The Many Growth Rates and Elasticities of Populations in Random Environments

Despite considerable interest in the dynamics of populations subject to temporally varying environments, alternate population growth rates and their sensitivities remain incompletely understood. For a Markovian environment, we compare and contrast the meanings of the stochastic growth rate ( \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage[OT2,OT1]{fontenc} \newcommand\cyr{ \renewcommand\rmdefault{wncyr} \renewcommand\sfdefault{wncyss} \renewcommand\encodingdefault{OT2} \normalfont \selectfont} \DeclareTextFontCommand{\textcyr}{\cyr} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} \landscape $$\lambda _{\mathrm{S}\,}$$ \end{document} ), the growth rate of average population ( \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage[OT2,OT1]{fontenc} \newcommand\cyr{ \renewcommand\rmdefault{wncyr} \renewcommand\sfdefault{wncyss} \renewcommand\encodingdefault{OT2} \normalfont \selectfont} \DeclareTextFontCommand{\textcyr}{\cyr} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} \landscape $$\lambda _{\mathrm{M}\,}$$ \end{document} ), the growth rate for average transition rates ( \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage[OT2,OT1]{fontenc} \newcommand\cyr{ \renewcommand\rmdefault{wncyr} \renewcommand\sfdefault{wncyss} \renewcommand\encodingdefault{OT2} \normalfont \selectfont} \DeclareTextFontCommand{\textcyr}{\cyr} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} \landscape $$\lambda _{\mathrm{A}\,}$$ \end{document} ), and the growth rate of an aggregate represented by a megamatrix (shown here to equal \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage[OT2,OT1]{fontenc} \newcommand\cyr{ \renewcommand\rmdefault{wncyr} \renewcommand\sfdefault{wncyss} \renewcommand\encodingdefault{OT2} \normalfont \selectfont} \DeclareTextFontCommand{\textcyr}{\cyr} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} \landscape $$\lambda _{\mathrm{M}\,}$$ \end{document} ). We distinguish these growth rates by the averages that define them. We illustrate our results using data on an understory shrub in a hurricane‐disturbed landscape, employing a range of hurricane frequencies. We demonstrate important differences among growth rates: \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage[OT2,OT1]{fontenc} \newcommand\cyr{ \renewcommand\rmdefault{wncyr} \renewcommand\sfdefault{wncyss} \renewcommand\encodingdefault{OT2} \normalfont \selectfont} \DeclareTextFontCommand{\textcyr}{\cyr} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} \landscape $$\lambda _{\mathrm{S}\,}< \lambda _{\mathrm{M}\,}$$ \end{document} , but \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage[OT2,OT1]{fontenc} \newcommand\cyr{ \renewcommand\rmdefault{wncyr} \renewcommand\sfdefault{wncyss} \renewcommand\encodingdefault{OT2} \normalfont \selectfont} \DeclareTextFontCommand{\textcyr}{\cyr} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} \landscape $$\lambda _{\mathrm{A}\,}$$ \end{document} can be < or > \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage[OT2,OT1]{fontenc} \newcommand\cyr{ \renewcommand\rmdefault{wncyr} \renewcommand\sfdefault{wncyss} \renewcommand\encodingdefault{OT2} \normalfont \selectfont} \DeclareTextFontCommand{\textcyr}{\cyr} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} \landscape $$\lambda _{\mathrm{M}\,}$$ \end{document} . We show that stochastic elasticity, \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage[OT2,OT1]{fontenc} \newcommand\cyr{ \renewcommand\rmdefault{wncyr} \renewcommand\sfdefault{wncyss} \renewcommand\encodingdefault{OT2} \normalfont \selectfont} \DeclareTextFontCommand{\textcyr}{\cyr} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} \landscape $$E^{\mathrm{S}\,}_{ij}$$ \end{document} , and megamatrix elasticity, \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage[OT2,OT1]{fontenc} \newcommand\cyr{ \renewcommand\rmdefault{wncyr} \renewcommand\sfdefault{wncyss} \renewcommand\encodingdefault{OT2} \normalfont \selectfont} \DeclareTextFontCommand{\textcyr}{\cyr} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} \landscape $$E^{\mathrm{M}\,}_{ij}$$ \end{document} , describe a complex perturbation of both means and variances of rates by the same proportion. Megamatrix elasticities respond slightly and stochastic elasticities respond strongly to changing the frequency of disturbance in the habitat (in our example, the frequency of hurricanes). The elasticity \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage[OT2,OT1]{fontenc} \newcommand\cyr{ \renewcommand\rmdefault{wncyr} \renewcommand\sfdefault{wncyss} \renewcommand\encodingdefault{OT2} \normalfont \selectfont} \DeclareTextFontCommand{\textcyr}{\cyr} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} \landscape $$E^{\mathrm{A}\,}_{ij}$$ \end{document} of \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage[OT2,OT1]{fontenc} \newcommand\cyr{ \renewcommand\rmdefault{wncyr} \renewcommand\sfdefault{wncyss} \renewcommand\encodingdefault{OT2} \normalfont \selectfont} \DeclareTextFontCommand{\textcyr}{\cyr} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} \landscape $$\lambda _{\mathrm{A}\,}$$ \end{document} does not predict changes in the other elasticities. Because \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage[OT2,OT1]{fontenc} \newcommand\cyr{ \renewcommand\rmdefault{wncyr} \renewcommand\sfdefault{wncyss} \renewcommand\encodingdefault{OT2} \normalfont \selectfont} \DeclareTextFontCommand{\textcyr}{\cyr} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} \landscape $$E^{\mathrm{S}\,}$$ \end{document} , although commonly utilized, is difficult to interpret, we introduce elasticities with a more direct interpretation: \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage[OT2,OT1]{fontenc} \newcommand\cyr{ \renewcommand\rmdefault{wncyr} \renewcommand\sfdefault{wncyss} \renewcommand\encodingdefault{OT2} \normalfont \selectfont} \DeclareTextFontCommand{\textcyr}{\cyr} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} \landscape $$E^{\mathrm{S}\,\mu }$$ \end{document} for perturbations of means and \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage[OT2,OT1]{fontenc} \newcommand\cyr{ \renewcommand\rmdefault{wncyr} \renewcommand\sfdefault{wncyss} \renewcommand\encodingdefault{OT2} \normalfont \selectfont} \DeclareTextFontCommand{\textcyr}{\cyr} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} \landscape $$E^{\mathrm{S}\,\sigma }$$ \end{document} for variances. We argue that a fundamental tool for studying selection pressures in varying environments is the response of growth rate to vital rates in all habitat states.

