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Collisions in Multiple Dimensions: Introduction

It is far more common for collisions to occur in two dimensions; that is, the initial velocity vectors are neither parallel nor antiparallel to each other. Let's see what complications arise from this. The first idea is that momentum is a vector. Like all vectors, it can be expressed as a sum of perpendicular components (usually, though not always, an x-component and a y-component, and a z-component if necessary). Thus, when the statement of conservation of momentum is written for a problem,...
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Related Experiment Video

Updated: Jul 13, 2026

Modeling and Imaging 3-Dimensional Collective Cell Invasion
07:08

Modeling and Imaging 3-Dimensional Collective Cell Invasion

Published on: December 7, 2011

Hierarchical spatiotemporal matrix models for characterizing invasions.

Mevin B Hooten1, Christopher K Wikle, Robert M Dorazio

  • 1Department of Mathematics and Statistics, Utah State University, Logan, Utah 84322-3900, USA. mevin.hooten@usu.edu

Biometrics
|August 11, 2007
PubMed
Summary

This study introduces a new Bayesian model to track invasive species population size, accounting for detection uncertainty and spatial variations. The model offers a more accurate ecological understanding of invasion dynamics.

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12:26

Integrating Remote Sensing with Species Distribution Models; Mapping Tamarisk Invasions Using the Software for Assisted Habitat Modeling (SAHM)

Published on: October 11, 2016

Area of Science:

  • Ecology
  • Population Dynamics
  • Invasive Species Research

Background:

  • Ecologists accurately model population survival and fecundity.
  • Invasive species significantly impact native ecosystems.
  • Current models often focus on relative abundance, not absolute population size.

Purpose of the Study:

  • To develop a hierarchical Bayesian framework for modeling invasive species population dynamics.
  • To address data discreteness and detection probability uncertainty.
  • To improve ecological understanding of invasion dynamics beyond relative abundance.

Main Methods:

  • Utilizing a hierarchical Bayesian framework.
  • Modeling nonlinear dynamics with an embedded deterministic population model.
  • Incorporating density-dependent growth and dispersal.
  • Accounting for spatially varying dispersal rates and detection probability.

Main Results:

  • The developed model accurately estimates population size dynamics.
  • The framework successfully addresses discrete data and detection uncertainty.
  • Spatially varying dispersal rates were shown to be crucial for accurate modeling.

Conclusions:

  • The hierarchical Bayesian approach provides a robust method for studying invasive species population size.
  • This model enhances ecological insights into invasion processes.
  • Accurate population size estimation is vital for effective ecological management and conservation efforts.