2.5 Autonomous DEs ,Population Dynamics

Autonmous

Ex1 (Exponential Growth)

$\phi (t)$is the popuation of the given species at time $t$

$$\frac{dy}{dt} = f(y)$$

consider $\frac{dy}{dt} = ry$

• r > 0 : increase
• r < 0 : decline

Assume r > 0 ,and $y(0) = y_0$ ,$\phi(t) = y_0 e^{rt}$

Ex2 (Logistic DEs)

$$r = h(y)$$

Condiser h(y) s.t. h(y) $\cong$r > 0,when y is small

• h(y) decrease as y grows larger
• h(y) < 0 when y is suff large

Assume that $h(y) = r-ay ,\quad a > 0$

Consider logistic equation:

$$\frac{dy}{dt} = (r-ay)y$$ $$\frac{dy}{dt} = r(1- \frac{y}{k})y , \ where \ k =\frac{r}{a}$$

• Find constant solution:

$$\frac{dy}{dy} = 0 \Longleftrightarrow r(1-\frac{y}{k})y = 0$$ $$the\ constant \ solutions \ are \quad y = \phi_1(t) \equiv0 ,\ y = \phi_2 \equiv k$$

• Study for

$$f( y) = r(1-\frac{y}{k})y = -\frac{r}{k}(y - \frac{k}{2})^2 + \frac{rk}{4}$$ $$\frac{dy}{dt} = \begin{cases} > 0 \quad if \ 0 < y < k \\ \quad \\ <0 \quad if \ y > k \\ \end{cases}$$

By Exsistence and Uniqueness Theorem ,no solution can intersect the equilibrium solution $y = k$

• Find $\frac{d^2y}{dt^2}$

$$\boxed{\frac{d^2y}{dt^2}} = \frac{d}{dt}(\frac{dy}{dt}) =\frac{d}{dt}f(y) = f'(y)\frac{dy}{dt} = \boxed{f'(y)f(y)}$$ $$\frac{d^2y}{dt^2} = \begin{cases} > 0 \quad if \ 0 < y < \frac{k}{2}\ or \ y > k \\ \quad \\ <0 \quad if \ \frac{k}{2}<y < k \\ \end{cases}$$

Note : $y = \frac{k}{2}$is an inflation points