Logistic Models with Differential Equations
Introduced| English | Meaning |
|---|---|
| logistic/ləˈdʒɪstɪk/ | logistic |
| carrying capacity/ˈkæriɪŋ kəˈpæsɪti/ | The largest population an environment can support sustainably under the prevailing conditions. |
Growth that hits a ceiling
- Pure exponential growth rises forever — unrealistic for real populations with limited resources.
- The logistic 逻辑斯蒂 model fixes this: growth that slows as it nears a maximum.
- It starts nearly exponential, then levels off toward a carrying capacity 环境容纳量 $L$.
- It's the standard model for constrained growth.
The logistic differential equation
-
$$\dfrac{dP}{dt}=kP\Big(1-\dfrac{P}{L}\Big)$$
- The $kP$ part is exponential-like growth; the $\big(1-\tfrac{P}{L}\big)$ factor brakes it as $P$ nears $L$.
- When $P$ is small, $1-\tfrac{P}{L}\approx1$: nearly exponential. When $P\approx L$, the factor $\to0$: growth stops.
- $L$ is the carrying capacity — the population's ceiling.
Logistic growth: nearly exponential at first then levelling off at the carrying capacity L
The logistic differential equation is $\dfrac{dP}{dt}=$
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The braking factor $(1-P/L)$ slows growth near $L$.
The maximum sustainable value $L$ that $P$ approaches is the ____ capacity.
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The carrying capacity is the ceiling.
Long-term behavior
- As $t\to\infty$, $P$ approaches the carrying capacity $L$ (for $0<P_0<L$).
- $P=L$ is a stable equilibrium; $P=0$ is an unstable one.
- Below $L$ the population rises toward $L$; above $L$ it falls back to $L$.
- So $L$ is the value the model settles at, regardless of the (positive) start.
The S-shaped logistic curve
y = a·tanh(bx) + d (S-shape)
A logistic curve rises fast near the middle and flattens toward the carrying capacity $L$ — steepest at $P=L/2$.
For $\dfrac{dP}{dt}=0.1P(1-\tfrac{P}{2000})$, what is the carrying capacity $L$?
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$L=2000$, the long-term limit.
As $t\to\infty$ (with $0<P_0<L$), the population $P$ approaches...
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$P\to L$ from below.
Fastest growth at half capacity
- The population grows fastest when $P=\dfrac{L}{2}$ — the midpoint of the S-curve.
- There the logistic curve has its inflection point: growth accelerates below $L/2$, decelerates above.
- Before $L/2$ the rate is climbing; after, it's easing off toward zero.
- This half-capacity peak is a favorite exam fact.
For the same model ($L=2000$), at what population is growth fastest?
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Fastest at $P=L/2=1000$.
Growth is fastest exactly at the carrying capacity $P=L$.
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At $P=L$ growth stops; the fastest rate is at $P=L/2$.
The carrying capacity $L$ is where growth stops ($\frac{dP}{dt}=0$ because the $\big(1-\tfrac{P}{L}\big)$ factor is $0$), not where it's fastest. The maximum rate is at $P=\tfrac{L}{2}$. Don't confuse the two: $L$ is the ceiling the population approaches; $\tfrac L2$ is where it's climbing quickest.
A fish population obeys $\dfrac{dP}{dt}=0.1P\big(1-\tfrac{P}{2000}\big)$.
- Carrying capacity: $L=2000$ fish — the long-term limit.
- Fastest growth at $P=\tfrac{L}{2}=1000$ fish.
- As $t\to\infty$, $P\to 2000$ from below (if it started under $2000$).
The logistic model $\frac{dP}{dt}=kP\big(1-\frac{P}{L}\big)$ describes growth limited by a carrying capacity $L$: near-exponential when small, leveling off toward $L$ as $t\to\infty$. Growth is fastest at $P=\tfrac L2$ (the inflection point), and stops at $P=L$.