Contribution to the logistic theory of growth: temporal structure and growth potential

The analysis of a structured population according to three (juvenile, mature and senescent) cellular states is carried out within the framework of Delattre's transformation systems theory. Growth in number, with the dissymmetry of cell divisions, is determined by an autocatalysis process under the constraint of the availability of a source. Two models are presented: their dynamics results in a growth of the exponential type or of the sigmoidal type, respectively. In the sigmoidal case, the logistic equation (Richards-Nelder's function with adjunction of a lower asymptote Y not equal to 0) fits satisfactorily the simulated data of the total cell number Y. The growth potential is defined as the instantaneous capacity of autocatalysis, which is expressed in relation to the present 'mitotic resources' (source + non-senescing mature cells). The acceleration variations d2Y/dt2 are in close agreement with the growth potential gradient. The analysis is then generalized to other population structuring. As a result, the logistic equation can be interpreted in terms of a formal model of growth of a structured population submitted to autocatalysis and competition.     

A mathematical derivation of size spectra in fish populations

Taking into account a predator/prey size ratio in a size-structured population model leads to a partial derivative equation of which we study the properties. By expliciting the structure of the attractor of this equation, it is shown that a simple mechanism, size-based opportunistic predation, can explain the stability in the shape of size spectra observed in various marine ecosystems.     

Effect of variable surrounding on species creation

We constructed a model of speciation from evolution in an ecosystem consisting of a limited amount of energy recources. The species possesses genetic information, which is inherited according to the rules of the Penna model of genetic evolution. The increase in the number of the individuals of each species depends on the quality of their genotypes and the available energy resources. The decrease in number of the individuals results from genetic death or maximum-age reaching by the individual. The amount of energy resources is represented by a solution of the differential logistic equation, where the growth rate of the amount of the energy resources has been modified to include the number of individuals from all species in the ecosystem under consideration. The fluctuating surrounding is modelled with the help of the function V(x, t) = 1/4 x4 + 1/2 b(t)x2, where x represents phenotype and the coefficient b(t) shows the cos(omega t) time dependence. The closer the value x of an individual to the minimum of V(x, t), the better adapted its genotype to the surrounding. We observed that the life span of the organisms strongly depends on the value of the frequency omega. It becomes shorter the more frequent the changes of the surrounding. However, there is a tendency for the species that have a higher value of the reproduction age aR to win the competition with the other species. Another observation is that small evolutionary changes of the inherited genetic information lead to spontaneous bursts of the evolutionary activity when many new species may appear in a short period.     

Individual-based spatially-explicit model of an herbivore and its resource: the effect of habitat reduction and fragmentation

We present an individual-based, spatially-explicit model of the dynamics of a small mammal and its resource. The life histories of each individual animal are modeled separately. The individuals can have the status of residents or wanderers and belong to behaviorally differing groups of juveniles or adults and males or females. Their territory defending and monogamous behavior is taken into consideration. The resource, green vegetation, grows depending on seasonal climatic characteristics and is diminished due to the herbivore's grazing. Other specifics such as a varying personal energetic level due to feeding and starvation of the individuals, mating preferences, avoidance of competitors, dispersal of juveniles, as a result of site overgrazing, etc., are included in the model. We determined model parameters from real data for the species Microtus ochrogaster (prairie vole). The simulations are done for a case of an enclosed habitat without predators or other species competitors. The goal of the study is to find the relation between size of habitat and population persistence. The experiments with the model show the populations go extinct due to severe overgrazing, but that the length of population persistence depends on the area of the habitat as well as on the presence of fragmentation. Additionally, the total population size of the vole population obtained during the simulations exhibits yearly fluctuations as well as multi-yearly peaks of fluctuations. This dynamics is similar to the one observed in prairie vole field studies.     

