Population Differential Equations and Laplace Transform

Malthus Model
$\displaystyle \frac{dN}{dt}=BN-DN=kN$

$N$ : Total population

$B$ : Birth-rate per capita

$D$ : Death-rate per capita

$k=B-D$

Solution to D.E.:
$\displaystyle \boxed{N(t)=\widehat{N}e^{kt}},$

where $\widehat{N}=N(0)$ .

Logistic Equation
$\begin{aligned} D&=sN\\ \frac{dN}{dt}&=BN-sN^2\\ \widehat{N}&=N(0)\\ N_\infty&=B/s \end{aligned}$

Logistic Case 1: Increasing population ( $\widehat{N}<N_\infty$ )
$\begin{aligned} N(t)&=\frac{B}{s+(\frac{B}{\widehat{N}}-s)e^{-Bt}}\\ &=\frac{N_\infty}{1+(\frac{N_\infty}{\widehat{N}}-1)e^{-Bt}} \end{aligned}$

The second expression can be derived from the first: divide by $s$ in both the numerator and denominator.

Logistic Case 2: Decreasing population ( $\widehat{N}>N_\infty$ )
$\begin{aligned} N(t)&=\frac{B}{s-(s-\frac{B}{\widehat{N}})e^{-Bt}}\\ &=\frac{N_\infty}{1-(1-\frac{N_\infty}{\widehat{N}})e^{-Bt}} \end{aligned}$

Logistic Case 3: Constant population ( $\widehat{N}=N_\infty$ )
$\displaystyle N(t)=N_\infty$

Harvesting
Basic Harvesting Model: $\displaystyle \boxed{\frac{dN}{dt}=(B-sN)N-E}.$

$E$ : Harvest rate (Amount harvested per unit time)

Maximum harvest rate without causing extinction: $\boxed{\dfrac{B^2}{4s}}$ .

$\displaystyle \boxed{\beta_1,\beta_2=\frac{B\mp\sqrt{B^2-4Es}}{2s}}.$

$\beta_1$ : Unstable equilibrium population

$\beta_2$ : Stable equilibrium population

Extinction Time: $\displaystyle \boxed{T=\int_{\widehat{N}}^0\frac{dN}{N(B-sN)-E}}.$

Laplace transform of $f$
$\displaystyle F(s)=L(f)=\int_0^\infty e^{-st}f(t)\,dt$

Tip: Use this equation when the questions contains the words “show from the definition”.

Inverse transform of $F(s)$
$\displaystyle f(t)=L^{-1}(F(s))$

Linearity
$\begin{aligned} L(af(t)+bg(t))&=aL(f)+bL(g)\\ L^{-1}(aF(s)+bG(s))&=aL^{-1}(F)+bL^{-1}(g) \end{aligned}$

List of common Laplace Transforms

$\begin{aligned} L(e^{at})&=\frac{1}{s-a}\\ L(1)&=\frac{1}{s}\\ L(\cos wt)&=\frac{s}{s^2+w^2}\\ L(\sin wt)&=\frac{w}{s^2+w^2}\\ L(t^n)&=\frac{n!}{s^{n+1}}\\ L(f')&=sL(f)-f(0)\\ L(f'')&=s^2L(f)-sf(0)-f'(0)\\ L(f^{(n)})&=s^nL(f)-s^{n-1}f(0)\\ &\quad -s^{n-2}f'(0)-\dots-f^{(n-1)}(0)\\ L\left(\int_0^t f(\tau)\,d\tau\right)&=\frac{1}{s}L(f) \end{aligned}$

$s$ -shifting
If $L(f)=F(s)$ , $s>a$ , then $\displaystyle \boxed{L(e^{ct}f(t))=F(s-c)},$
$s-c>a$ .

Tip: Use this when doing Laplace Transform of a function with an exponential factor $e^{ct}$ . Note that the reverse direction can sometimes be used as well: $\displaystyle L^{-1}[F(s-c)]=e^{ct}f(t).$

$t$ -shifting
If $L(f(t))=F(s)$ , then $\displaystyle \boxed{L(f(t-a)u(t-a))=e^{-as}F(s)}.$

Tip: Frequently, we use the reverse direction $\displaystyle L^{-1}[e^{-as}F(s)]=f(t-a)u(t-a).$

Delta function
$\delta(t)$ : infinitely tall and narrow spike at $t=0$ .

$\delta(t-a)$ : infinitely tall and narrow spike at $t=a$ .

$\boxed{L[\delta(t-a)]=e^{-as}}$

Two properties of delta function
$\begin{aligned} \int_0^\infty\delta(t-a)\,dt&=1\\ \int_0^\infty \delta(t-a)g(t)\,dt&=g(a) \end{aligned}$
for $a\geq 0$ .