Logistic Growth

2019 ◽  
pp. 287-318
Keyword(s):  
Logistic Growth

scholarly journals The Probability of Fixation in Populations of Changing Size

Genetics ◽  
1997 ◽  
Vol 146 (2) ◽  
pp. 723-733 ◽  
Author(s):  
Sarah P Otto ◽  
Michael C Whitlock
Keyword(s):  
Diffusion Equation ◽  
Adaptive Evolution ◽  
Process Model ◽  
Branching Process ◽  
Approximate Solutions ◽  
Logistic Growth ◽  
Beneficial Mutations ◽  
Demographic Models ◽  
Deleterious Alleles ◽  
Probability Of Fixation

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.

scholarly journals The influence of density in population dynamics with strong and weak Allee effect

2021 ◽  
Vol 29 (1) ◽  
Author(s):  
Kamrun Nahar Keya ◽  
Md. Kamrujjaman ◽  
Md. Shafiqul Islam
Keyword(s):  
Population Dynamics ◽  
Allee Effect ◽  
Global Bifurcation ◽  
Reaction Diffusion ◽  
Logistic Growth ◽  
Allee Effects ◽  
Reaction Diffusion Model ◽  
Positive Steady States ◽  
The Stability ◽  
The Impact

AbstractIn this paper, we consider a reaction–diffusion model in population dynamics and study the impact of different types of Allee effects with logistic growth in the heterogeneous closed region. For strong Allee effects, usually, species unconditionally die out and an extinction-survival situation occurs when the effect is weak according to the resource and sparse functions. In particular, we study the impact of the multiplicative Allee effect in classical diffusion when the sparsity is either positive or negative. Negative sparsity implies a weak Allee effect, and the population survives in some domain and diverges otherwise. Positive sparsity gives a strong Allee effect, and the population extinct without any condition. The influence of Allee effects on the existence and persistence of positive steady states as well as global bifurcation diagrams is presented. The method of sub-super solutions is used for analyzing equations. The stability conditions and the region of positive solutions (multiple solutions may exist) are presented. When the diffusion is absent, we consider the model with and without harvesting, which are initial value problems (IVPs) and study the local stability analysis and present bifurcation analysis. We present a number of numerical examples to verify analytical results.

Predator-dependent transmissible disease spreading in prey under Holling type-II functional response

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Dipankar Ghosh ◽  
Prasun K. Santra ◽  
Abdelalim A. Elsadany ◽  
Ghanshaym S. Mahapatra
Keyword(s):  
Functional Response ◽  
Prey Species ◽  
Linearization Method ◽  
Logistic Growth ◽  
Type Ii ◽  
Disease Spreading ◽  
Predator Interactions ◽  
The Stability ◽  
Infected Prey ◽  
Type Ii Functional Response

Abstract This paper focusses on developing two species, where only prey species suffers by a contagious disease. We consider the logistic growth rate of the prey population. The interaction between susceptible prey and infected prey with predator is presumed to be ruled by Holling type II and I functional response, respectively. A healthy prey is infected when it comes in direct contact with infected prey, and we also assume that predator-dependent disease spreads within the system. This research reveals that the transmission of this predator-dependent disease can have critical repercussions for the shaping of prey–predator interactions. The solution of the model is examined in relation to survival, uniqueness and boundedness. The positivity, feasibility and the stability conditions of the fixed points of the system are analysed by applying the linearization method and the Jacobian matrix method.

scholarly journals Stability and Hopf Bifurcation for a Delayed SIR Epidemic Model with Logistic Growth

2013 ◽  
Vol 2013 ◽  
pp. 1-11 ◽  
Author(s):  
Yakui Xue ◽  
Tiantian Li
Keyword(s):  
Incidence Rate ◽  
Epidemic Model ◽  
Threshold Value ◽  
Endemic Equilibrium ◽  
Logistic Growth ◽  
Global Dynamics ◽  
Asymptotically Stable ◽  
Sir Epidemic Model ◽  
Globally Asymptotically Stable ◽  
Sir Epidemic

We study a delayed SIR epidemic model and get the threshold value which determines the global dynamics and outcome of the disease. First of all, for anyτ, we show that the disease-free equilibrium is globally asymptotically stable; whenR0<1, the disease will die out. Directly afterwards, we prove that the endemic equilibrium is locally asymptotically stable for anyτ=0; whenR0>1, the disease will persist. However, for anyτ≠0, the existence conditions for Hopf bifurcations at the endemic equilibrium are obtained. Besides, we compare the delayed SIR epidemic model with nonlinear incidence rate to the one with bilinear incidence rate. At last, numerical simulations are performed to illustrate and verify the conclusions.

