Mô hình Logistic với Phương trình vi phân

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

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.

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.