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.     

Selection of certain dichotomous experiments

Summary The problem of allocating a single observation to one of the two available populations is considered. Suppose that a certain characteristic has densityf in one population, and has densityg in the other. On the basis of the value observed, one must specify which population has densityf. It is assumed that when a wrong population is chosen, a certain known loss is incurred. The problem is to allocate the observation so as to minimize the expected loss. General conditions onf andg are derived to decide which population should be selected for taking the observation. Research was supported in part by the National Science Foundation under grant No. SOC79-06386. Work done while on sabbatical leave at Carnegie-Mellon University.     

A New Approach to Cycling in a 2-locus 2-allele Genetic Model

In the reproductive process new genetic types arise due to crossing over and recombination at the meiotic stage. A simplified biological model will be developed which incorporates this effect and the effect of selection. Although a chromosome may contain thousands of genes we will consider a simplified model consisting of two genetic loci, each containing two alleles of some gene.

The model will be then turned into a difference equation or mapping model x* = G(x,r) where x represents the frequency distribution of genotypes in a certain infinite population, x* is this distribution one generation later and r is the recombination parameter. For a certain choice of fitness and recombination parameters the mapping exhibits several fixed points. As r is varied one of the fixed points of the mapping loses its stability due to a conjugate pair of eigenvalues of the linearized mapping leaving the unit disk. It is shown that the required non-resonance conditions and “nonlinear damping” condition are satisfied and thus the fixed point undergoes a Neimark–Sacker bifurcation to a cycling or oscillatory state.

Once a cycling orbit is established one can conclude that genetic variation (over time) of the population can be maintained. This work reformulates and proves earlier observations of Alan Hastings in a way that makes the treatment of chromosomes with more genetic loci more straightforward.     

Nonlinear Leray-Schauder alternatives and application to nonlinear problem arising in the theory of growing cell population

Motivated by a mathematical model of an age structured proliferating cell population, we state some new variants of Leray-Schauder type fixed point theorems for (ws)-compact operators. Further, we apply our results to establish some new existence and locality principles for nonlinear boundary value problem arising in the theory of growing cell population in L ₁-setting. Besides, a topological structure of the set of solutions is provided.     

A genetic filtering problem ∗

Members of a population of fixed size N can be in any one of n states. In discrete time the individuals jump from one state to another, independently of each other, and with probabilities described by a homogeneous Markov chain. At each time a sample of size M is withdrawn, (with replacement). Based on these observations, and using the techniques of Hidden Markov Models, recursive estimates for the distribution of the population are obtained     

Nonparametric Estimations about Species Not Observed in a Random Sample

A brief survey of work done on two long-standing and important problems in statistics is given. In a simple random sample (with replacement) of size n from a population divided into species, if N distinct species are observed, what is the probability that, on the next trial, a species not observed before is discovered? And what is the total number of species not observed? Interesting in many applied areas, these problems have been discussed in a great number of papers. We survey some of the related publications as well as a Bayes-like estimator recently devised by the authors, together with results on the estimation of the distribution of the probability of discovering a new species. Dedicated to the memory of Bruno Forte, mentor and friend to one of us Lecture held by C.C.A. Sastri in the Seminario Matematico e Fisico on May 12, 2003 Received: June, 2004 An erratum to this article is available at .     

Solutions to the XXX type Bethe ansatz equations and flag varieties

We consider a version of the A _N Bethe equation of XXX type and introduce a reporduction procedure constructing new solutions of this equation from a given one. The set of all solutions obtained from a given one is called a population. We show that a population is isomorphic to the sl _N +1 flag variety and that the populations are in one-to-one correspondence with intersection points of suitable Schubert cycles in a Grassmanian variety. We also obtain similar results for the root systems B _N and C _N . Populations of B _N and C _N type are isomorphic to the flag varieties of C _N and B _N types respectively.     

Regular population monotonic allocation schemes

The paper introduces a refinement of the notion of population monotonic allocation scheme, called regular population monotonic allocation scheme (regularpmas). This refinement is based on economic situations in which players may have to select new partners from a set of potential players and in which there exist certain capacity constraints. A sufficient condition for the existence of a regularpmas is given. For the class of games with regularpmas, we prove that the core coincides with the Davis and Maschler and the Mas-Colell bargaining sets.     

