ࡱ> lnkz3 WHjbjb^^ h<h<?lbbbb8D b.(:::, !-:@::.:J.: &VDؐbb  !!Part iI. Basic procedures in systems modeling BUILDING Blocks for Systems Models Quote of the Day To do science is to search for repeated patterns, not simply to accumulate facts..." - Robert H. MacArthur (1972) I. Why Building Blocks? Building blocks, or simulation modules are simple models that represent some basic system structures and dynamics. These modules are very important for understanding many fundamental processes common in biological, physical, and socioeconomic phenomena. One certainly needs to understand them well before attempting to deal with complex feedback systems. In the same time, the model building blocks demonstrate how these commonly occurring basic processes are represented in systems simulation, and often become convenient and effective for constructing complex systems models. Linear Growth in STELLA  X(t) = X(t - dt) + (dX_dt) * dt INIT X = 0 INFLOWS: dX_dt = 2 II. Building Blocks for Systems Models (I) First-Order Systems First-order: One state variable (or 1 stock) Linear: No non-linear combinations of the state variable of any sort in the algebraic equation of rates. 1. Linear Change One state variable. One rate variable.  The rate of change is a constant, not dependent on the state variable. Thus, there is no feedback loop. (1) Linear Growth The rate of change is positive, and thus represented as an inflow. Examples: Increase in water level in a tank with a constant inflow of water into the tank. Plant biomass sometimes is found to increase linearly with certain soil nutrients. (2) Linear Decline The rate of change is negative, and thus represented as an outflow. Linear Decline in STELLA  X(t) = X(t - dt) + (- dX_dt) * dt INIT X = 40 OUTFLOWS: dX_dt = 2 Examples: Decrease in the amount of materials left in a stock when the materials are taken out of the stock in the same amount per unit of time. Decrease in crop yield with declining annual precipitation sometimes exhibits linear pattern. (3) Linear Growth and Decline Combination of the linear growth and decline module.  One state variable. Either two uniflows (one inflow and one outflow) or one biflow (compare the structural diagrams below). The uniflow version gives more details, whereas the biflow version is more concise. The choice of the two forms is dependent on the degree of details is needed. Linear growth occurs when the total rate of change (inflow minus outflow) is positive, and linear decline does otherwise. Linear Growth and Decline in STELLA (1) Two-uniflow model: (2) One-biflow model:   X(t) = X(t - dt) + Y(t) = Y(t - dt) + (dX_dt_in - dX_dt_out) * dt (dY_dt) * dt INIT X = 0 INIT Y = 40 INFLOWS: INFLOWS: dX_dt_in = 2 dY_dt = -1 OUTFLOWS: dX_dt_out = 1 2. Exponential Growth and Decay Exponential Growth in STELLA  N(t) = N(t - dt) + (dN_dt) * dt INIT N = 2 INFLOWS: dN_dt = r*N r = 0.1 (1) Exponential Growth Also known as compounding process One state variable. One rate variable. One auxiliary variable. The rate of change is equal to a constant proportion of the state variable. Because the rate variable depends on the state variable itself, they form a positive feedback loop, in which the state variable and the rate of change reinforce each other, generating an accelerating, run-away behavior.  The rate of change is always positive (i.e., the state variable increases monotonically), and is represented it as an inflow. Time Constant Tc: The reciprocal of the constant r in the exponential model, Tc = 1/r, with the dimension of units of time. For exponential growth, Tc is the time required for the state variable to become e (= 2.7183) times of its current value. This can be seen from the following simple manipulations:  EMBED Equation.2 . When t is equal to Tc, we have   EMBED Equation.2  In general, if t = nTc,  EMBED Equation.2  Note: For continuous systems, the time step, "t, in simulation must be smaller than the smallest time constant in the model. Doubling Time Td: The time required for the state variable in the exponential model to double in value. To calculate the doubling time, let N = 2N0 and let t = Td. Thus, from  EMBED Equation.2 , we have:   EMBED Equation.2  Therefore, the doubling time for exponential growth is a constant. It is conversely proportional to the rate of change, and is about 70% of the time constant. This is where the rule of 70 in population demography comes from (i.e., population doubling time = 70 / percent natural growth rate).  