\relax \citation{Bazzaz1996} \citation{Bazzaz1996} \@writefile{toc}{\contentsline {chapter}{\numberline {5}Lotka--Volterra Interspecific Competition}{135}} \@writefile{lof}{\addvspace {10\p@ }} \@writefile{lot}{\addvspace {10\p@ }} \newlabel{cha:lotka-volt-intersp}{{5}{135}} \@writefile{lof}{\contentsline {figure}{\numberline {5.1}{\ignorespaces Changes in abundances of six species of \emph {Aster}, \emph {Euthamia}, and \emph {Solidago} during early secondary succession. 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Both species have the same intraspecific competition coefficient, $\alpha _{ii}=0.01$. In the absence of interspecific competition, both species would reach $K$. However, they both have negative effects on each other ($\alpha _{ij} >0$), and species 1 (solid line) has a greater negative effect on species 2 ($\alpha _{21} > \alpha _{12}$). }}{138}} \newlabel{fig:ic1}{{5.2}{138}} \@writefile{toc}{\contentsline {paragraph}{Discrete logistic competition dynamics (Fig. 5.2\hbox {})}{138}} \@writefile{toc}{\contentsline {subsection}{\numberline {5.1.3}Continuous time model}{139}} \newlabel{eq:N2a}{{5.5}{139}} \newlabel{eq:N2b}{{5.6}{139}} \@writefile{lot}{\contentsline {table}{\numberline {5.1}{\ignorespaces Parameters of the continuous 2 species Lotka-Volterra competition model. 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($\alpha _{ii} = 0.01$)}}{156}} \newlabel{fig:RT}{{5.8}{156}} \@writefile{toc}{\contentsline {paragraph}{Effect of $r$ on stability and return time}{157}} \newlabel{fig:LVeig3DAll}{{5.9a}{157}} \newlabel{sub@fig:LVeig3DAll}{{(a)}{a}} \newlabel{fig:eig3DAllhalf}{{5.9b}{157}} \newlabel{sub@fig:eig3DAllhalf}{{(b)}{b}} \@writefile{lof}{\contentsline {figure}{\numberline {5.9}{\ignorespaces The dominant eigenvalue of the Jacobian matrix varies with $r$ as well as with $\beta $ --- higher $r$ causes greater responsiveness to perturbations around an internal equilibrium. \subref {fig:LVeig3DAll} $r=1$, \subref {fig:eig3DAllhalf} $r=0.5$.}}{157}} \@writefile{lof}{\contentsline {subfigure}{\numberline{(a)}{\ignorespaces {$\lambda _1$ with $r=1$}}}{157}} \@writefile{lof}{\contentsline {subfigure}{\numberline{(b)}{\ignorespaces {$\lambda _1$ with $r=0.5$}}}{157}} \newlabel{fig:LVeigR}{{5.9}{157}} \@writefile{toc}{\contentsline {section}{\numberline {5.5}Summary}{157}} \@writefile{toc}{\contentsline {section}{Problems}{158}} \newlabel{prob:LVbasic}{{5.1}{158}} \newlabel{prob:relabundance}{{5.2}{158}} \newlabel{prob:partderivcomp}{{5.3}{158}} \newlabel{prob:LVTotal}{{5.4}{158}} \newlabel{prob:Routh}{{5.5}{158}} \newlabel{prob:LVJac}{{5.6}{158}} \@setckpt{Chap05/Chapter05}{ \setcounter{page}{160} \setcounter{equation}{27} \setcounter{enumi}{4} \setcounter{enumii}{0} \setcounter{enumiii}{0} \setcounter{enumiv}{0} \setcounter{footnote}{7} \setcounter{mpfootnote}{0} \setcounter{part}{2} \setcounter{section}{5} \setcounter{subsection}{0} \setcounter{subsubsection}{0} \setcounter{paragraph}{0} \setcounter{subparagraph}{0} \setcounter{figure}{9} \setcounter{table}{2} \setcounter{chapter}{5} \setcounter{theorem}{0} \setcounter{prob}{6} \setcounter{merk}{0} \setcounter{pp@next@reset}{0} \setcounter{parentequation}{0} \setcounter{float@type}{4} \setcounter{KVtest}{0} \setcounter{subfigure}{0} \setcounter{subfigure@save}{2} \setcounter{lofdepth}{1} \setcounter{subtable}{0} \setcounter{subtable@save}{0} \setcounter{lotdepth}{1} \setcounter{FancyVerbLine}{1} }