Effect of the population heterogeneity on growth behavior and its estimation

Different types of the Logistic model are constructed based on a simple assumption that the microbial populations are all composed of homogeneous members and consequently, the condition of design for the initial value of these models has to be rather limited in the case of N(t ₀)=N ₀. Therefore, these models cannot distinguish the dynamic behavior of the populations possessing the same N ₀ from heterogeneous phases. In fact, only a certain ratio of the cells in a population is dividing at any moment during growth progress, termed as θ, and thus, dN / dt not only depends on N, but also on θ. So θ is a necessary element for the condition design of the initial value. Unfortunately, this idea has long been neglected in widely used growth models. However, combining together the two factors (N ₀ and θ) into the initial value often leads to the complexity in the mathematical solution. This difficulty can be overcome by using instantaneous rates (V _inst) to express growth progress. Previous studies in our laboratory suggested that the V _inst curve of the bacterial populations all showed a Guassian function shape and thus, the different growth phases can be reasonably distinguished. In the present study, the Gaussian distribution function was transformed approximately into an analytical form ( Y_i = ae^{[ - 0.5( \fracx_i - x₀ b )2 ]} Y_i = \alpha e^{\left[ { - 0.5\left( {\frac{{x_i - x_0 }}{b}} \right)^2 } \right]} ) that can be conveniently used to evaluate the growth parameters and in this way the intrinsic growth behavior of a bacterial species growing in heterogeneous phases can be estimated. In addition, a new method has been proposed, in this case, the lag period and the double time for a bacterial population can also be reasonably evaluated. This approach proposed could thus be expected to reveal important insight of bacterial population growth. Some aspects in modeling population growth are also discussed.     

Effect of physiological heterogeneity of E. coli popula-tion on antibiotic susceptivity test

According to the instantaneous growth rate (dN/dt) of E. coli CVCC249 growing in batch culture, the entire growth progress was distinguished into four phases: accelerating growth phase, constant growth phase, decelerating growth phase and declining phase, in each of which obvious variation in physiological and biochemical properties was detected, including total DNA, total protein, and MTT-dehydrogenase activity, etc., that led to difference in their antibiotic susceptivity. Antibiotic susceptivity of the population sampled from each phase was tested by Concentration-killing Curve (CKC) approach following the formula N=N ₀/{1+exp[r·(x-BC ₅₀)]}, showing as normal distribution at the individual cell level for an internal population, in which the median bactericidal concentration BC ₅₀ represents the mean level of susceptivity, while the bactericidal span BC ₁₋₉₉=(2lnN ₀)/r indicates the variation degree of the antibiotic susceptivity. Furthermore, tested by CKC approach, the antibiotic susceptivity of E. coli CVCC249 population in each physiological phase to gentamicin or enoxacin was various: susceptivity of the population in the constant growth phase and declining phase all increased compared with that in the accelerating growth phase for gentamicin but declined for enoxacin. The primary investigations revealed that the physiological phase should be taken into account in the context of antibiotic susceptivity and research into antimicrobial mechanism. However there are few reports concerned with this study. Further research using different kinds of antibiotics with synchronized continuous culture of different bacterial strains is required. Supported by the Natural Science Foundation of Shandong Province, China (Grant No. Y2005C58), the Natural Key Technology R&D Program of China (Grant No. 2006BAK02A03-6) and the Youth Scientific Research Foundation of Shandong Academy of Agricultural Science (2005YQ035)     

Stochastic models of tumor growth and the probability of elimination by cytotoxic cells

The probability of tumor extinction due to the action of cytotoxic cell populations is investigated by several one dimensional stochastic models of the population growth and elimination processes of a tumor. The several models are made necessary by the nonlinearity of the processes and the different parameter ranges explored. The deterministic form of the model is where γ₀, k′₆ and k ₁ are positive constants. The parameter of most import is which determines the stability of the T = 0 equilibrium. With an initial tumor size of one, a (linear) branching process is used to estimate the extinction probability. However, in the case λ = 0 when the linearization of the deterministic model gives no information (T = 0 is actually unstable) the branching model is unsatisfactory. This makes necessary the utilization of a density-dependent branching process to approximate the population. Through scaling a diffusion limit is reached which enables one to again compute the probability of extinction. For populations away from one a sequence of density-dependent jump Markov processes are approximated by a sequence of diffusion processes. In limiting cases, the estimates of extinction correspond to that computed from the original branching process. Table 1 summarizes the results.     

