2.2 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) or Differential 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

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 2 Consider the differential equation dQ/dt = Q, with Q 0 = 200. a.Using delta notation, give a finite difference equation corresponding to the differential equation. a.At time t = 9.0 sec, give the time at the previous time step, where  t = 0.5 sec. a.If Q(t-  t) = and Q(t) = , 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 Table Table of Estimated Populations, Where the Initial Population is 100, the Continuous Growth Rate is 10% per Hour, and the Time Step is hr ___________________________________________________________________________ tpopulation(t)= population(t-  t)+(growth)*  t = *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 )

Algorithm 1: Unconstrained Growth initialize simulationLength initialize population initialize growthRate initialize length of time step  t numIterations  simulationLength /  t for i going from 1 through numIterations do: growth  growthRate * population population  population + growth*  t t  i*  t display 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 simulationLength initialize population initialize growthRate initialize length of time step  t numIterations  simulationLength /  t growthRatePerStep  growthRate *  t for i going from 1 through numIterations do: population  population + growthRatePerStep* population t  i*  t display 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 P 0 = 100 is P = 100e 0.10t 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 : Some tools use log for ln Don’t forget to add constant C Solve for P using algebra + fact that e ln(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 P 0 is

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)

Exponential Decay Example Radium-226 has a continuous decay rate of about % per year. Determine its half-life in whole years.

Exponential Decay Example Radium-226 has a continuous decay rate of about % per year. Determine its half-life in whole years. Answer: 1620 years