City Clinic) cohort. Witht _i=4.5 years, this scaling property—r→λ^-2r undert→[λt+(λ−1)t _i] for all parameter values λ≧1—encapsulates the effects of relevant biological and sociological changes in the key epidemiological variables during the 8-year seropositive rise period, 1978–1985 inclusive.     

Stability of the analytical solution of Penna model of biological aging

There are some analytical solutions of the Penna model of biological aging; here, we discuss the approach by Coe et al. (Phys. Rev. Lett. 89, 288103, 2002), based on the concept of self-consistent solution of a master equation representing the Penna model. The equation describes transition of the population distribution at time t to next time step (t + 1). For the steady state, the population n(a, l, t) at age a and for given genome length l becomes time-independent. In this paper we discuss the stability of the analytical solution at various ranges of the model parameters—the birth rate b or mutation rate m. The map for the transition from n(a, l, t) to the next time step population distribution n(a + 1, l, t + 1) is constructed. Then the fix point (the steady state solution) brings recovery of Coe et al. results. From the analysis of the stability matrix, the Lyapunov coefficients, indicative of the stability of the solutions, are extracted. The results lead to phase diagram of the stable solutions in the space of model parameters (b, m, h), where h is the hunt rate. With increasing birth rate b, we observe critical b ₀ below which population is extinct, followed by non-zero stable single solution. Further increase in b leads to typical series of bifurcations with the cycle doubling until the chaos is reached at some b _c. Limiting cases such as those leading to the logistic model are also discussed.     

Sublethal effects of fenpyroximate and pyridaben on two predatory mite species, <Emphasis Type="Italic">Neoseiulus womersleyi</Emphasis> and <Emphasis Type="Italic">Phytoseiulus persimilis</Emphasis> (Acari,Phytoseiidae)

Laboratory bioassays were conducted to evaluate the sublethal effects of fenpyroximate and pyridaben on life-table parameters of two predatory mites species, Neoseiulus (= Amblyseius) womersleyi and Phytoseiulus persimilis. In these assays, young adult females were treated with three sublethal concentrations of each acaricide. The life-table parameters were calculated at each acaricide concentration, and compared using bootstrap procedures. For each acaricide, the LC₅₀ estimates for both species were similar, yet the two species exhibited completely different susceptibility when the population growth rate was used as the endpoint. Exposure to both acaricides reduced the net reproduction rate (R _o ) in a concentration-dependent manner and their EC₅₀s were equivalent to less than LC₇. Two different scales of population-level endpoints were estimated to compare the total effect between the species and treatments: the first endpoint values were based on the net reproductive rate (fecundity λ) and the second endpoint values incorporated the mean egg hatchability into the net reproductive rate (vitality λ). The fecundity λ decreased in a concentration-dependent manner for both acaricide treatments, but the vitality λ decreased abruptly after treatment of N. womersleyi with pyridaben. The change in the patterns of λ revealed that the acaricide effects at the population level strongly depended on the life-history characteristics of the predatory mite species and the chemical mode of action. When the total effects of the two acaricides on N. womersleyi and P. persimilis were considered, fenpyroximate was found to be the most compatible acaricide for the augmentative release of N. womersleyi after treatment.     

A fitness function for optimal life history in relation to density effects,competition, predation and stability of the environment

A fitness function (function maximized under natural selection) is studied in a population model in which the growth of a population is suppressed by crowding, density-independent continuous mortality (by euryphagous predators) and periodic disturbances. The dynamics of the population density between occurrence of disturbance can be expressed as,dN/dt=(F(N/K)−D)N, whereN is the population density,K is the carrying capacity,D is the density-independent continuous mortality, andF is the growth regulation factor described as a function of crowding (N/K). The period of disturbance isS. The survival rate under disturbance isu. It is concluded that the fitness function is (approximately) a product of competitive ability (C), carrying capacity, and degree of saturation, and is given byCKF ⁻¹(D−(lnu)/S). The degree of saturation is the inverse function of regulation factor (F) at the death rate due to predators and disturbance. I assume a population in which density is regulated only through survival. In this case, a low survival rate at the critical age-group means a high value ofCKF ⁻¹(D−(lnu)/S). Therefore, the reciprocal of the density-dependent survival rate at critical age-group is a measure of the fitness function. Using this measure, I predict the optimal age (body size) at first reproduction of a species of salamander. I also found that fitness calculated from observed values ofl(x) andm(x) includes a tautology. When the concept of fitness function is compared with the ESS method, the latter is more flexible. However, there is a possibility that an ESS is at the minimum of fitness function.     