The notion of the internal and external limitations of monotonic growth functions. A reformulation of the logistic equation

The boundary value (plateau) of non-periodic growth functions constitutes one of the parameters of various usual models such as the logistic equation. Its double interpretation involves either a limit of an internal or endogenous nature or an external environment-dependent limit. Using the autocatalytic model of structured cell populations (Buis, model II, 2003), a reformulation of the logistic equation is put forward and illustrated in the case of three cell classes (juvenile, mature, senescing). The agonistic component corresponds exactly to the only active fraction of the population (non-senescing mature cells), whereas the antagonistic component is interpreted in terms of an external limit (available substrate or source). The occurrence and properties of an external limit are investigated using the same autocatalytic model with two major modifications: the absence of competition (non-limiting source) and the occurrence of a maximum number of mitoses per cell filiation (Lück and Lück, 1978). The analysis, which is carried out according to the principle of deterministic cell automata (L-systems), shows the flexibility of the model, which exhibits a diversity of kinetic properties: shifts from the sigmoidal form, number and position of growth rate extremums, number of phases of the temporal structure. These characteristics correspond to the diversity of the experimental growth curves where the singularities of the growth rate gradient are often not accounted for satisfactorily by the usual global models.     

A mathematical model of population dynamics for Batesian mimicry system

We analyse a mathematical model of the population dynamics among a mimic, a corresponding model, and their common predator populations. Predator changes its search-and-attack probability by forming and losing its search image. It cannot distinguish the mimic from the model. Once a predator eats a model individual, it comes to omit both the model and the mimic species from its diet menu. If a predator eats a mimic individual, it comes to increase the search-and-attack probability for both model and mimic. The predator may lose the repulsive/attractive search image with a probability per day. By analysing our model, we can derive the mathematical condition for the persistence of model and mimic populations, and then get the result that the condition for the persistence of model population does not depend on the mimic population size, while the condition for the persistence of mimic population does depend the predator's memory of search image.     

数学判别模型在预测害虫种群动态上的应用

根据两个总体的Fisher判别准则,建立了预测害虫种群动态的数学判别模型,对山东省惠民县1967～1977年共11年二代棉铃虫发生程度的两类资料进行了数量分析,建立了数学模型：y＝0．0127x1－0．023X2,对历史资料的回代验证与独立样本的预测,符合率在90％以上。     

A mathematical model to study the effects of drugs administration on tumor growth dynamics

A mathematical model for describing the cancer growth dynamics in response to anticancer agents administration in xenograft models is discussed. The model consists of a system of ordinary differential equations involving five parameters (three for describing the untreated growth and two for describing the drug action). Tumor growth in untreated animals is modelled by an exponential growth followed by a linear growth. In treated animals, tumor growth rate is decreased by an additional factor proportional to both drug concentration and proliferating cells. The mathematical analysis conducted in this paper highlights several interesting properties of this tumor growth model. It suggests also effective strategies to design in vivo experiments in animals with potential saving of time and resources. For example, the drug concentration threshold for the tumor eradication, the delay between drug administration and tumor regression, and a time index that measures the efficacy of a treatment are derived and discussed. The model has already been employed in several drug discovery projects. Its application on a data set coming from one of these projects is discussed in this paper.     

An approach to the nonlinear dynamics of Russian wheat aphid population growth with the cusp catastrophe model

Many insect field populations, especially aphids, often exhibit irregular and even catastrophic fluctuations. The objective of the present study is to explore whether or not the population intrinsic rates of growth ( r _m) obtained under laboratory conditions can shed some light on the irregular changes of insect field populations. We propose to use the catastrophe theory, one of the earliest nonlinear dynamics theories, to answer the question. To collect the necessary data, we conducted a laboratory experiment to investigate population growth of the Russian wheat aphid (RWA), Diuraphis noxia (Mordvilko), in growth chambers. The experiment was designed as the factorial combinations of five temperatures and five host plant-growth stages (25 treatments in total): 1800 newly born RWA nymphs arranged in the 25 treatments (each treatment with 72 repetitions) were observed for their development, reproduction and survival through their entire lifetimes. After obtaining the population intrinsic rates of growth ( r _m) from the experimental data under various environmental conditions, we built a cusp catastrophe model for RWA population growth by utilizing r _m as the system state variable, and temperature and host plant-growth stage as control variables. The cusp catastrophe model suggests that RWA population growth is intrinsically catastrophic , and dramatic jumps from one state to another might occur even if the temperature and plant-growth stage change smoothly . Other basic behaviors of the cusp catastrophe model, such as catastrophic jumps , hystersis and divergence , are also expected in RWA populations. These results suggest that the answer to the previously proposed question should be yes.     