Effect of physiological heterogeneity of <Emphasis Type="Italic">E. coli</Emphasis> population on antibiotic susceptivity test

According to the instantaneous growth rate (dN/dt) of E. coli CVCC249 growing in batch culture, the entire growth progress was distinguished into four phases: accelerating growth phase, constant growth phase, decelerating growth phase and declining phase, in each of which obvious variation in physiological and biochemical properties was detected, including total DNA, total protein, and MTT-dehydrogenase activity, etc., that led to difference in their antibiotic susceptivity. Antibiotic susceptivity of the population sampled from each phase was tested by Concentration-killing Curve (CKC) approach following the formula N=N ₀/{1+exp[r·(x-BC ₅₀)]}, showing as normal distribution at the individual cell level for an internal population, in which the median bactericidal concentration BC ₅₀ represents the mean level of susceptivity, while the bactericidal span BC ₁₋₉₉=(2lnN ₀)/r indicates the variation degree of the antibiotic susceptivity. Furthermore, tested by CKC approach, the antibiotic susceptivity of E. coli CVCC249 population in each physiological phase to gentamicin or enoxacin was various: susceptivity of the population in the constant growth phase and declining phase all increased compared with that in the accelerating growth phase for gentamicin but declined for enoxacin. The primary investigations revealed that the physiological phase should be taken into account in the context of antibiotic susceptivity and research into antimicrobial mechanism. However there are few reports concerned with this study. Further research using different kinds of antibiotics with synchronized continuous culture of different bacterial strains is required.     

Identification of conservation units in the European Mergus merganser based on nuclear and mitochondrial DNA markers

The conservation status of small breeding areas of the Goosander (Mergus merganser merganser) in Central Europe is unclear. Geographic isolation of these areas suggests restricted gene flow to and from large North-European populations. On the other hand, migrating Goosanders from northern Europe join the Central European breeding population for wintering. To evaluate the conservation status of the small breeding areas we assessed the genetic structure of M. merganser populations in Europe by examining two nuclear marker systems (microsatellites and Single Nucleotide Polymorphisms, SNP) and mitochondrial (mtDNA) control region sequence variation for Goosanders in 11 sampling areas representing three of five distinct breeding areas and two subspecies (M. m. merganser and M. m. americanus). Overall population differentiation estimates including both subspecies were high, both based on mtDNA () and nuclear markers (θ _ST = 0.219; 95% CI 0.088–0.398, SNP and microsatellites combined). Within Europe, mtDNA revealed a strong overall () and significant pairwise population differentiation between almost all comparisons. In contrast, both nuclear marker systems combined revealed only a small overall genetic differentiation (θ _ST = 0.022; 95% CI 0.003–0.041). The strong genetic differentiation based on female-inherited mtDNA but not on biparentally inherited nuclear markers can be explained by sex-biased dispersal and strong female philopatry. Therefore, small breeding areas in Europe are endangered despite large male-mediated gene-flow, because when these populations decline, only males—but due to strong philopatry not females—can be efficiently supplemented by migration from the large North European populations. We therefore propose to manage the small breeding areas independently and to strengthen conservation efforts for this species in Central Europe.     

Growth parameters and total mortality in Oreochromis niloticus (Linnaeus) from Nyanza Gulf,Lake Victoria

Length-frequency data collected from fish landings in the Kenya waters of Lake Victoria were used to estimate the growth parameters, total mortality rate and growth performance index in Oreochromis niloticus. The asymptotic length, (L _∞) and the ratio of the total mortality rate (Z) to the growth constant (K), were estimated to be 64.6 cm and 3.219 respectively. K was 0.254 y^-1, Z was 0.818 y^-1 and the growth performance index θ′ = Log₁₀ K + 2 log₁₀ L∞ = 3.025, which is rather high as compared to other tilapia populations in natural waters.     

