When N≪K, (1-N/K)≈1 and growth is exponential. As N→K, dN/dt→0 and growth stops. The growth rate dN/dt is greatest at N=K/2, the inflection point.
Lotka-Volterra (Predator-Prey) Equations
Prey: $\dfrac{dN}{dt} = \alpha N - \beta NP$
Predator: $\dfrac{dP}{dt} = \delta NP - \gamma P$
The solutions produce periodic oscillations with offset phases.
FAQ
How does logistic growth differ from exponential growth?
Exponential growth (dN/dt=rN) assumes unlimited resources. Logistic growth slows as population approaches carrying capacity K, producing an S-shaped curve.
What is carrying capacity K?
It is the maximum population an environment can sustain long-term, determined by food, space, predation and other limiting factors.
Can a population overshoot and crash?
With high r values in discrete models, oscillations or chaos occur. This continuous model stabilizes near K without overshoot.
Does human population follow logistic growth?
In the long term, technological and resource limits suggest convergence toward K, but estimating a global carrying capacity remains hotly debated.
🙋
I can see the simulation updating, but what exactly is being calculated here?
🎓
Great question! The simulator solves the governing equations in real time as you move the sliders. Each parameter you control directly affects the physical outcome you see in the graph. The key is to build an intuitive feel for how each variable influences the result — that's how engineers develop physical judgment.
🙋
So when I increase this parameter, the curve shifts significantly. Is that a linear relationship?
🎓
It depends on the model. Some relationships are linear, but many engineering phenomena are nonlinear. Try moving the sliders to extreme values and see if the output changes proportionally — if the graph shape changes, that's a sign of nonlinearity. This hands-on exploration is exactly what simulations are best for.
🙋
Where is this kind of analysis actually used in practice?
🎓
Constantly! Engineers run these calculations during the design phase to quickly screen parameters before investing in expensive physical tests or detailed finite element simulations. Getting comfortable with these simplified models is a real engineering skill.
Enter initial population (N0) in the sN0Num field—typical values range from 10 to 10,000 individuals depending on species
Set carrying capacity (K) in the field—this represents maximum sustainable population given resource constraints
Adjust intrinsic growth rate (R) using sRNum, typically 0.1 to 2.0 for ecological systems
Execute simulation to observe population trajectory over time intervals
Analyze stabilization behavior when dN/dt approaches zero at equilibrium
Worked Example
A fishery models cod stock with N0=500 individuals, K=8000 fish (sustainable harvest level), and R=1.2 (annual growth rate). The logistic equation dN/dt=RN(1−N/K) predicts rapid growth for 3–4 years reaching approximately 6500 individuals, then asymptotic approach to 8000 as resource limitation intensifies. At year 10, population stabilizes at 7980 fish with growth rate under 0.5% annually, enabling sustainable fishing quotas of 150–200 fish/year without destabilizing equilibrium.
Practical Notes
Overshoot occurs when R exceeds 2.0—population oscillates above K before crashing; reduce growth rate or increase carrying capacity estimates
For invasive species management, lower K values (30–50% baseline) simulate culling interventions and predict minimum population thresholds
Allee effect (instability at low N) not captured here; verify real populations remain above 5–10% of K for model validity
Carrying capacity changes seasonally in agriculture; rerun quarterly simulations with updated K for accuracy