[1]  J. Pascarella.  The effects of Hurricane Andrew on the population dynamics and mating system of the tropical understory shrub Ardisia escallonioides (Myrsinaceae) , 1995 .

[2]  P. Dixon,et al.  Sensitivity Analysis of Structured-Population Models for Management and Conservation , 1997 .

[3]  S. Orzack.  Population Dynamics in Variable Environments. V. The Genetics of Homeostasis Revisited , 1985, The American Naturalist.

[4]  J. Pascarella.  Resiliency and response to hurricane disturbance in atropical shrub, Ardisia escallonioides (Myrsinaceae), in southFlorida. , 1998, American journal of botany.

[5]  P. V. Tienderen,et al.  ELASTICITIES AND THE LINK BETWEEN DEMOGRAPHIC AND EVOLUTIONARY DYNAMICS , 2000 .

[6]  Amy W. Ando,et al.  On the Use of Demographic Models of Population Viability in Endangered Species Management , 1998 .

[7]  S. Tuljapurkar,et al.  Environmental uncertainty and variable diapause. , 1993, Theoretical population biology.

[8]  Shripad Tuljapurkar,et al.  Structured-Population Models in Marine, Terrestrial, and Freshwater Systems , 1997, Population and Community Biology Series.

[9]  D. Whigham,et al.  A model of patch dynamics, seed dispersal, and sex ratio in the dioecious shrub Lindera benzoin (Lauraceae) , 1994 .