ECOLOGICAL‐ECONOMIC MODELS OF SUSTAINABLE HARVEST FOR AN ENDANGERED BUT EXOTIC MEGAHERBIVORE IN NORTHERN AUSTRALIA

ABSTRACT. How can one manage wildlife under a suite of competing values? In isolation, the ecological economics of native wildlife harvest, threatened species conservation and control of exotic species are all well established sub‐disciplines of wildlife management. However, the wild banteng (Bos javanicus) population of northern Australia represents an interesting combination of these aspirations. A native bovid of Southeast Asia now ‘endangered’ in its native range, banteng were introduced into northern Australia in 1849. Today, a population of 8,000–10,000 resides on one small, isolated peninsula in western Arnhem Land, Northern Territory and is harvested by both recreational (trophy) and aboriginal subsistence hunters. Indigenous, industry and conservationist stakeholders differ in their requirements for population management. Here we analyze the ecological and economic costs/benefits of a series of potential harvest management options for Australia's banteng population, with the aim being either to: (1) maximize sustainable yield (MSY); (2) maximize harvest of trophy males; (3) maximize indigenous off‐take; (4) suppress density or completely eradicate the population; (5) minimize risk of extinction whilst limiting range expansion; (6) scenarios incorporating two or more of options 1–5. The modeling framework employed stochastic, density‐regulated matrix population models with life‐history parameters derived from (i) allometric relationships (for estimating r_max, generation length, fecundity and densities for a banteng‐sized mammal) and (ii) measured vital rates for wild and captive banteng and other Bos spp. For each management option, we present a simple economic analysis that incorporates estimated costs of management implementation and associated profits projected. Results demonstrate that revenue of >Ä$200,000 is possible from meat production and safari hunting without compromising long‐term population stability or the conservation status of this endangered bovid.     

A Preliminary Test in Discriminant Analysis

One of the aims of discriminant analysis is the allocation of unknown entities to populations that are knowna priori. Considerkpopulations. LetXdenote the vector of observations on an experimental unit, whose origin is uncertain. For the general parametric case, a test is proposed to verify the hypothesis thatXis coming from a new population. The case of normal population with equal variance is fully derived.     

Improvement of random coefficient differential models of growth of anaerobic photosynthetic bacteria by combining Bayesian inference and gPC

The time evolution of microorganisms, such as bacteria, is of great interest in biology. In the article by D. Stanescu et al. [Electronic Transactions on Numerical Analysis, 34, 44–58 (2009)], a logistic model was proposed to model the growth of anaerobic photosynthetic bacteria. In the laboratory experiment, actual data for two species of bacteria were considered: Rhodobacter capsulatus and Chlorobium vibrioforme. In this paper, we suggest a new nonlinear model by assuming that the population growth rate is not proportional to the size of the bacteria population, but to the number of interactions between the microorganisms, and by taking into account the beginning of the death phase in the kinetic curve. Stanescu et al. evaluated the effect of randomness into the model coefficients by using generalized polynomial chaos (gPC) expansions, by setting arbitrary distributions without taking into account the likelihood of the data. By contrast, we utilize a Bayesian inverse approach for parameter estimation to obtain reliable posterior distributions for the random input coefficients in both the logistic and our new model. Since our new model does not possess an explicit solution, we use gPC expansions to construct the Bayesian model and to accelerate the Markov chain Monte Carlo algorithm for the Bayesian inference.     

Covariance information based on the <Emphasis Type="Italic">t</Emphasis> prior

The exact distribution of the covariance between two populations is derived. It is assumed that one of the population means has a Student's t prior while the other is taken to come from one of normal, Student's t, Laplace, logistic or Bessel families (the five well-known symmetric distributions). The exact distribution is given in terms of the characteristic function. The calculations involve several special functions.     

Asymptotic distribution of the increase of the largest canonical correlation when one of the vectors is augmented

Let r₁ > r₂ > … be the sample canonical correlations in a sample of size n from a multivariate normal population partitioned into two subvectors with population canonical correlations ₁ > ₂ > …. Let one of the subvectors be augmented by adding one or more variables to it. For the increase in the largest canonical correlation, Δr in the sample and Δ in the population, it is shown that √n(Δr − Δ) → ^DN(0, σ²) and a formula for σ² is derived.     

Efficient Computation of the Joint Sample Frequency Spectra for Multiple Populations

A wide range of studies in population genetics have employed the sample frequency spectrum (SFS), a summary statistic which describes the distribution of mutant alleles at a polymorphic site in a sample of DNA sequences and provides a highly efficient dimensional reduction of large-scale population genomic variation data. Recently, there has been much interest in analyzing the joint SFS data from multiple populations to infer parameters of complex demographic histories, including variable population sizes, population split times, migration rates, admixture proportions, and so on. SFS-based inference methods require accurate computation of the expected SFS under a given demographic model. Although much methodological progress has been made, existing methods suffer from numerical instability and high computational complexity when multiple populations are involved and the sample size is large. In this article, we present new analytic formulas and algorithms that enable accurate, efficient computation of the expected joint SFS for thousands of individuals sampled from hundreds of populations related by a complex demographic model with arbitrary population size histories (including piecewise-exponential growth). Our results are implemented in a new software package called momi (MOran Models for Inference). Through an empirical study, we demonstrate our improvements to numerical stability and computational complexity.     

