Post on 10-Feb-2021
localbehaviourofbirthanotterprocesses
An birth rate
Mn deathrate
Given Xlt n what isX Ltth forh small
P No events happen betweent and tth t Hutten htoch
Tn n Explintun P Tn he Hutten h
1 AntMan ht 01h
I IPfone birth event happens
in Ltitth duh och
Puente P TnCh Tutti h
gfohldntfne Hnttht.PCTn.ph Hdt
du Joh e HutteriteduntMute h t dt
du Joh r Cdutantttoch lCduntyunakh H tochDdt
Ian Soh DL CLEEK cduqq.eettoChDdt
dnhtoChof we death event in 4th gun ht och
aP More than two events in Ct tthD o.chh Ch Quinn htoChl ht och pinhtoch och
Given Xltl n
xu.tn fitI'ituentis i 1T
n L with prob gunht ochUtz
Iz with proboch
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Jun ngu n 21 dund t 0 neo
So Xctth Xlt Iw p Xctlpehtoch
Xlt Wrp I CXCHHtml to ht och
theatingw
p fitoshtochiQFind Mitt _FIXED
A idea we want to use theequations above to find an
ODE satisfied by ME
Math E XHthD EfECXCtthllXCtD
and E XHth1lXltD Xlt 1 Xltlphtoch Xl 4 1Xlt l o h och Xlt I Xlt
ult 8 ht och och
X h Xltyubtxltfdhtxfttohtxctldhtot.tkCH
NEIGH X t och
Xlt t Xlt Ayu 03h t och
So MltthhM µ f Kyu to t old
o as h so
As h O M Ct kyu Mlt t O 1st order linear ODEa If you havedoneMATH215
DM solve withat AM
0 Integrating factors
It is a so called separable equationVariation of parametersmethod
Mlt t
J adM JDS t us the depends on the initialconditionIa enfants to tt K
Assuming MCO i IL Iaen Caito
so Ia en GMHSto ft LaluLaito
iaenfaa.MY
taMLHto Caito eatMLts iecdmttffyufecd.int y
Remark If 174 then Mlt grows exponentiallyCsingle birth rate single
death rate
If pead then fiveMets
single birth mate singledeath rate
If pi X M Ct Otto
condition on the value of Tn
P Tn Ch TntTnt hJohppft PCTnt Tnt h Tn t dt
Joh guy eHint M p fT tT
dt
Tn andTate areindependent
Joh pin et HM p Tnt h t dt