Mat 3 Lecture Notes Modeling Warren Weckesser Department of Matematics Colgate University 5 7 January 006 Classifying Matematical Models An Example We consider te following scenario. During a storm, a large tree wit several mice is blown into te ocean. Te storm carries te tree many miles until it wases asore on an island tat, until now, as ad no mice. Tis island as many seed-bearing plants tat mice love, and a nice climate, so te mice ave a good cance to survive and prosper. How will te population of mice on tis island cange over time? For simplicity, we will consider just te population of female mice. We will also assume tat a new generation is produced eac year. We begin by making te following assumptions. In eac generation, eac female produces tree offspring (along wit some number of males).. Te offspring can reproduce after one year. 3. Mice live forever. Clearly tese are not realistic assumptions. We will accept tem for now, in order to develop a simple model. Later we will look at some more realistic variations. To start, we will work wit discrete time. Let p(n) be te population at te end of te nt year, were n is an integer. Wit tis notation, p(0) is te initial population. Let s suppose tat p(0) = ; tat is, te initial population contained just one female mouse. At te end of te first year, tis mouse as produced tree female offspring, so p() =. At te end of te second year, eac of te four mice as produced tree more offspring, so p() = 6. In general, we ave p(n + ) = p(n) () Equation () is te rule tat describes ow te population canges over time. Suc a rule involving discrete time is sometimes called a difference equation. Tis terminology is a bit clearer if we rewrite te equation as p(n + ) p(n) = 3p(n) () Tis gives te rule for computing te difference between successive generations. We can easily verify tat () as te solution p(n) = p(0) n (3)
Tis is a solution in te sense tat te population at time n is given directly as a function of n and te initial population. If we plot tis solution, we will see a stair-step plot, wit te size of te steps getting larger as n increases. If all te mice produce teir offspring at exactly te same time, ten tis stair-step sape is reasonable. But we don t really expect tat to be te case. Presumably mice will be born trougout te year, and we expect te actual grap of te population to ave many smaller steps. In fact, wen te population is large (and if we blur our vision a bit), we migt expect te grap to look like a smoot curve. Let p(t) be te population at time t, were now t is a real number. Wat matematical rule is obeyed by p(t)? If we still believe our assumptions, we still expect tat in one year, te population increases four-fold. Tat is, we still ave p(t + ) = p(t) () Wat is te corresponding rule for increments of time less tan one year? Tat is, wat can we say about p(t + ), p(t + 3 ), or in general, p(t + )? I claim tat te correct rule is p(t + ) = p(t) (5) If = 0, we obtain p(t) = p(t), as we sould, and if =, we obtain (). If = /k, were k is an integer, we ave p(t + ) = p(t + k + k = /k p(t + k = /k p(t + k. = k k p(t + = p(t) so te repeated application of (5) wit = /k also agrees wit (). We now subtract p(t) from bot sides of (5), and divide by : ( p(t + ) p(t) = p(t) p(t) ) = p(t). (7) Take te limit 0. On te left we obtain p (t). On te rigt, we apply L Hopital s Rule (and recall tat d dx [ax ] = ln(a)a x ) to obtain ln()p(t). Tus we ave (6) p (t) = ln()p(t) (8) Tis is a differential equation. Tis equation says tat te instantaneous rate of cange of te population at time t is proportional to te population at time t; te proportionality constant is ln(). Equations () and (8) bot give rules for determining te population. Te first is a discrete time model, and te second is a continuous time model. Tis distinction is one of te fundamental categorizations of models.
We now consider a more complicated discrete model, in wic we no longer assume tat te mice live forever. Suppose te mice only live tree years. Moreover, eac female mouse produces two female offspring during its second year and its tird year. At time n, we need tree quantities to describe te state of te population. We define p 0 (n) is te number of new female offspring in year n; p (n) is te number of one-year-old females in year n; and p (n) is te number of two-year-old females in year n. Ten we ave p 0 (n + ) = p (n) + p (n) p (n + ) = p 0 (n) p (n + ) = p (n) Suppose tat in te inital population, p 0 (0) = 0, p (0) = and p (0) = 0. Let s compute te population for a few generations: n p 0 (n) p (n) p (n) Total 0 0 0 0 6 0 8 3 8 6 6 8 8 We appear to ave a growing population, but unlike te simpler model, a formula for te solution is not obvious. (We will see ow to solve a problem like tis later in te course.) A key observation to make about te model is tat te state of te population is tree dimensional. In order to write down te rules tat determine ow te population canges, we needed to keep track of tree quantities. We can put tese in a vector: p 0 (n) x(n) = p (n) (0) p (n) Ten te rules given in (9) can be written more concisely as (9) x(n + ) = f( x(n)) () were f is te vector-valued function (or map, or mapping) given by te rigt side of (9). Equation () is a general form for multi-dimensional discrete time models. We will often call suc an equation an iterated map. (Just like te one-dimensional case, tese are also often called difference equations.) We will also study multi-dimensional systems of differential equations: were x (t) = f( x(t)) () x (t) x x (t) (t) x(t) =. and x x (t) = (t). x m (t) x m(t) (3) 3
Five Instances of a Stocastic Model 0 Population 8 6 0 0 3 5 Year Figure : Five instances of te population in a stocastic model. Deterministic vs. Stocastic Models. We ave one more important categorization to discuss. Bot () and () are deterministic. Tat is, for a given starting state (e.g. x(0)), te fate of te population is determined; te equations produce only one possible solution. Tere are no random events incorporated in te model. Models tat explicitly include random events are called stocastic. For example, suppose we model a population of mice wit te rule tat in eac year, tere is a one in ten cance tat a mouse will die, and a one in two cance tat te mouse will produce one offspring. (Tis means tere is a four in ten cance tat te mouse will not die and will not produce an offspring.) In suc a model, if te initial population is, ten in te next year te population could be 0,, or. Te following year it could be 0,,, 3 or. Figure sows five instances of te population for tis model. In all five cases, te initial population is. In suc models, te question tat we ask is not Wat is te population in year n? Rater, we usually ask Wat is te probability distribution of te population in year n? If we ave te probability distribution at year n, we can ten answer questions suc as Wat is te probability tat te population is zero? or Wat is te probability tat te population is greater tan 500 in year 0? Figure sows a numerically computed distribution of te population after five years. Tis was computed by running 50000 simulations, and adding up te number of times eac possible final population occurred. For example, in te figure we see tat approximately 3.6% of te time, te population is zero after five years.
7 Results from 50000 trials. 6 Percent Occurrences 5 3 0 0 5 0 5 0 5 30 35 0 Final Population Figure : Numerically computed population probability distribution for te population after five years. Tis was computed by tallying te results of 50000 individual simulations. 5