A population dynamic model of Batesian mimicry

A population dynamic model of Batesian mimicry, in which populations of both model and mimetic species were considered, was analyzed. The probability of a predator catching prey on each encouter was assumed to depend on the frequency of the mimic. The change in population size of each species was considered to have two components, growth at the intrinsic growth rate and carrying capacity, and reduction by predation. For simplicity in the analyses, three assumptions were made concerning the carrying capacities of each population: (1) with no density effects on the mimic population growth rate; (2) with no density effects on the model species; and (3) with density effects on both species. The first and second cases were solved analytically, whereas the last was, for the most part, investigated numerically. Under assumption (1), two stable equilibria are possible, in which both species either coexist or go to extinction. Under assumption (2), there are also two stable equilibria possible, in which either only the mimic persists or both go to extinction. These results explain the field records of butterflies (Pachliopta aristolochiae and its mimic Papilio polytes) in the Ryukyu Islands, Japan.     

A simple model of a Gyrodactylus population

Experimental populations of 20 Gyrodactylus alexanderi Mizelle &; Kritsky, 1967, on 19 isolated Gasterosteus aculeatus at 15°C increased for 2 weeks to a mean of 61, then decreased in 2 further weeks to a mean of 9. Fish that lost their fluke infestations were refractory to further infestation for about 3 weeks.The chief factors affecting fluke abundance were measured, including reproduction and mortality rates of flukes on fish, rate of shedding by the fish, mortality rates of flukes while off fish, and the rate of reattachment of the flukes. Data on these individual factors were combined to form a simple deterministic model which simulated the population changes on isolated fish. This was later made more realistic by the introduction of a random variable. When the model was tested in a multiple-host situation it predicted results close to those observed experimentally.     

Plant population growth and competition in a light gradient: a mathematical model of canopy partitioning

Can a difference in the heights at which plants place their leaves, a pattern we call canopy partitioning, make it possible for two competing plant species to coexist? To find out, we examine a model of clonal plants living in a nonseasonal environment that relates the dynamical behavior and competitive abilities of plant populations to the structural and functional features of the plants that form them. This examination emphasizes whole plant performance in the vertical light gradient caused by self-shading. This first of three related papers formulates a prototype single species Canopy Structure Model from biological first principles and shows how all plant properties work together to determine population persistence and equilibrium abundance. Population persistence is favored, and equilibrium abundance is increased, by high irradiance, high maximum photosynthesis rate, rapid saturation of the photosynthetic response to increased irradiance, low tissue respiration rate, small amounts of stem and root tissue necessary to support the needs of leaves, and low density of leaf, stem, and root tissues. In particular, equilibrium abundance decreases as mean leaf height increases because of the increased cost of manufacturing and maintaining stem tissue. All conclusions arise from this formulation by straightforward analysis. The argument concludes by stating this formulation's straightforward extension, called a Canopy Partitioning Model, to two competing species.     

A population and energy model of a forest insect outbreak

A model is proposed for the dynamics of a forest insect population with account of food consumption and the response of plants to damage. Equations are derived relating the propagation coefficient, female mass, pest conversion efficacy, and plant reaction. Outbreak scenarios are analyzed as dependent on steady-state female weight. The results are compared with the data of observations in nature.     

A hierarchical matrix model to assess the impact of habitat fragmentation on population dynamics: an elasticity analysis

To better understand the role of habitat quality and boundaries on population dynamics at the landscape scale, we develop a model combining a spatially implicit approach, a spatial population Leslie-type model and an implicit model of habitat fragmentation. An original approach of elasticity permits to identify which types of element and boundary influence the most population viability according to the wood fragmentation degree. The studied species is a corridor forest insect sensitive to fragmentation (Abax parallelepipedus, Coleoptera, Carabidae). We show that a single large patch of wood is better than several small patches for the population viability.     