Exponential growth is a common type of dynamics that exists in all different disciplines. Any phenomena described by words such as snowball effect, vicious circles, virtuous circles, and bandwagon effect can be represented as a positive feedback loop structure. Examples: Savings increase in a bank account due to interest income Population growth when resources are not limiting. Drug addiction. Arms race (proliferation). Superexponential (or supraexponential) growth: In many positive feedback systems, doubling time decreases, rather than remain constant, as the state variable increases. This means that the rate of change is a nonlinear function of the state variable (see Figure). For example,  N(t) = N(t - dt) + (dN_dt) * dt INIT N = 10 INFLOWS: dN_dt = r*N Super_N(t) = Super_N(t - dt) + (dN_dt_super) * dt INIT Super_N = 10 INFLOWS: dN_dt_super = (r_2/10)*Super_N^2 r = 0.1 r_2 = if (time<10) then 0.1 else 0  EMBED Equation.2  The dynamics generated by this nonlinear system, with a rate of change faster than a fixed proportion of the state variable, is called superexponential growth. The superexponential growth has the same feedback structure as the exponential. (2) Exponential Declay Also known as draining process One state variable. One rate variable. The rate of change is equal to a constant proportion of the state variable, but in contrast with exponential growth it is negative. Exponential Decay in STELLA  N(t) = N(t - dt) + (- dN_dt) * dt INIT N = 100 OUTFLOWS: dN_dt = r*N r = 0.1 The rate variable depends on the state variable itself, and form a feedback loop between them. Because an increase in the value of the state variable increases the rate of change, but an increase in the rate decreases the value of the state variable, the feedback is negative (or goal-seeking). Time constant Tc: The reciprocal of the constant r in the exponential model, Tc = 1/r, with the dimension of units of time. For exponential decay, Tc is the time required for the state variable to become e -1 (=  EMBED Equation.2 ) times of its current value, or the time required for 63% of the contents in the stock to vanish. This is also called the relaxation time, which is sometimes used as a measure of the speed with which a system is absorbing disturbances. Mathematically, relaxation time can be derived as follows:   EMBED Equation.2 , For a time period from t=0 to t=Tc,  EMBED Equation.2 , In general, when t = nTc, we have  EMBED Equation.2  Half-life Th: The time required for the state variable to reduce its current value by a half. It is an analog of the doubling time in exponential growth. It can be calculated as follows:  EMBED Equation.2 ,  EMBED Equation.2 , thus,  EMBED Equation.2  Therefore, the half-life time for exponential decay is a constant (about 70% of the time constant), independent of the value of the state variable. Examples: Draining water from a tank Population-death rate process Redioactive decay process (3) Exponential Growth and Decay Combined Simply a combination of exponential growth and exponential decay. The behavior of the first-order linear system exhibits three different patterns: exponential growth when growth constant is larger than decay constant; exponential decay when growth constant is smaller than decay constant; and remaining unchanged (unstable equilibrium) when the two constants are equal. Exponential Growth/Decline in STELLA  N(t) = N(t - dt) + (inflow - outflow) * dt INIT N = 150 INFLOWS: inflow = growth_constant*N OUTFLOWS: outflow = decay_constant*N decay_constant = 0.1 growth_constant = 0.1  Exponential Collapse in STELLA  level(t) = level(t - dt) + (- outflow) * dt INIT level = 99 OUTFLOWS: outflow = collapse_constant*(M-level) collapse_constant = 0.15 M = 100 3. Exponential Collapse Also known as accelerated decay, indicating that the rate of change gets faster as the level goes down. A simple exponential collapse model may consist of 1 state variable, 1 outflow, and 2 constants. A positive feedback: increasing level --> increasing rate of change --> increasing level --> ... A simple mathematical model for exponential collapse:  EMBED Equation.2  where N (< M) is the state variable, and r and M are two constants. Examples: Population shrinking when smaller than MVP. Change in the interactive force between molecules when water is heated up and boils.  Decline of species diversity with increasing habitat fragmentation. Some other threshold phenomena in physical and biological processes may exhibit behavior that resembles exponential collapse. 