Three types of mathematical representational translations: Comparing empirical and theoretical results

Previous research has investigated the representational translation practices of high school students, high school teachers, and college preservice teachers in various mathematical contexts including linear functions. Findings from qualitative research has frequently led to new notions about participant work and understanding. Many quantitative research has investigated the degree to which some in these populations correctly perform these translations. However, it seems that only infrequently have empirical research investigated findings from qualitative studies and vice versa, and findings regarding one population are rarely compared with findings of another population. This study (a) empirically explores the frequency of success of preservice teachers (N = 80) regarding representational translations in the context of linear functions, (b) quantifies results from previous qualitative, literature‐based research regarding high school students and teachers, and (c) quantitatively compares the results. This study demonstrates that some mathematical representational translations are more difficult than others.     

GEVA: geometric variability-based approaches for identifying patterns in data

This paper, arising from population studies, develops clustering algorithms for identifying patterns in data. Based on the concept of geometric variability, we have developed one polythetic-divisive and three agglomerative algorithms. The effectiveness of these procedures is shown by relating them to classical clustering algorithms. They are very general since they do not impose constraints on the type of data, so they are applicable to general (economics, ecological, genetics...) studies. Our major contributions include a rigorous formulation for novel clustering algorithms, and the discovery of new relationship between geometric variability and clustering. Finally, these novel procedures give a theoretical frame with an intuitive interpretation to some classical clustering methods to be applied with any type of data, including mixed data. These approaches are illustrated with real data on Drosophila chromosomal inversions.     

Periodic solution of a neutral delay model of single-species population growth on time scales

By using a fixed point theorem of strict-set-contraction, we present some sufficient conditions for the existence of at least one positive ω-periodic solution to a neutral delay model of single-species population growth on time scales. The significance of the present paper is that we address an open problem introduced by Kuang [Kuang Y. Delay differential equations with applications in population dynamics. New York: Academic Press; 1993] on time scales.     

A fast steady-state <Emphasis Type="Italic">ε</Emphasis>-dominance multi-objective evolutionary algorithm

Multi-objective evolutionary algorithms (MOEAs) have become an increasingly popular tool for design and optimization tasks in real-world applications. Most of the popular baseline algorithms are pivoted on the use of Pareto-ranking (that is empirically inefficient) to improve the convergence to the Pareto front of a multi-objective optimization problem. This paper proposes a new ε-dominance MOEA (EDMOEA) which adopts pair-comparison selection and steady-state replacement instead of the Pareto-ranking. The proposed algorithm is an elitist algorithm with a new preservation technique of population diversity based on the ε-dominance relation. It is demonstrated that superior results could be obtained by the EDMOEA compared with other algorithms: NSGA-II, SPEA2, IBEA, ε-MOEA, PESA and PESA-II on test problems. The EDMOEA is able to converge to the Pareto optimal set much faster especially on the ZDT test functions with a large number of decision variables.     

A chemostat model of resource competition and allelopathy

Presented is a chemostat model in which one microbial population excretes a poison that increases the mortality of another, with toxin production increasing as the growth rate of the toxic species decreases. The model is intended to explore the role of allelopathy in blooms of harmful algae, such as red tide (Karenia brevis) and golden algae (Prymnesium parvum). This study introduces the model and its biological basis, and proceeds to the analysis of its asymptotic states. All theoretical results are supported by a set of numerical simulations. A discussion of biological conclusions and similarities to other mathematical models is also presented.     

Affinity genetic algorithm

Based on some phenomena from human society and nature, we propose a binary affinity genetic algorithm (aGA) by adopting the following strategies: the population is adaptively updated to avoid stagnation; the newly generated individuals will be ensured to survive for some generations in order for them to have time to show their good genes; new individuals and the old ones are balanced to have the advantages of both. In order to quantitatively analyze the selective pressure, the concept of selection degree and a simple linear control equation are introduced. We can maintain the diversity of the evolutionary population by controlling the value of the selection degree. Performance of aGA is further enhanced by incorporating local search strategies. Partially supported by a National Key Basic Research Project of China and by a USA NSF grant CCR-0201253.