10: Population modeling
- Page ID
- 78193
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)- Review the basic arithmetic and algebra needed to think quantitatively about populations using mathematical models.
- Unpack the concept of the demographic rates of a population, including survival, birth rate, immigration, and emigration, and how these rates can be used to determine the growth rate of a population.
- Introduce two key patterns of population growth: exponential and logistic growth.
- Show how information on population growth can be used to project change in a population into the future.
- Detail a classic tool used by human demographers and ecologists to study population change: life tables.
- 10.1: Prelude - Learning the Math of Population Models
- Ecologists frequently use mathematical models to describe population dynamics. These models can be used to describe the trajectory of population growth when resources are abundant, its maximum size when resources are limited, or how rapidly in space it expands into new territory.
- 10.2: Demographic rates
- Four processes contribute to changes in population size: mortality, birth, immigration, and emigration. These and related quantities such as survival are called demographic rates. Understanding the conceptual importance and mathematical characterization of these rates allows us to understand how and why a population changes over time.
- 10.4: Overview of Population Growth Models
- Although life histories describe the way many characteristics of a population (such as their age structure) change over time in a general way, population ecologists make use of a variety of methods to model population dynamics mathematically. These more precise models can then be used to accurately describe changes occurring in a population and better predict future changes.
- 10.5: Geometric and Exponential Growth
- Populations of all species - whether fruit flies or elephants - have the potential to increase rapidly when resources are abundant. This results in a classic "J"-shaped graph of population size over time. For populations that can breed throughout the year this is called exponential growth. Most organisms, however, are restricted to breeding at certain times of the year, which results in geometric growth.
- 10.6: Projecting population growth
- Once the population growth rate of a population has been estimated we can predict how the population will change in the future. This is called projecting the population into the future and is a key step in population analysis for understanding the prospects of a population to increase, decrease, or even go extinct.
- 10.7: Life Tables
- Life tables are a classic technique used to understand how populations change. By determining the life span of individuals in a population and how many offspring they produce human demographers and ecologists can summarize key aspects of a population, such as average life span, generation time, and growth rate. This allows comparisons between different populations and species.
Central to population ecology is the mathematical concept -- and biological reality -- of exponential population growth. The centrality of exponential growth in ecology and evolution was recognized by Darwin and plays a key role today in applied ecological decisions such as the management of invasive species, harvest limits for hunted species, and the management of endangered species. To appreciate the ecology of populations fully we therefore need to do some math. In this section, we will gently ramp up the skills and concepts we need to use math to think quantitatively about how populations change and can be effectively managed and preserved. First, we'll review the basic math we'll use throughout the chapter. Second, we'll build up the ecological concepts of demographic rates such as survival and birthrate and how they are represented mathematically. These demographic rates will then be combined into a mathematical model of population growth that shows how populations change over time due to survival and reproduction. Third, we'll show how once you have determined a population growth rate, you can predict how a population will change over time and demonstrate that it will grow exponentially. Fourth, we'll lay out a classic mathematical tool used both by human demographers and population ecologists: life tables.