Wild genetic diversity preservation in a small-sized first generation breeding population of <Emphasis Type="Italic">Allanblackia floribunda</Emphasis> (Clusiaceae)

Founder group size is of prime importance in tree breeding programs. We determined whether sampling 20-plus trees for breeding in Allanblackia floribunda, a tropical forest tree species that has been recently enrolled in tree improvement program for fruit and seed production, would affect neutral genetic diversity and inbreeding level in both breeding and production populations. Using eight informative microsatellite loci, we: (a) assessed the nuclear genetic diversity of ten natural populations, and of the breeding population in the humid forest zone of Cameroon; (b) investigated temporal effective-size fluctuations in A. floribunda natural populations, with a view to identifying the role of past demographic events in the genetic structure of the studied species; and (c) tested the hypothesis that genetic diversity in a founder group of 20 individuals is not different from that existing in the wild. The eight loci were variable. High levels of genetic diversity (A = 4.96; H _E = 0.59) and moderate differentiation (R _ST = 0.061) were found within and among populations in wild stands. High genetic distances existed between populations ( \textaverage chord distance = 0.\text2769 ±0.00\text554 ) \left( {{\text{average chord distance}} = 0.{\text{2769}} \pm 0.00{\text{554}}} \right) . Eight of the ten surveyed populations showed signs of deviation from mutation-drift equilibrium, suggesting Pleistocene population bottlenecks and fluctuations in effective population size. Mantel tests did not reveal any relationships between genetic and geographic distances. A neighbor-joining dendrogram showed a population structure that could be explained by historical factors. The hypothesis tested has been accepted. However, a slight increase in inbreeding was observed in the breeding population.     

Nitrogen use efficiency of sugarcane in relation to its BNF potential and population of endophytic diazotrophs at different N levels

The nitrogen fixing bacterial endophytes Gluconacetobacter diazotrophicus and Herbaspirillum spp. have been proposed to benefit sugarcane (Saccaharum spp. hybrids) growth. Variable populations of these endophytes exist depending upon ontogenic and climatic variations as well. This study investigates the effect of variable chemical nitrogen application in soil on the population of endophytic diazotrophs, acetylene reduction ability of excised roots, plant N-nutrient use efficiency and probable interactions among different parameters in eight commercial sugarcane varieties of subtropical India. Recovery efficiency (RE), agronomic efficiency (AE), partial factor productivity (PFP) and physiologic efficiency (PE) indicators were used for accounting N-nutrient use efficiency. The population of G. diazotrophicus was more at N₇₅ compared to N₀ and N₁₅₀, whereas Herbaspirillum population increased from N₀ to N₁₅₀. ARA was positively correlated with Gluconacetobacter population in rhizosphere and root, whereas it had poor correlation with Herbaspirillum population. Positive correlation of RE and AE with ARA of roots, Gluconacetobacter and Herbaspirillum populations in roots and stems indicate their positive contribution in total nitrogen uptake by the plant per kg of N applied. Average PFP was 808.9 at N₇₅ compared to 408.7 at N₁₅₀ indicating that N was utilized efficiently at low N input status in sugarcane. Strong positive correlations of AE₇₅ (agronomic efficiency from 75 kg N ha⁻¹ to 150 kg N ha⁻¹) with N-uptake (r ² = 0.615), cane yield (r ² = 0.758) and PFP (r ² = 0.758) and other parameters compared to AE (agronomic efficiency from 0 kg N ha⁻¹ to 75 kg N ha⁻¹ or 150 kg N ha⁻¹) correlations with N-uptake (r ² = 0.111), cane yield (r ² = 0.368) and PFP (r ² = 0.190) indicated that the AE of sugarcane was strongly directed towards producing more cane yield per unit of N fertilizer once the sugarcane plant has established using initial dose of nitrogen and thus AE₇₅ seems to be a more appropriate indicator for accounting N-nutrient use efficiency in sugarcane.     

