Carrying_capacity - Wikiparency

# Carrying capacity ## Microsim (three.js) <div class="microsim-player"> <iframe src="https://wikitube-3d-microsims.netlify.app/Carrying_capacity.html" width="100%" height="620" frameborder="0" loading="lazy" sandbox="allow-scripts allow-same-origin"></iframe> </div> *A cornerstone of [[Population_growth|population growth]] and [[Ecology|ecology]]: the logistic curve tames the runaway Malthusian exponential with a resource ceiling — the same braking that flattens a bacterial growth curve as a culture nears stationary phase.* > Carrying capacity (**K**) is the largest population an environment can support indefinitely given its finite resources — food, space, water, light. In the logistic model, a population grows almost exponentially while it is small, then decelerates as it fills the niche, tracing an S-shaped curve that levels off at K. This microsim lets you dial the growth rate, the capacity, and the starting number, then watch the population climb, brake, and settle — or start it above K and see it fall back. ## About this microsim The panel exposes three sliders — **Growth rate r** (0.02–1.2), **Capacity K** (10–180), and **Start pop. N₀** (1–200) — plus a **Sim speed** control (0.1–4×) that only changes how fast the animation plays, not the underlying biology. Buttons let you **⏸ Pause**, **↻ Reset** to the initial condition, and **Try overshoot**, which presets N₀ above K so you can watch a population fall back toward its ceiling. There is no hidden data: every curve is generated live from the values you set. ## Related microsims - Malthusian growth model — the unlimited exponential this model corrects (the $N \ll K$ limit). - [[Population_growth]] — broader framework of birth, death, and regulation. - Bacterial growth — lag, log, and stationary phases as a real logistic-like curve. - Chemostat — continuous culture where dilution rate fixes a steady-state population. - Invasive species — populations that boom, overshoot, and settle in new ranges. - [[Ecology]] — the discipline where carrying capacity organizes population regulation. - Agar plate — related GENOMICS microsim - Biofilm — related GENOMICS microsim - Bioremediation — related GENOMICS microsim - [[Ecological_footprint]] — related GENOMICS microsim - Enrichment culture — related GENOMICS microsim ## Links (Wikipedia order)  `7_Billion_Actions` · `Abiotic_component` · `Abiotic_stress` · `Abundance_(ecology)` · `Afforestation` · `Agriculture` · 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`World_population` · `World_population_milestones` · `Xerosere` · `Zero_population_growth` ## Overview The concept was formalized by Pierre-François Verhulst in 1838 as the *logistic* correction to unbounded exponential growth. Real populations cannot grow forever: as numbers rise, per-individual access to resources falls, birth rates drop and death rates climb. This **density dependence** is the heart of carrying capacity, and it underpins fisheries management, conservation targets, pest control, and microbial culture. K is not a fixed constant of a species but a property of a species *in an environment* — it shifts when resources, predators, or conditions change. ## The mechanism The continuous logistic equation is $\frac{dN}{dt} = rN\left(1 - \frac{N}{K}\right),$ whose per-capita rate $\frac{1}{N}\frac{dN}{dt} = r\left(1 - N/K\right)$ falls linearly as N rises. Several regimes follow directly: | Condition | Factor $(1-N/K)$ | Behaviour | |---|---|---| | $N \ll K$ | ≈ 1 | near-exponential growth ($dN/dt \approx rN$) | | $N = K/2$ | 0.5 | steepest climb — maximum $dN/dt$ | | $N = K$ | 0 | equilibrium; growth stops | | $N > K$ | < 0 | population declines back toward K | The closed-form solution, $N(t) = K / \left(1 + \frac{K-N_0}{N_0}e^{-rt}\right)$, is the familiar sigmoid. Its inflection point sits at $N = K/2$, which is why the *maximum sustainable yield* of a harvested stock is classically taken at half of carrying capacity. Because K is a **stable equilibrium**, trajectories converge to it from below (growth) and from above (decline) — the behaviour the overshoot button makes visible. ## Controls -> what each maps to | Control | Maps to | Range / values | Meaning | |---|---|---|---| | Growth rate r | intrinsic rate of increase in $rN(1-N/K)$ | 0.02–1.2 | speed of growth when resources are abundant | | Capacity K | carrying capacity (equilibrium) | 10–180 | the population ceiling the environment sustains | | Start pop. N₀ | initial condition $N(0)$ | 1–200 | where the trajectory begins | | Sim speed | animation playback rate | 0.1–4× | speeds/slows the display, not the model | | ⏸ Pause | freeze animation | button | hold the current frame | | ↻ Reset | restart at $t=0$ | button | return N to N₀ | | Try overshoot (N₀>K) | preset N₀ above K | button | start above capacity to watch the decline | ## Learning objective Understand how a single density-dependent term converts unlimited exponential growth into an S-shaped approach to a stable carrying capacity K. ## Limits and connections The classic logistic assumes constant K, instant density feedback, no age structure, and no time lags; adding a lag or discrete time steps can produce overshoot, oscillations, and even chaos not shown here. It also ignores immigration, stochasticity, and interacting species. Even so, it is the minimal model behind chemostat steady states, harvesting theory, and the global debate over the human [[Ecological_footprint|ecological footprint]]. ## Poster & source <div class="microsim-fallback">  <p><em>Live microsim · <a href="https://wikitube-3d-microsims.netlify.app/Carrying_capacity.html">open full</a> · source: Microsims for Dissemination/GENOMICS_ThreeJS_Microsims/Carrying_capacity.html</em></p> </div>  *Built to the [[WT!Three_js_Microsim_Master_Class|three.js Master Class]].*   *Connected to the Apex Spine:* Carrying capacity → [[Ecological_footprint|Ecological footprint]] → [[Hydropower|Hydropower]] — [[WT!Thury_Hydrodynamics_Compendium|Compendium]] section 15, *Hydropower*.  ## Wikipedia : Wikitube **Strict pair:** [Wikipedia](https://en.wikipedia.org/wiki/Carrying_capacity) : [Wikitube](https://en.wikitube.io/wiki/Carrying_capacity) ## Previous hub tags Tree parent: [[System_dynamics]]. Legacy hubs: `GENOMICS`. --- *Sources: 1 legacy note. Minted wave 1, 2026-07-30 (v1.6 order).*