A study of some diffusion models of population growth

The stochastic differential equations of many diffusion processes which arise in studies of population growth in random environments can be transformed, if the Stratonovich stochastic calculus is employed, to the equation of the Wiener process. If the transformation function has certain properties then the transition probability density function and quantities relating to the time to first attain a given population size can be obtained from the known results for the Wiener process. Some other random growth processes can be derived from the Ornstein-Uhlenbeck process. These transformation methods are applied to the random processes of Malthusian growth, Pearl-Verhulst logistic growth and a recent model of density independent growth due to Levins.     

Modelling logistic growth by a new diffusion process: Application to biological systems

The present paper introduces a new diffusion process for the purpose of modelling logistic-type behaviour patterns. Unlike other processes in the same context, this one verifies that its mean function is a logistic curve. In addition, its transition density can be found explicitly, which allows to analyse inference from the discrete sampling of trajectories. The main features of the process will be analysed and the maximum likelihood estimation of parameters will be carried out through discrete sampling. Regarding the numerical problems found to solve the likelihood equations, several strategies are developed for obtaining initial solutions for the usual numerical procedures. Such strategies are compared by means of a simulation example. Also, another simulation study is carried out in order to compare the estimation in this process to that developed by means of continuous sampling in the logistic diffusion model considered by Giovanis and Skiadas (1999). Finally an example is given for the growth of a microorganism culture. This example illustrates the predictive possibilities of the new process, as well as its ability to study time variables formulated as first-passage-times.     

Microbial growth kinetics of fed-batch fermentations

Summary Fed-batch fermenters are generally operated with the addition of small doses of nutrients, therefore the volume of the fermentation broth increases with time. Batch fermenters generally contain and almost constant volume of broth and a logistic equation has been commonly employed to simulate microbial growth in them. Mass balances were determined with fed-batch fermentation to obtain expressions which account for the effect of volume increase and the subsequent dilution of the biomass. A growth rate expression was obtained for fed-batch fermentations. The new expression was very similar to the logistic equation. Therefore an anology between the values of the parameters of the new model and of the logistic equation was sought. The new model was also employed to simulate six sets of data from the literature and satisfactory results were obtained.     

Growth with regulation in random environment

The diffusion model for a population subject to Malthusian growth is generalized to include regulation effects. This is done by incorporating a logarithmic term in the regulation function in a way to obtain, in the absence of noise, an S-shaped growth law retaining the qualitative features of the logistic growth curve. The growth phenomenon is modeled as a diffusion process whose transition p.d.f. is obtained in closed form. Its steady state behavior turns out to be described by the lognormal distribution. The expected values and the mode of the transition p.d.f. are calculated, and it is proved that their time course is also represented by monotonically increasing functions asymptotically approaching saturation values. The first passage time problem is then considered. The Laplace transform of the first passage time p.d.f. is obtained for arbitrary thresholds and is used to calculate the expected value of the first passage time. The inverse Laplace transform is then determined for a threshold equal to the saturation value attained by the population size in the absence of random components. The probability of absorption for an arbitrary barrier is finally calculated as the limit of the absorption probability in a two-barrier problem.     

The expected extinction time of a population within a system of interacting biological populations

The quasi-stationary distribution of a population within a system of interacting populations is approximated by a stochastic logistic process. The parameters of this process can be expressed in the parameters of the full system. Using the diffusion approximation, an expression for the expected extinction time is derived from this logistic process. Since the expected extinction time is expressed in the parameters of the full system, the effect of these parameters on the extinction risk can be easily evaluated, which may be of use for studies in ecology, conservation biology and epidemiology. The outcome is compared with simulation results for the case of a prey-predator system.     

Stochastic model of population growth and spread

A stochastic model for the population regulated by logistic growth and spreading in a given region of two-or three-dimensional space has been introduced. For many-species population the interactions among the species have also been icorporated in this model. From the random variables that describe stochastic processes of a Wiener type the space-dependent random population densities have been formed and shown to satisfy the Langevin equations. The Fokker-Planck equation corresponding to these Langevin equations has been approximately solved for the transition probability of the population spreading and it has been found that such approximate expressions of the transition probability depend on the solutions of the deterministic equations of the diffusion model with logistic growth and interactions. Also, the stationary or equilibrium solutions of the Fokker-Planck equation together with the special discussion on the pattern of single-species population spreading have been made.     