Population genetic structure of wild-growing ginseng (<Emphasis Type="Italic">Planax ginseng</Emphasis> C.A. Meyer) assessed using AFLP markers

Genetic variability in ten populations of wild-growing ginseng was assessed using AFLP markers with the application of fragment analysis on a genetic analyzer. The variation indices were high in the populations (P = 55.68%, H _S = 0.1891) and for the species (P = 99.65%; H _S = 0.2857). Considerable and statistically significant population differentiation was demonstrated (θ_B = 0.363; Bayesian approach, “full model”; F _ST = 0.36, AMOVA). The results of AMOVA and Bayesian analysis indicate that 64.46% of variability is found within the populations. Mantel test showed no correlation between the genetic and geographic distances among the populations (r = −0.174; p = 0.817). Hierarchical AMOVA and analysis of genetic relationships based on Euclidean distances (NJ, PCoA, and MST) identified two divergent population groups of ginseng. Low gene flow between these groups (N _m = 0.4) suggests their demographic independence. In accordance to the concept of evolutionary significant units (ESU), these population groups, in terms of the strategy and tactics for conservation and management of natural resources, should be treated as management units (MUs). The MS tree topology suggests recolonization of southern Sikhote-Alin by ginseng along two directions, from south and west.     

Importance of adult survival, local recruitment and immigration in a declining boreal forest passerine, the willow tit Parus montanus

Population growth rate (λ) and its components (adult survival, local recruitment, immigration and their relative contributions to λ) were studied in the declining willow tit Parus montanus in Northern Finland. Capture–recapture models for open populations were used to estimate the population parameters and their process variation. Adult survival was fairly high with low variation (0.593, CV=0.067). As expected, local recruitment was lower and more variable (0.063, CV=0.610). During the 12-year study, the population growth rate averaged to one (0.988, CV=0.197; calculated as However, if the present processes continue, population projections show that the population is likely to decline. There was considerable temporal variation in the relative contributions of demographic parameters to λ. In all years, adult survival had the highest relative contribution (mean 64%) to the population growth rate and it was the least variable trait. Immigration had a higher relative contribution (22%) to λ than local recruitment (14%). Based on the results for the contributions to λ, the main conservation concern for willow tits is adult survival. Due to low variation, adult survival may be difficult to enhance, but at least it should be prevented from declining. High stochasticity in local recruitment and immigration is probably an inherent characteristic of highly seasonal environments, making these traits difficult to address for conservation practices.     

Effect of mutation on helper T-cells and viral population: A computer simulation model for HIV

Summary A Monte Carlo simulation is proposed to study the dynamics of helper T-cells (N _H) and viral (N _V) populations in an immune response model relevant to HIV. Cellular states are binary variables and the interactions are described by logical expressions. Viral population shows a nonmonotonic growth before reaching a constant value while helper T-cells grow to a constant after a relaxation/reaction time. Initially, the population of helper cells grows with time with a power-law, N _H ∼t ^β, before reaching the steady-state; the growth exponent β increases systematically (β ≈ 1 – 2) with the mutation rate (P _mut≈0.1–0.4). The critical recovery time (t _c) increases exponentially with the viral mutation, t _c≈Ae ^αP _mut , with α=4.52±0.29 in low mutation regime and α=15.21±1.41 in high mutation regime. The equilibrium population of helper T-cell declines slowly with P _mut and collapses at ∼ 0.40; the viral population exhibits a reverse trend, i.e., a slow increase before the burst around the same mutation regime.     

Bacterial growth and substrate degradation by BTX-oxidizing culture in response to salt stress

Interactions between microbial growth and substrate degradation are important in determining the performance of trickle-bed bioreactors (TBB), especially when salt is added to reduce biomass formation in order to alleviate media clogging. This study was aimed at quantifying salinity effects on bacterial growth and substrate degradation, and at acquiring kinetic information in order to improve the design and operation of TBB. Experiment works began by cultivating a mixed culture in a chemostat reactor receiving artificial influent containing a mixture of benzene, toluene, and xylene (BTX), followed by using the enrichment culture to degrade the individual BTX substrates under a particular salinity, which ranged 0–50 g l⁻¹ in batch mode. Then, the measured concentrations of biomass and residual substrate versus time were analyzed with the microbial kinetics; moreover, the obtained microbial kinetic constants under various salinities were modeled using noncompetitive inhibition kinetics. For the three substrates the observed bacterial yields appeared to be decreased from 0.51–0.74 to 0.20–0.22 mg mg⁻¹ and the maximum specific rate of substrate utilization, declined from 0.25–0.42 to 0.07–0.11 h⁻¹, as the salinity increased from 0 to 50 NaCl g l⁻¹. The NaCl acted as noncompetitive inhibitor, where the modeling inhibitions of the coefficients, K _T(S), were 22.7–29.7 g l⁻¹ for substrate degradation and K _T(μ), 13.0–19.0 g l⁻¹, for biomass formation. The calculated ratios for the bacterial maintenance rate, m _S, to further indicated that the percentage energy spent on maintenance increased from 19–24 to 86–91% as salinity level increased from 0 to 50 g l⁻¹. These results revealed that the bacterial growth was more inhibited than substrate degradation by the BTX oxidizers under the tested salinity levels. The findings from this study demonstrate the potential of applying NaCl salt to control excessive biomass formation in biotrickling filters.     

