Thenbackin the first equation, 10000 =P 0 (2). SoP 0 = 5000.Thereare 5000initialimmigrants.
Example 2 (Logistic Growth) The population on an island is given by the equation
,P 0 = 1000.Find the populationsizesP(20),P(30).At whattime will the
populationfirst exceed4000?
Solution.The solutionis givenby where.
Solvefor time, gives. Sot= 56. The population
first exceed 4000 in the 56thyear.
Exercise
- (ExponentialGrowth)The populationof a suburbancity increasedfrom 10000in 2005to 30000in 2007.
Assumingan exponentialgrowthmodelon the population,by whichyear will the populationfirst exceeds
100000? - (LogisticGrowth)The populationof a city is givenby the equation P 0
= 25000.Find the populationsizesP(10),P(25).At whattime will the populationfirst exceed90000?
NumericalMethods(Euler's,ImprovedEuler, Runge-Kutta)
The Euler'smethodis a numericalapproximationto a solutioncurvestartingfrom the point(a,b) through
the algorithm:
where and is the step size.
The shorterstep size, the betteris the approximationto the solutioncurve.
Improved Euler (Heun) method adapts on Euler's method by using both end point values:
Since also appearson the right side,we replaceit by Euler'sformula,