Unconstrained Growth and Decay

CS 170:Computing for the Sciences

and Mathematics

Administrivia

Last time Error and Basic differential calculus Assigned HW 2

Today HW2 due! Unconstrained Growth

WE HAVE CLASS ON MONDAY

Unconstrained Growth

Population growth without constraints

Examples?

Example of Unconstrained Growth

Rate of change of population is directly proportional to number of individuals in the population (P)

dP/dt = rP

where r is the growth rate.

Analytic Solution

“Closed Form” solution

Can determine with a computer algebra system Like Maple

P = P0ert

Exponential Decay

Rate of change of mass of radioactive substance proportional to mass of substance

Constant of proportionality(rate) is negativeRadioactive Carbon-14: -0.000120968

(about .0120968% per year)

dQ/dt = -0.000120968 Q

Q = Q0 e-0.000120968t

Why Carbon-14?

Where’s The Computation?

An analytic solution is always preferable! But…finding it can be very hard

Instead of solving the relationship, we’ll approximate it.

Finite difference equation

new = old + change

population(t) = population(t - ∆t) + ∆population

If I repeat this calculation a lot (moving the time up a bit each pass), I can see the trend of population over time

Approximating Unconstrained Growth

initialize simulationLength, population, growthRate, ∆t

numIterations simulationLength / ∆tfor i going from 1 to numIterations do the following:

growth growthRate * populationpopulation population + growth * ∆tt i * ∆tdisplay t, growth, and population

UNITS ARE IMPORTANTDoes this give me the exact answer?

Systems Dynamics Tool

Helps to model

Performs simulation

What happens at one time step influences what happens at next

Stock/Box Variable/Reservoir

Anything that accumulates, buffer, resourceExamples

Population Radioactivity Phosphate Body fat Labor

Flow

Represents activitiesExamples

Birthing, dying with population Intaking & expending calories with body fat Moving from one population to another Diffusion Reactions

Converter/Variable/Formula

Contains equations that generate output for each time period

Converts inputs into outputsTakes in information & transforms for use by

another variableExamples

Growth rate with population & growth Calories in a food Rates of reaction/diffusion

Connector/Arrow/Arc

LinkTransmits information & inputsRegulates flows

Shows dependence

With system dynamics tool

Enter equationsRun simulationsProduce graphsProduce tables

HOMEWORK!

READ Module 3.2 in the textbook

YES CLASS on Monday