Integral projection models perform better for small demographic data sets than matrix population models: a case study of two perennial herbs

1. Matrix population models are widely used to describe population dynamics, conduct population viability analyses and derive management recommendations for plant populations. For endangered or invasive species, management decisions are often based on small demographic data sets. Hence, there is a need for population models which accurately assess population performance from such small data sets.
2. We used demographic data on two perennial herbs with different life histories to compare the accuracy and precision of the traditional matrix population model and the recently developed integral projection model (IPM) in relation to the amount of data.
3. For large data sets both matrix models and IPMs produced identical estimates of population growth rate (λ). However, for small data sets containing fewer than 300 individuals, IPMs often produced smaller bias and variance for λ than matrix models despite different matrix structures and sampling techniques used to construct the matrix population models.
4. Synthesis and applications . Our results suggest that the smaller bias and variance of λ estimates make IPMs preferable to matrix population models for small demographic data sets with a few hundred individuals. These results are likely to be applicable to a wide range of herbaceous, perennial plant species where demographic fate can be modelled as a function of a continuous state variable such as size. We recommend the use of IPMs to assess population performance and management strategies particularly for endangered or invasive perennial herbs where little demographic data are available.     

A mathematical model of a crocodilian population using delay-differential equations

The crocodilia have multiple interesting characteristics that affect their population dynamics. They are among several reptile species which exhibit temperature-dependent sex determination (TSD) in which the temperature of egg incubation determines the sex of the hatchlings. Their life parameters, specifically birth and death rates, exhibit strong age-dependence. We develop delay-differential equation (DDE) models describing the evolution of a crocodilian population. In using the delay formulation, we are able to account for both the TSD and the age-dependence of the life parameters while maintaining some analytical tractability. In our single-delay model we also find an equilibrium point and prove its local asymptotic stability. We numerically solve the different models and investigate the effects of multiple delays on the age structure of the population as well as the sex ratio of the population. For all models we obtain very strong agreement with the age structure of crocodilian population data as reported in Smith and Webb (Aust. Wild. Res. 12, 541-554, 1985). We also obtain reasonable values for the sex ratio of the simulated population.     

Invasion of an intermediate predator: Fish population dynamics in a mathematical model of a trophic chain (As applied to Syamozero lake)

A mathematical model is presented for the dynamics of a spatially heterogeneous predator-prey population system; a prototype is the Syamozero lake fish community. We show that the invasion of an intermediate predator can evoke chaotic oscillations in the population densities. We also show that different dynamic regimes (stationary, nonchaotic oscillatory, and chaotic) can coexist. The “choice” of a particular regime depends on the initial invader density. Analysis of the model solutions shows that invasion of an alien species is successful only in the absence of competition between the juvenile invaders and the native species.     

A mathematical model for the growth of mycelial pellet populations

In liquid culture, filamentous organisms often grow in the form of pellets. Growth result in an increase in radius, whereas shear forces result in release of hyphal fragments which act as centers for further pellet growth and development. A previously published model for pellet growth of filamentous microorganisms has been examined and is found to be unstable for certain parameter values. This instability has been identified as being due to inaccuracies in estimating the numbers of fragments which seed the pellet population. A revised model has been formulated, based on similar premises, but adopting a finite element approach. This considers the population of pellets to be distributed in a range of size classes. Growth results in movement to classes of increasing pellet size, while fragments enter the smallest size class, from which they grow to form further pellets. The revised model is stable and predicts changes in the distribution of pellet sizes within a population growing in liquid batch culture. It considers pellet growth and death, with fragmentation providing new centers of growth within the pellet population, and predicts the effects of shear forces on pellet growth and size distribution. Predictions of pellet size distributions are tested using previously published data on the growth of fungal pellets and further predictions are generated which are suitable for experimental testing using cultures of filamentous fungi or actinomycetes. (c) 1995 John Wiley & Sons, Inc.