Population Differential Equations and Laplace Transform

Singapore Maths Tuition

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

$latex N$: Total population

$latex B$: Birth-rate per capita

$latex D$: Death-rate per capita

$latex k=B-D$

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

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

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

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

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

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

Logistic Case 3: Constant population ($latex widehat{N}=N_infty$)
$latex displaystyle N(t)=N_infty$

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

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

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

$latex displaystyle boxed{beta_1,beta_2=frac{Bmpsqrt{B^2-4Es}}{2s}}.$

$latex beta_1$: Unstable equilibrium population

$latex beta_2$: Stable equilibrium population

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

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

Tip: Use this equation when the questions…

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