Growth with regulation in fluctuating environments

In this paper deterministic growth laws of a logistic-like type are initially introduced. The growth equations are expressed by first order differential equations containing a third order nonlinear term. Such equations are then parameterized in a way to allow for random fluctuations of the intrinsic fertility and of the environmental carrying capacity, thus leading to diffusion processes of new types. Their transition p.d.f. and asymptotic moments are then obtained and a detailed study of the extinction problem is performed within the framework of the first passage time problem through arbitrarily fixed threshold values. Some statistically significant quantities, such as the mean time necessary for the process to attain an assigned state, are obtained in closed form. The behavior of the diffusion processes here derived is finally compared with that of the well known diffusion processes obtained by parameterizing logistic and Gompertz growth equations.Work supported in part by the Group for Mathematical Information Sciences (GNIM) of the National Research Council and by Progetto Finalizzato Sofmat, Contract No. 82.00845.97     

A new Gompertz-type diffusion process with application to random growth

Stochastic models describing growth kinetics are very important for predicting many biological phenomena. In this paper, a new Gompertz-type diffusion process is introduced, by means of which bounded sigmoidal growth patterns can be modeled by time-continuous variables. The main innovation of the process is that the bound can depend on the initial value, a situation that is not provided by the models considered to date. After building the model, a comprehensive study is presented, including its main characteristics and a simulation of sample paths. With the aim of applying this model to real-life situations, and given its possibilities in forecasting via the mean function, discrete sampling based inference is developed. The likelihood equations are not directly solvable, and because of difficulties that arise with the usual numerical methods employed to solve them, an iterative procedure is proposed. The possibilities of the new process are illustrated by means of an application to real data, concretely, to growth in rabbits.     

Pattern formation in prey-taxis systems

In this paper, we consider spatial predator–prey models with diffusion and prey-taxis. We investigate necessary conditions for pattern formation using a variety of non-linear functional responses, linear and non-linear predator death terms, linear and non-linear prey-taxis sensitivities, and logistic growth or growth with an Allee effect for the prey. We identify combinations of the above non-linearities that lead to spatial pattern formation and we give numerical examples. It turns out that prey-taxis stabilizes the system and for large prey-taxis sensitivity we do not observe pattern formation. We also study and find necessary conditions for global stability for a type I functional response, logistic growth for the prey, non-linear predator death terms, and non-linear prey-taxis sensitivity.     

Pattern formation in prey-taxis systems

In this paper, we consider spatial predator-prey models with diffusion and prey-taxis. We investigate necessary conditions for pattern formation using a variety of non-linear functional responses, linear and non-linear predator death terms, linear and non-linear prey-taxis sensitivities, and logistic growth or growth with an Allee effect for the prey. We identify combinations of the above non-linearities that lead to spatial pattern formation and we give numerical examples. It turns out that prey-taxis stabilizes the system and for large prey-taxis sensitivity we do not observe pattern formation. We also study and find necessary conditions for global stability for a type I functional response, logistic growth for the prey, non-linear predator death terms, and non-linear prey-taxis sensitivity.     

A stochastic metapopulation model accounting for habitat dynamics

A stochastic metapopulation model accounting for habitat dynamics is presented. This is the stochastic SIS logistic model with the novel aspect that it incorporates varying carrying capacity. We present results of Kurtz and Barbour, that provide deterministic and diffusion approximations for a wide class of stochastic models, in a form that most easily allows their direct application to population models. These results are used to show that a suitably scaled version of the metapopulation model converges, uniformly in probability over finite time intervals, to a deterministic model previously studied in the ecological literature. Additionally, they allow us to establish a bivariate normal approximation to the quasi-stationary distribution of the process. This allows us to consider the effects of habitat dynamics on metapopulation modelling through a comparison with the stochastic SIS logistic model and provides an effective means for modelling metapopulations inhabiting dynamic landscapes.     

The determination of the parameter of Bertalanffy's growth differential equations by means of nonlinear internal regression

A short survey is given on various parameterized versions of the logistic law of growth and of Bertalanffy's growth differential equations. To examine the validity of these various growth expressions internal nonlinear regressions were performed, and the results of the calculations are presented. The body length growth of man within the embryonic development serves as examples of a growth process. The parameters in the differential equations will be adjusted to the course of the divided central differences calculated from means of measured values of this growth process.     

The Probability of Fixation in Populations of Changing Size

The rate of adaptive evolution of a population ultimately depends on the rate of incorporation of beneficial mutations. Even beneficial mutations may, however, be lost from a population since mutant individuals may, by chance, fail to reproduce. In this paper, we calculate the probability of fixation of beneficial mutations that occur in populations of changing size. We examine a number of demographic models, including a population whose size changes once, a population experiencing exponential growth or decline, one that is experiencing logistic growth or decline, and a population that fluctuates in size. The results are based on a branching process model but are shown to be approximate solutions to the diffusion equation describing changes in the probability of fixation over time. Using the diffusion equation, the probability of fixation of deleterious alleles can also be determined for populations that are changing in size. The results developed in this paper can be used to estimate the fixation flux, defined as the rate at which beneficial alleles fix within a population. The fixation flux measures the rate of adaptive evolution of a population and, as we shall see, depends strongly on changes that occur in population size.     

On a conjecture concerning population growth in random environment

Discrete stochastic models are constructed and their limit diffusion processes are derived to shed light on a controversial conjecture regarding the effects of environmental variance on the asymptotic behavior of a population subject to logistic growth in random environment.Work supported in part by the National Group for Mathematical Information Sciences (GNIM) of the National Council for Research     

Modeling the Dynamics of Populations in a Heterogeneous Environment: Invasion and Multistability

Biophysics - Abstract—We model the interaction of two populations based on evolutionary equations that consider diffusion, taxis, and logistic growth. Scenarios of biological invasion are...     

