01. First-Order Equations¶

ToC¶

01.01. The Simplest Example
01.02. The Logistic Population Model
01.03. Constant Harversting and Bifurcations

01.01. The Simplest Example¶

$\frac{dx}{dt} = ax$

$x'(t) = ax(t)$

obtained from calculus
real num $k$ , $x(t) = ke^{at}$ is solution

$x'(t) = ake^{at} = ax(t)$

let $u(t)$ be any solution, compute the derivertiev of $u(t) e^{-at}$

$\begin{align*} \frac{d}{dt} ( u(t)e^{-at} ) &= u'(t) e^{-at} + u(t) (-a e^{-at}) \\ &= a u(t) e^{-at} - a u(t) e^{-at} &= 0 \end{align*}$

initial condition $x(t_0) = u_0$

in the form of an initial valuee problem

$x' = ax, \, x(0) = u_0$

when $k = 0$ , constant solution $x(t) \equiv 0$
equilibrium solution, equilibrium point

if $a > 0$ , $\lim_{t\rightarrow\infty}{ke^{at}} = \begin{cases} \infty &\text{when } k > 0 \\ -\infty &\text{when } k < 0 \end{cases}$
if $a = 0$ , $k e^{at} = \text{constant}$
if $a < 0$ , $\lim_{t \rightarrow \infty} k e^{at} = 0$

source: away from it
sink: toward it

phase plane

bifurcation at $a = 0$

01.02. The Logistic Population Model¶

$x' = ax$ when $a > 0$

Logistic population growth model

$x' = ax (1 - \frac{x}{N})$

$a$ , $N$ : positive param

$x' = f_a(x) = ax(1 - x)$

first order

autnomous: depends on $x$ alone not $t$

$\begin{align*} \int{\frac{dx}{x(1-x)}} &= \int{a} \, dt \\ \int{\Big( \frac{1}{x} + \frac{1}{1-x} \Big)} \, dx &= \int{a} \, dt \\ \log{x} - \log{(1-x)} &= at \\ \log{\frac{x}{1-x}} &= at \\ \frac{x}{1-x} &= e^{at} \\ (1 + e^{at}) x &= e^{at} \\ x &= \frac{e^{at}}{1 + e^{at}} \\ x(t) &= \frac{K e^{at}}{1 + K e^{at}} \end{align*}$

$t = 0$ ,

$K = \frac{x(0)}{1 - x(0)}$

rewrite

$x(t) = \frac{x(0) e^{at}}{1 - x(0) + x(0) e^{at}}$