Geographical variation in the influence of density dependence and climate on the recruitment of Norwegian moose

The effects of variation in climate on population dynamics are likely to differ within the distributional range of a species, yet the consequences of such regional variation on demography and population dynamics are rarely considered. Here we examine how density dependence and different climate variables affect spatio-temporal variation in recruitment rates of Norwegian moose using data collected over a large geographical area during the hunting season. After accounting for observation error by a Bayesian Markov chain Monte Carlo technique, temporal variation in recruitment rates was relatively independent of fluctuations in local population size. In fact, a positive relationship was as common as a density-dependent decrease in fecundity rates. In general, high recruitment rates were found during autumn 1 year after years with a warm February, and after a warm May or cold June in year t − 1 or in year t. Large regional variation was also found in the effects of some of the weather variables, especially during spring. These patterns demonstrate both direct and delayed effects of weather on the recruitment of moose that possibly operate through an effect of body mass on the proportion of the females that sexually mature as 1.5 or 2.5 years old.     

Detecting and managing an overgrazing-drought synergism in the threatened <Emphasis Type="Italic">Echeveria longissima</Emphasis> (Crassulaceae): the role of retrospective demographic analysis

Echeveria longissima, a threatened herb whose habitat has been severely overgrazed and eroded, was studied for three years in a currently grazed and a fenced area. Matrix population models were used to assess if livestock elimination provides a proper management strategy. The merits of retrospective perturbation analyses in terms of management planning have been debated. Nevertheless, they may prove useful when applied in combination with exclosures because they may detect the effects of anthropogenic disturbance on population dynamics. Thus, the results of retrospective and prospective methods were compared. A rapid decrease in population size was projected in both areas, even though it was faster in the exposed one. The demographic processes that were favourable or detrimental in a given year were magnified outside of the fence, but buffered in the exclosure, showing a strong drought-disturbance synergism. Thus, the largest difference in the population growth rate λ between areas was observed in the driest year. Higher nurse-plant density inside the fence seems to alleviate drought effects. The use of prospective analysis alone may lead to erroneous management decisions, since the highest elasticities corresponded to transitions that were favoured by human activities. While allowing for an increased λ in the short term, intervention aimed at increasing these transitions further without attending others that are lessened by disturbance may introduce large changes in the population dynamics, with negative long-term consequences. Retrospective methods can detect which processes have been altered by disturbance and its synergisms, so we may more efficiently restore healthy population dynamics.     

Scattering spectrum properties and their relationship to biogeochemical parameters: a case study in Taihu Lake

The scattering spectrum properties of highly turbid and eutrophic inland case 2 water from Taihu Lake were studied during three cruises from 2006 to 2007. The scattering [b _p(λ)] and backscattering [b _bp(λ)] coefficients and the backscattering probability (B) for Taihu Lake were found to show a clear spectral dependence, and this dependence was well simulated by a power-law function. This dependence, however, became weak when algae dominated the sample points. The mean values of the power-law index for b _p(λ), v, in Oct 2006, Mar 2007 and Nov 2007 were −0.6712, −0.8129 and −0.7600, respectively. To interpret the spectral characteristics and mechanisms of b _p(λ) and b _bp(λ), water samples were collected simultaneously for the biogeochemical characterization of suspended particles. The average values of the specific scattering coefficients for total suspended matter, inorganic suspended matter (ISPM) and organic suspended matter (OSPM) were 0.634 (550 nm), 1.057 (532 nm), and 0.396 g m⁻² (532 nm), respectively. The power-law index of b _bp(λ) (Y) was significantly related to ISPM/OSPM and b _bp(532 nm), but only weakly related to the particle size distribution index. The mean (spatial and wavelength) values of B in Oct 2006, Mar 2007, and Nov 2007 were 0.0108, 0.0138, and 0.0125, respectively. B decreases with increasing ISPM concentration because of the large contribution of ISPM to b _b(λ) and the strong restraint on b _bp(λ) caused by the multi-scattering effect under high-turbidity conditions.