scholarly journals Ensemble bootstrap methodology for forecasting dynamic growth processes using differential equations: application to epidemic outbreaks

2021 ◽  
Vol 21 (1) ◽  
Author(s):  
Gerardo Chowell ◽  
Ruiyan Luo
Keyword(s):  
Differential Equations ◽  
Ensemble Method ◽  
Transmission Model ◽  
Logistic Growth ◽  
Ensemble Modeling ◽  
Growth Processes ◽  
Probabilistic Forecasts ◽  
Forecasting Performance ◽  
Dynamic Growth ◽  
The Individual

AbstractBackgroundEnsemble modeling aims to boost the forecasting performance by systematically integrating the predictive accuracy across individual models. Here we introduce a simple-yet-powerful ensemble methodology for forecasting the trajectory of dynamic growth processes that are defined by a system of non-linear differential equations with applications to infectious disease spread.MethodsWe propose and assess the performance of two ensemble modeling schemes with different parametric bootstrapping procedures for trajectory forecasting and uncertainty quantification. Specifically, we conduct sequential probabilistic forecasts to evaluate their forecasting performance using simple dynamical growth models with good track records including the Richards model, the generalized-logistic growth model, and the Gompertz model. We first test and verify the functionality of the method using simulated data from phenomenological models and a mechanistic transmission model. Next, the performance of the method is demonstrated using a diversity of epidemic datasets including scenario outbreak data of theEbola Forecasting Challengeand real-world epidemic data outbreaks of including influenza, plague, Zika, and COVID-19.ResultsWe found that the ensemble method that randomly selects a model from the set of individual models for each time point of the trajectory of the epidemic frequently outcompeted the individual models as well as an alternative ensemble method based on the weighted combination of the individual models and yields broader and more realistic uncertainty bounds for the trajectory envelope, achieving not only better coverage rate of the 95% prediction interval but also improved mean interval scores across a diversity of epidemic datasets.ConclusionOur new methodology for ensemble forecasting outcompete component models and an alternative ensemble model that differ in how the variance is evaluated for the generation of the prediction intervals of the forecasts.

scholarly journals Spatial and Temporal Features of the Growth of a Bacterial Species Colonizing the Zebrafish Gut

mBio ◽  
2014 ◽  
Vol 5 (6) ◽  
Author(s):  
Matthew Jemielita ◽  
Michael J. Taormina ◽  
Adam R. Burns ◽  
Jennifer S. Hampton ◽  
Annah S. Rolig ◽  
...  
Keyword(s):  
Fluorescence Microscopy ◽  
Spatial Structure ◽  
Growth Kinetics ◽  
Spatial Organization ◽  
Temporal Dynamics ◽  
Bacterial Species ◽  
Bacterial Load ◽  
Light Sheet ◽  
Logistic Growth ◽  
Light Sheet Fluorescence Microscopy

ABSTRACTThe vertebrate intestine is home to microbial ecosystems that play key roles in host development and health. Little is known about the spatial and temporal dynamics of these microbial communities, limiting our understanding of fundamental properties, such as their mechanisms of growth, propagation, and persistence. To address this, we inoculated initially germ-free zebrafish larvae with fluorescently labeled strains of anAeromonasspecies, representing an abundant genus in the zebrafish gut. Using light sheet fluorescence microscopy to obtain three-dimensional images spanning the gut, we quantified the entire bacterial load, as founding populations grew from tens to tens of thousands of cells over several hours. The data yield the first ever measurements of the growth kinetics of a microbial species inside a live vertebrate intestine and show dynamics that robustly fit a logistic growth model. Intriguingly, bacteria were nonuniformly distributed throughout the gut, and bacterial aggregates showed considerably higher growth rates than did discrete individuals. The form of aggregate growth indicates intrinsically higher division rates for clustered bacteria, rather than surface-mediated agglomeration onto clusters. Thus, the spatial organization of gut bacteria both relative to the host and to each other impacts overall growth kinetics, suggesting that spatial characterizations will be an important input to predictive models of host-associated microbial community assembly.IMPORTANCEOur intestines are home to vast numbers of microbes that influence many aspects of health and disease. Though we now know a great deal about the constituents of the gut microbiota, we understand very little about their spatial structure and temporal dynamics in humans or in any animal: how microbial populations establish themselves, grow, fluctuate, and persist. To address this, we made use of a model organism, the zebrafish, and a new optical imaging technique, light sheet fluorescence microscopy, to visualize for the first time the colonization of a live, vertebrate gut by specific bacteria with sufficient resolution to quantify the population over a range from a few individuals to tens of thousands of bacterial cells. Our results provide unprecedented measures of bacterial growth kinetics and also show the influence of spatial structure on bacterial populations, which can be revealed only by direct imaging.

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