3.2 Unconstrained Growth

Malthusian Population Model

The power of population is indefinitely greater than the power in the earth to produce subsistence for man – T. Malthus

Mathematically:

Thomas Malthus(1766 – 1834)

orDifferential equation

Malthusian Population Model

(instantaneous, continuous) growth rate;constant of proportionality

r = 0.1

Initial condition:P(0) = 100

Finite Difference Equation

• In a system-dynamics tool like Vensim (or in a computer program), we simulate continuous time via small, discrete steps.

• So instead of using dP/dt for growth, we have P/t:

• Then solve for population(t) ….

Finite Difference Equation

Finite Difference Equation

• In other words:new_population = old_population + change_in_population

• In general, a finite difference equation has the form new_value = old_value + change_in_value

• Such an equation is an approximation to a differential equation (equal in the limit as t approaches 0)

Quick Review Question 1• Consider the differential equation dQ/dt = -0.0004Q, with Q0 = 200.

a. Using delta notation, give a finite difference equation corresponding to the differential equation.

b. At time t = 9.0 sec, give the time at the previous time step, where t = 0.5 sec.

c. If Q(t-t) = 199.32 and Q(t) = 199.28, give Q.

Quick Review Question 2• Evaluate to six decimal places population(.045), the population at the next time interval after the end of Table 3.2.1.

Table 3.2.1Table of Estimated Populations, Where the Initial Population is 100, the Continuous Growth Rate is 10% per Hour, and the Time Step is 0.005 hr___________________________________________________________________________t population(t) = population(t-t) + (growth) * t0.000 100.0000000.005 100.050000 = 100.050000 + 10.000000 * 0.005

• Let’s do it in Excel….

Simulation Algorithms: Background

• An algorithm is an explicit step-by-step procedure for solving a problem.

• Basic building blocks are

• Sequencing (one instruction after another)

• Conditionals (IF … THEN … ELSE)

• Looping (For 100 steps, do the following:)

• Assignment statements use left arrows: x x + 1

Al-Khwarizmi(ca. 780-850)

Algorithm 1: Unconstrained Growth

initialize simulationLengthinitialize populationinitialize growthRateinitialize length of time step tnumIterations simulationLength / tfor i going from 1 through numIterations do:

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

Removing Loop Invariants

• If we don’t need to display growth, we can remove the implicit, loop-invariant product growthRate * t used to compute population:

population population + growthRate* t* population

Removing Loop Invariants• Then we save time by computing growthRate * t just once, before the loop:

initialize simulationLengthinitialize populationinitialize growthRateinitialize length of time step tnumIterations simulationLength / tgrowthRatePerStep growthRate * tfor i going from 1 through numIterations do:

population population + growthRatePerStep* populationt i* tdisplay t and population

Analytical Solutions• Some problems can be solved analytically, without simulation

• For example, calculus tells us that the solution to dP/dt = 0.10P with initial condition P0 = 100 is P = 100e0.10t

• If such solutions exist, we should use them. But the point of modeling a system is usually that no analytical solution exists.

Analytical Solution via Indefinite Integrals

• Separation of variables: move dependent variable (P(t)) and independent variable (t) to opposite sides of equal sign:

• Then integrate both sides:

Analytical Solution via Indefinite Integrals

• Solve the integral, using e.g. a free online tool like http://integrals.wolfram.com :

• Some tools use log for ln• Don’t forget to add constant C

• Solve for P using algebra + fact that eln(x) = x

Completion of Analytical Solution

• Need constant k in

• We know P = 100 at t = 0, so

• So analytical solution is

General Solution to Differential Equation for Unconstrained Growth

In general, the solution to

with initial population P0

is

Quick Review Question 3• Give the solution to the differential equation

where P0 = 57

Unconstrained Decay• For some systems r is negative

• E.g., radioactive decay of carbon-14:

Unconstrained Decay

half-life (time to decay to half original amount)

Quick Review Question 4• Radium-226 has a continuous decay rate of about 0.0427869% per year. Determine its half-life in whole years.

Quick Review Question 4• Radium-226 has a continuous decay rate of about 0.0427869% per year. Determine its half-life in whole years.

• Answer: 1620 years