Skip to main content
Biology LibreTexts

12.6: Independent carrying capacities

  • Page ID
    25497
  • \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)

    Self-limitation or self-enhancement of population growth are within the \(r\,+\,sN\) framework. Below these terms are in red.

    \(\frac{1}{N_1}\frac{dN_1}{dt}\,=\,r_1\,s_{1,2}N_2\,\color{red}{+\,s_{1,1}N_1}\)

    \(\frac{1}{N_2}\frac{dN_2}{dt}\,=\,r_2\,s_{2,1}N_1\,\color{red}{+\,s_{2,2}N_2}\)

    The self-feedback term for the prey, \(s_{1,1}\), is typically negative, reflecting a carrying capacity for the prey in the absence of predators, \(K_1\,=\,-r_1\,/\,s_{1,1}\). This tends to stabilize the system, dampening oscillations and leading to a joint equilibrium of predator and prey.

    On the other hand, the self-feedback term for the predator, \(s_{2,2}\), is typically zero, meaning the predators vanish in the absence of prey. But it could be positive, indicating benefits from group hunting and the like. A positive value for \(s_{2,2}\) tends to destabilize the system, leading to enlarging oscillations.


    This page titled 12.6: Independent carrying capacities is shared under a CC BY-NC 4.0 license and was authored, remixed, and/or curated by Clarence Lehman, Shelby Loberg, & Adam Clark (University of Minnesota Libraries Publishing) via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.