Photosynthesis Inhibition During Gas Exchange Oscillations in ABA-Treated Helianthus annuus: Relative Role of Stomatal Patchiness and Leaf Carboxylation Capacity

Environmental factors that induce spatial heterogeneity of stomatal conductance, g _s, called stomatal patchiness, also reduce the photochemical capacity of CO₂ fixation, yet current methods cannot distinguish between the relative effect of stomatal patchiness and biochemical limitations on photosynthetic capacity. We evaluate effects of stomatal patchiness and the biochemical capacity of CO₂ fixation on the sensitivity of net photosynthetic rate (P _N) to stomatal conductance (g _s), θ (θ = δP _N/g _s). A qualitative model shows that stomatal patchiness increases the sensitivity θ while reduced biochemical capacity of CO₂ fixation lowers θ. We used this feature to distinguish between stomatal patchiness and mesophyll impairments in the photochemistry of CO₂ fixation. We compared gas exchange of sunflower (Helianthus annuus L.) plants grown in a growth chamber and fed abscisic acid, ABA (10⁻⁵ M), for 10 d with control plants (-ABA). P _N and g _s oscillated more frequently in ABA-treated than in control plants when the leaves were placed into the leaf chamber and exposed to a dry atmosphere. When compared with the initial CO₂ response measured at the beginning of the treatment (day zero), both ABA and control leaves showed reduced P _N at particular sub-stomatal CO₂ concentration (c _i) during the oscillations. A lower reduction of P _N at particular g _s indicated overestimation of c _i due to stomatal patchiness and/or omitted cuticular conductance, g _c. The initial period of damp oscillation was characterised by inhibition of chloroplast processes while stomatal patchiness prevailed at the steady state of gas exchange. The sensitivity θ remained at the original pre-treatment values at high g _s in both ABA and control plants. At low g _s, θ decreased in ABA-treated plants indicating an ABA-induced impairment of chloroplast processes. In control plants, g _c neglected in the calculation of g _s was the likely reason for apparent depression of photosynthesis at low g _s. This revised version was published online in August 2006 with corrections to the Cover Date.     

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     

Estimation of population structure in coastal Douglas-fir [Pseudotsuga menziesii (Mirb.) Franco var. menziesii] using allozyme and microsatellite markers

Characterizing population structure using neutral markers is an important first step in association genetic studies in order to avoid false associations between phenotypes and genotypes that may arise from non-selective demographic factors. Population structure was studied in a wide sample of ∼1,300 coastal Douglas-fir [Pseudotsuga menziesii (Mirb.) Franco var. menziesii] trees from Washington and Oregon. This sample is being used for association mapping between cold hardiness and phenology phenotypes and single-nucleotide polymorphisms in adaptive-trait candidate genes. All trees were genotyped for 25 allozyme and six simple sequence repeat (SSR) markers using individual megagametophytes. Population structure analysis was done separately for allozyme and SSR markers, as well as for both datasets combined. The parameter of genetic differentiation (θ or F _ST) was standardized to take into account high within-population variation in the SSR loci and to allow comparison with allozyme loci. Genetic distance between populations was positively and significantly correlated with geographic distance, and weak but significant clinal variation was found for a few alleles. Although the STRUCTURE simulation analysis inferred the same number of populations as used in this study and as based on previous analysis of quantitative adaptive trait variation, clustering among populations was not significant. In general, results indicated weak differentiation among populations for both allozyme and SSR loci (θ _s = 0.006–0.059). The lack of pronounced population subdivision in the studied area should facilitate association mapping in this experimental population, but we recommend taking the STRUCTURE analysis and population assignments for individual trees (Q-matrix) into account in association mapping.     

