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Modeling with Differential Equations01:25

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Population dynamics can be described mathematically by considering the population size P(t) as a function of time. The rate of change of the population is then represented by the derivative of P(t). A simple assumption is that the rate of growth is proportional to the size of the population itself. This leads to an exponential growth model, where the population increases rapidly without bound. While this is a useful first approximation, it does not reflect realistic long-term...
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Population Growth00:57

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Population size is dynamic, increasing with birth rates and immigration, and decreasing with death rates and emigration. In ideal conditions with unlimited resources, populations can increase exponentially, which plots as a J-shaped growth rate curve of population size against time. This type of curve is characteristic of newly-introduced invasive species, or populations that have suffered catastrophic declines and are rebounding.
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In population modeling, integration provides a systematic way to determine accumulated quantities from known rates of change. One such application arises in ecology, where the total weight of a fish population in a body of water is referred to as its biomass. When the rate of growth of this biomass is known as a function of time, calculus can be used to determine the total biomass at a future date.Growth Rate and Biomass FunctionLet the growth rate of the fish population be represented by a...
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Mechanistic Models: Compartment Models in Individual and Population Analysis01:23

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Mechanistic models are utilized in individual analysis using single-source data, but imperfections arise due to data collection errors, preventing perfect prediction of observed data. The mathematical equation involves known values (Xi), observed concentrations (Ci), measurement errors (εi), model parameters (ϕj), and the related function (ƒi) for i number of values. Different least-squares metrics quantify differences between predicted and observed values. The ordinary least...
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Exponential Equations for Modeling Growth01:26

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Exponential models are essential for describing rapid, multiplicative changes in natural systems, such as population growth. When a population doubles at regular intervals, the process can be modeled using a suitable base. For instance, a bacterial culture that doubles every three hours follows the model n(t)=n0⋅2t/3, where n(t) is the population at the time t.A more general model uses the natural base e, especially for continuous growth. This takes the form n(t)=n0⋅ert, where r is...
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Conservation of declining population focuses on ways of detecting, diagnosing, and halting a population decline. The approach uses methods to prevent populations from going extinct.
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Predicting the Effectiveness of Population Replacement Strategy Using Mathematical Modeling
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Population balance modeling: current status and future prospects.

Doraiswami Ramkrishna1, Meenesh R Singh

  • 1School of Chemical Engineering, Purdue University, West Lafayette, Indiana 47907;

Annual Review of Chemical and Biomolecular Engineering
|March 11, 2014
PubMed
Summary
This summary is machine-generated.

Population balance modeling is rapidly expanding. This review strengthens its foundations and explores new applications like crystal morphology, cell processes, and drug resistance transfer using various population balance equations.

Keywords:
biofilm growthcrystal morphologygene regulatory processespersonalized medicinestem cell differentiationstochastic internal coordinates

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Area of Science:

  • Multiphysics modeling
  • Chemical engineering
  • Biotechnology

Background:

  • Population balance modeling (PBM) is a powerful tool for simulating systems with many particles.
  • Existing reviews often focus on specific applications, leaving a gap in fundamental understanding and broad application scope.
  • The field is experiencing significant growth, necessitating updated reviews.

Purpose of the Study:

  • To reinforce the foundational principles of population balance modeling.
  • To highlight the diverse and emerging applications of PBM.
  • To present the various forms of population balance equations relevant to these applications.

Main Methods:

  • Comprehensive literature review of population balance modeling.
  • Synthesis of fundamental concepts across different PBM formulations.
  • Categorization of new and existing applications.

Main Results:

  • Detailed exposition of the core mathematical framework of PBM.
  • Identification of key advancements in PBM applications.
  • Demonstration of PBM's versatility in modeling phenomena such as crystal morphology, cell growth and differentiation, gene regulatory networks, and drug resistance transfer.

Conclusions:

  • Population balance modeling offers a unified approach to complex systems.
  • The fundamental understanding and broad applicability of PBM are crucial for scientific advancement.
  • This review provides a robust resource for researchers entering or advancing within the field of PBM.