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.

Tip: Use delta function when the keywords “suddenly”, “burst”, etc. appear.

Unit step function
\displaystyle u(t-a)=\begin{cases}  0, &t<a\\  1, &t>a.  \end{cases}

For 0<a<b, \displaystyle u(t-a)-u(t-b)=\begin{cases}  0, &t<a\\  1, &a<t<b\\  0, &t>b.  \end{cases}

Tip: Use unit step function for questions that require a force to “switch on / switch off” at certain times.

\displaystyle \boxed{L(u(t-a))=\frac{e^{-as}}{s}}

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