Population genetic structure of the endangered and endemic medicinal plant <Emphasis Type="Italic">Commiphora</Emphasis><Emphasis Type="Italic">wightii</Emphasis>

Commiphora wightii is a medicinally important endangered species endemic to the Thar Desert of Rajasthan, India and adjoining areas of Pakistan. The populations of this species are declining sharply because of its extensive use as a natural herb. Random amplified polymorphic DNA analysis was conducted to find the genetic variation among 7 populations of C. wightii. Of the 100 random primers screened, 44 primers yielded 220 loci. Statistical analysis indicated low genetic diversity (H _pop = 0.0958; I = 0.1498; mean polymorphic loci = 14.28%), and high genetic differentiation among the populations (G _ST = 0.3990; AMOVA Φ _ST of 0.3390; Bayesian θ ^(II) = 0.3002). The low genetic diversity may be due to geographic isolation and restricted gene flow (N _m = 0.7533) between the fragmented populations. Unsustainable utilization of the plant has fragmented the population continuum which served the purpose of genetic exchange between populations. Mantel’s test was performed which revealed a highly significant positive correlation between genetic and geographic distance (r ² = 0.614, P = 0.023) among the populations studied. Low variation can also be attributed to poor seed setting and the slow growth pattern of the species, which is also an apomict. In UPGMA dendrogram the Commiphora wightii samples were divided into two major and one minor cluster. These findings can serve as a guide to preserving the genetic resources of this medicinal plant species.     

Small effective population sizes of two remnant ocelot populations (<Emphasis Type="Italic">Leopardus pardalis albescens</Emphasis>) in the United States

Threatened populations are vulnerable to the effects of genetic drift and inbreeding, particularly when gene flow is low and the effective population size is small. Estimates of effective population size (N _e ) provide important information on the status of endangered populations that have experienced severe fragmentation and serve as indicators of genetic viability. Genetic data from microsatellite loci were used to estimate N _e for the 2 remaining populations of the endangered ocelot (Leopardus pardalis albescens) occurring in the United States. Several methods were used to calculate N _e , resulting in estimates ranging from N _e  = 8.0 (95% CI: 3.2–23.1) to 13.9 (95% CI: 7.7–25.1) for the population located at the Laguna Atascosa Wildlife Refuge in Cameron County, Texas. The ocelot population in Willacy County, Texas, had N _e estimates of 2.9 (95% CI: 1.7–5.6) and 3.1 (95% CI: 1.9–13.5), respectively. Estimates of N _e in both populations were below the critical value recommended for short-term viability.     

Studies in imitative behavior: A generalization of the rashevsky model; Its mathematical properties