[10]  S. Engen,et al.  STOCHASTIC POPULATION MODELS : SOME CONCEPTS, DEFINITIONS AND RESULTS , 1998 .

[11]  Stephen P. Ellner,et al.  When is it meaningful to estimate an extinction probability , 2000 .

[12]  Paulette Bierzychudek,et al.  The Demography of Jack‐in‐the‐Pulpit, a Forest Perennial that Changes Sex , 1982 .

[13]  Mark S. Boyce,et al.  Stochastic Demography for Conservation Biology , 1997 .

[14]  J. E. Cohen,et al.  Comparative statics and stochastic dynamics of age-structured populations. , 1979, Theoretical population biology.

[15]  Shripad Tuljapurkar,et al.  Structured-Population Models in Marine, Terrestrial, and Freshwater Systems , 1998 .

[16]  E. Álvarez-Buylla.  Density Dependence and Patch Dynamics in Tropical Rain Forests: Matrix Models and Applications to a Tree Species , 1994, The American Naturalist.

[17]  J. Hajnal,et al.  On products of non-negative matrices , 1976, Mathematical Proceedings of the Cambridge Philosophical Society.

[18]  A. Grant,et al.  Elasticity analysis as an important tool in evolutionary and population ecology. , 1999, Trends in ecology & evolution.

[19]  J. Pascarella.  Hurricane Disturbance, Plant‐Animal Interactions, and the Reproductive Success of a Tropical Shrub 1 , 1998 .

[20]  P. V. Tienderen,et al.  Life Cycle Trade-Offs in Matrix Population Models , 1995 .

[21]  S. Tuljapurkar.  Population dynamics in variable environments. II. Correlated environments, sensitivity analysis and dynamics , 1982 .

[22]  H. Caswell,et al.  A general formula for the sensitivity of population growth rate to changes in life history parameters. , 1978, Theoretical population biology.

[23]  Joel E. Cohen,et al.  Ergodicity of age structure in populations with Markovian vital rates, III: Finite-state moments and growth rate; an illustration , 1977, Advances in Applied Probability.

[24]  S. W. Christensen,et al.  A stochastic age-structured population model of striped bass (Morone saxatilis) in the Potomac River , 1983 .

[25]  J. Cohen.  Ergodicity of age structure in populations with Markovian vital rates. II. General states , 1977, Advances in Applied Probability.

[26]  H. Kroon,et al.  ELASTICITIES: A REVIEW OF METHODS AND MODEL LIMITATIONS , 2000 .

[27]  C. Horvitz,et al.  HURRICANE DISTURBANCE AND THE POPULATION DYNAMICS OF A TROPICAL UNDERSTORY SHRUB: MEGAMATRIX ELASTICITY ANALYSIS , 1998 .

[28]  Michael J. Wisdom,et al.  Life Stage Simulation Analysis: Estimating Vital-Rate Effects on Population Growth for Conservation , 2000 .

[29]  T. Clutton‐Brock,et al.  Does environmental stochasticity matter? Analysis of red deer life-histories on Rum , 1995, Evolutionary Ecology.

[30]  R. Lande,et al.  Extinction dynamics of age-structured populations in a fluctuating environment. , 1988, Proceedings of the National Academy of Sciences of the United States of America.

[31]  A. Grant,et al.  How to Keep Fit in the Real World: Elasticity Analyses and Selection Pressures on Life Histories in a Variable Environment , 1996, The American Naturalist.

[32]  Shripad Tuljapurkar,et al.  Population Dynamics in Variable Environments , 1990 .

[33]  Shripad Tuljapurkar,et al.  Population Dynamics in Variable Environments. VII. The Demography and Evolution of Iteroparity , 1989, The American Naturalist.

[34]  C. Horvitz,et al.  Seed dispersal and environmental heterogeneity in a neotropical herb: a model of population and patch dynamics , 1986 .

[35]  E. Menges,et al.  Population viability analyses in plants: challenges and opportunities. , 2000, Trends in ecology & evolution.