Logistic Proliferation of Cells in Scratch Assays is Delayed

Scratch assays are used to study how a population of cells re-colonises a vacant region on a two-dimensional substrate after a cell monolayer is scratched. These experiments are used in many applications including drug design for the treatment of cancer and chronic wounds. To provide insights into the mechanisms that drive scratch assays, solutions of continuum reaction–diffusion models have been calibrated to data from scratch assays. These models typically include a logistic source term to describe carrying capacity-limited proliferation; however, the choice of using a logistic source term is often made without examining whether it is valid. Here we study the proliferation of PC-3 prostate cancer cells in a scratch assay. All experimental results for the scratch assay are compared with equivalent results from a proliferation assay where the cell monolayer is not scratched. Visual inspection of the time evolution of the cell density away from the location of the scratch reveals a series of sigmoid curves that could be naively calibrated to the solution of the logistic growth model. However, careful analysis of the per capita growth rate as a function of density reveals several key differences between the proliferation of cells in scratch and proliferation assays. Our findings suggest that the logistic growth model is valid for the entire duration of the proliferation assay. On the other hand, guided by data, we suggest that there are two phases of proliferation in a scratch assay; at short time, we have a disturbance phase where proliferation is not logistic, and this is followed by a growth phase where proliferation appears to be logistic. These two phases are observed across a large number of experiments performed at different initial cell densities. Overall our study shows that simply calibrating the solution of a continuum model to a scratch assay might produce misleading parameter estimates, and this issue can be resolved by making a distinction between the disturbance and growth phases. Repeating our procedure for other scratch assays will provide insight into the roles of the disturbance and growth phases for different cell lines and scratch assays performed on different substrates.     

Diffusion of hydrogen peroxide across DPPC large unilamellar liposomes

The decomposition of hydrogen peroxide catalyzed by catalase entrapped in the pool of di-palmitoylphosphatidyl choline unilamellar liposomes has been studied. The rate of the process was evaluated by following the production of oxygen as a function of time. Under the experimental conditions employed the rate of oxygen production was controlled by the diffusion of hydrogen peroxide, allowing for the estimation of the diffusion coefficient of hydrogen peroxide across the liposome bilayer. The rate of diffusion across the bilayer increases with the temperature and the presence of fluidizers (n-nonanol), according with changes in the bilayer fluidity, as sensed by 1,6-diphenyl hexatriene (DPH) fluorescence anisotropy. A peculiar aspect of the data is the fast hydrogen peroxide diffusion observed at the bilayer phase transition temperature. This fast diffusion is associated to rafts fluctuations that take place in the partially melted bilayer. These fluctuations have no effect on the microviscosity sensed by DPH.     

A master equation for a spatial population model with pair interactions

We derive a closed master equation for an individual-based population model in continuous space and time. The model and master equation include Brownian motion, reproduction via binary fission, and an interaction-dependent death rate moderated by a competition kernel. Using simulations we compare this individual-based model with the simplest approximation, the spatial logistic equation. In the limit of strong diffusion the spatial logistic equation is a good approximation to the model. However, in the limit of weak diffusion the spatial logistic equation is inaccurate because of spontaneous clustering driven by reproduction. The weak-diffusion limit can be partially analyzed using an exact solution of the master equation applicable to a competition kernel with infinite range. This analysis shows that in the case of a top-hat kernel, reducing the diffusion can increase the total population. For a Gaussian kernel, reduced diffusion invariably reduces the total population. These theoretical results are confirmed by simulation.     

双碳源单细胞蛋白间歇培养及流加培养的动力学模型

应用非结构的逻辑增殖模型研究了两种酵母的单碳源和双碳源单细胞蛋白间歇培养的动力学,用改进的逻辑增殖模型研究了双碳源流加培养过程的动力学,从实验数据拟合了动力学模型参数,模型计算值与实验数据吻合良好。     

Molecular diffusion into horse spleen ferritin: a nitroxide radical spin probe study.

Electron paramagnetic resonance spectroscopy and gel permeation chromatography were employed to study the molecular diffusion of a number of small nitroxide spin probes (approximately 7-9 A diameter) into the central cavity of the iron-storage protein ferritin. Charge and polarity of these radicals play a critical role in the diffusion process. The negatively charged radical 4-carboxy-2,2,6,6-tetramethylpiperidine-N-oxyl (4-carboxy-TEMPO) does not penetrate the cavity whereas the positively charged 4-amino-TEMPO and 3-(aminomethyl)-proxyl radical and polar 4-hydroxy-TEMPO radical do. Unlike the others, the apolar TEMPO radical does not enter the cavity but instead binds to ferritin, presumably at a hydrophobic region of the protein. The kinetic data indicate that diffusion is not purely passive, the driving force coming not only from the concentration gradient between the inside and outside of the protein but also from charge interactions between the diffusant and the protein. A model for diffusion is derived that describes the observed kinetics. First-order half-lives for diffusion into the protein of 21-26 min are observed, suggesting that reductant molecules with diameters considerably larger than approximately 9 A would probably enter the protein cavity too slowly to mobilize iron efficiently by direct interaction with the mineral core.