This paper is based on N. Rashevsky's theory of imitative behavior, the underlying idea being that performance of one reaction by a given individual produces an increased stimulation (or tendency) toward the same reaction in other individuals. For simplicity, consideration is limited to cases in which each individual may choose only between two (or two main categories of) reactions, denoted byA andB in the following. However, upon suggestion from Dr. Rashevsky and certainly in better agreement with actual facts, the strength of imitative interaction is assumed to vary from individual to individual. More precisely, if Ψ_i denotes the additional excitation caused by imitation in theith individual,PA_i the probability for performance of reactionA, andPB_i the probability for performance of reactionB by theith individual, we postulate that where the constants α _ik ^′ and β _ik ^′ (coefficients of imitative interaction) measure the amount of imitative influence exerted by thekth individual upon theith,N being the total number of individuals in the population. The term — α_i Ψ_i accounts for the spontaneous decay of excitation, and the quantities α _ik ^′ and α _ik ^′ are assumed to benon-negative. The expressions forPA_i andPB_i are obtained from H. D. Landahl's theory of conflicting stimuli; they depend non-linearly on the values Ψ_i. It is implicit in this formulation that the theory can only be applied if the frequency of contacts between individuals is not too small. Some further shortcomings and limitations of the model are outlined, and the discussion includes suggestions for reinterpretation and improvement of the theory. If all the quantities α _ik ^′ and α _ik ^′ have the same value, sayA, we return to the case treated by Rashevsky (and Landau, 1950); these authors, however, replace the sums in the equation above by integrals, which automatically restricts the validity of their results to very large values ofN. Their work may therefore be characterized by the assumption of uniform interaction in large populations. Our equations, on the other hand, are applicable even to very small groups, and therein lies one of their main advantages. In this paper the mathematical properties of the non-linear system of equations above are studied with particular reference to the existence and stability of steady states [dΨ_i/dt ≡ 0;, i = 1 , 2, . . . N]. A sufficient condition for the existence of only one stable steady state is derived. It may be formulated roughly by stating that all the coefficients of interaction should be sufficiently small. It that is not the case, there may exist a greater number of stationary states. In particular, two of them (called “extremal”) have the following properties: they arestable and such that the average number of individuals in the group performing one or the other reaction is the largest (or smallest) possible as compared with the other steady states. Hence the situation is qualitatively similar to that found by Rashevsky and Landau.Quantitatively, however, important differences may arise, depending on the nature of the matrix specifying the interaction. A stable state may be approached through damped oscillations, but this effect is important only if the damping is sufficiently small for the oscillations to become practically observable. Little information could be obtained on this point, due to mathematical difficulties. As mentioned above, the most interesting applications of this theory will be with respect to small populations or to populations partitioned into subgroups with varying amounts of imitative interaction within as well as between groups.     

Demography of the cycad Ceratozamia mirandae (Zamiaceae) under disturbed and undisturbed conditions in a biosphere reserve of Mexico

The cycad Ceratozamia mirandae is endemic to Chiapas, Mexico. Demographic studies were made in two of its populations in the Sepultura Biosphere Reserve under different conservation conditions; in the nucleus zone “Tres Picos” (conserved) and buffer zone “La Sombra” (disturbed and under management). Spatial distribution of C. mirandae was aggregated, showed a clumped local distribution on shallow soils on steep slopes and male and female cones appear to be synchronous in both populations. The population structure was of type I (Bongers) for both sites. Individuals between the sites showed differences in growth pattern. The oldest plants (80–90 cm tall) were estimated to be about 490 years at “La Sombra”. The finite growth rate () in the buffer zone population showed a tendency for decrease whilst in the nucleus zone this estimate remained stable. The highest elasticity values lied in the transition of the first three classes of the “La Sombra” population, in “Tres Picos” this corresponded to adult plants between 20 and 30 cm tall. Given the above, it is proposed that in the nucleus zone, reproductive adults should be of highest conservation priority, whereas in the buffer zone seedling reintroduction should be carried out regularly until the population increases. We recommend an IUCN Red List category of Vulnerable (VU C, 2a), largely due to difficult-to-control destructive annual forest fires that occur in this Reserve.     

Mapping dominant markers using F2 matings

The development of efficient methods for amplifying random DNA sequences by the polymerase chain reaction has created the basis for mapping virtually unlimited numbers of mixed-phase dominant DNA markers in one population. Although dominant markers can be efficiently mapped using many different kinds of matings, recombination frequencies and locus orders are often mis-estimated from repulsion F₂ matings. The major problem with these matings, apart from excessive sampling errors of recombination frequency () estimates, is the bias of the maximum-likelihood estimator (MLE) of ( _ML). when the observed frequency of double-recessive phenotypes is 0 and the observed frequency of double-dominant phenotypes is less than 2/3 — the bias for those samples is — . We used simulation to estimate the mean bias of _ML. Mean bias is a function of n and and decreases as n increases. Valid maps of dominant markers can be built by using sub-sets of markers linked in coupling, thereby creating male and feamle coupling maps, as long as the maps are fairly dense (about 5 cM) — the sampling errors of increase as increases for coupling linkages and are equal to those for backcross matings when =0. The use of F₂ matings for mapping dominant markers is not necessarily proscribed because they yield twice as many useful markers as a backcross population, albeit in two maps, for the same number of DNA extractions and PCR assays; however, dominant markers can be more effeciently exploited by using doubled-haploid, recombinant-inbred, or other inbred populations.