4. Exponential Growth and Collapse Combined  A simple exponential collapse consists of 1 state variable, 1 bi-directional flow, and two constants. This simple model can be mathematically expressed as:  EMBED Equation.2  The simple structure exhibits 3 different behavioral patterns: Exponential growth when N > M; Exponential collapse when N < M; and Remaining constant when N = M. Exponential Growth/Collapse in STELLA  level(t) = level(t - dt) + (- outflow)*dt INIT level = 99 OUTFLOWS: outflow = collapse_constant*(M-level) collapse_constant = 0.15 M = 100  5. Goal-Seeking Behavior Simple Goal-Seeking in STELLA  level(t) = level(t - dt) + (Inflow) * dt INIT level = 2 INFLOWS: Inflow = constant*discrepancy constant = 0.1 discrepancy = goal-level goal = 100 (1) Simple Goal-Seeking A simple goal-seeking model may consist of 1 state variable, 1 biflow, and 2 constants. The state variable and the flow form a negative feedback. A simple goal-seeking model may take the form:  EMBED Equation.2  where L is the level, G is the goal, and c is a constant. Apparently, exponential decay is a special case of goal-seeking behavior, in which the goal is zero! Question: What is the time constant Tc for this simple model?  (2) S-Shaped Growth Also known as logistic growth or sigmoidal growth. It is a first-order nonlinear system (see the Rate-Level graphs). The state variable and the flow form a negative feedback that ultimately gives rise to the goal seeking behavior (see the Forrester diagram).  level(t) = level(t - dt) + (Inflow) * dt INIT level = 300 INFLOWS: Inflow = Rate_Value Rate_Value = GRAPH(level)... It may take different forms (uniflow or biflow versions). [For example, in population regulation, crowding may affect only the per capita birth rate, or only the per capita death rate, or both.]  Rate-Level Graph The simple structure that generates S-shaped dynamics may look like this: The behavior of this simple structure has three patterns: Increasing and approaching the goal if simulation starts with a value of the level smaller than the goal; Decreasing and approaching the goal if simulation starts with a value of the level larger than the goal; Remaining unchanged if simulation starts with the value of the level equal to the goal.  A familiar example of sigmoidal growth is the logistic equation:   EMBED Equation.2    N(t) = N(t - dt) + (dN_dt) * dt INIT N = 2 INFLOWS: dN_dt = r*N*(K-N)/K K = 100 r = 0.2 This is a first-order nonlinear differential equation. The following is an equivalent simple systems model.  Dynamics from the Logistic Equation-Based Module Feedback loop analysis of the S-shaped growth: Positive vs negative feedback regions Characteristics of the rate-level curve: several similar, but distinctive forms Goal-seeking does NOT always guarantee goal-achieving!   (II) Second-Order Systems First-order (continuous) systems do not generate overshoot and collapse or oscillations unless significant time delays are introduced. Such behavioral patterns are commonly found in second- or higher-order systems (with 2 or more stocks). The following section illustrates some simple systems structures that give rise to non-monotonic dynamics. Overshoot and Collapse in STELLA  food(t) = food(t - dt) + (- consumption_rate) * dt INIT food = 1000 {tones} OUTFLOWS: consumption_rate = per_capita_consumption_rate*population population(t) = population(t - dt) + (birth_rate - death_rate) * dt INIT population = 100 INFLOWS: birth_rate = per_capita_birth_rate*population OUTFLOWS: death_rate = per_capita_death_rate*population per_capita_birth_rate = 0.1 {1/time} per_capita_consumption_rate = 0.1 {tones/(individual * time)} per_capita_death_rate = GRAPH(food/INIT(food)) (0.00, 0.995), (0.1, 0.645), (0.2, 0.45), (0.3, 0.3), (0.4, 0.21), (0.5, 0.14), (0.6, 0.095), (0.7, 0.06), (0.8, 0.03), (0.9, 0.01), (1, 0.00) 6. Overshoot and Collapse A simple structure that generates overshoot and collapse behavior includes 2 stocks (e.g., population and food resource) and a few rates associated with them (see the Forrester diagram). In this example, population grows exponentially when food is not limited. More and more food resource is consumed by the growing population, resulting in less and less food resource. At some point, death rate will rise above birth rate, and the population eventually collapse due to the lack of food.    Oscillation in STELLA  Stock_1(t) = Stock_1(t - dt) + (inflow_1 - outflow_1) * dt INIT Stock_1 = 10 INFLOWS: inflow_1 = Stock_2*productivity_1 OUTFLOWS: outflow_1 = 10 Stock_2(t) = Stock_2(t - dt) + (inflow_2 - outflow_2) * dt INIT Stock_2 = 15 INFLOWS: inflow_2 = 10 OUTFLOWS: outflow_2 = Stock_1*productivity_2 productivity_1 = 1 productivity_2 = 1 7. Oscillations Two stocks and four rates associated. The first stock promotes the inflow to the second stock, and the second stock accelerates the depletion of the first stock. This negative feedback is essential for oscillatory behavior. With STELLA, you must use one of the Runge-Kutta simulation algorithms. (1) A Simple General Structure (cf. Richmond et al. 1993).  (2) A Predator-Prey Model Oscillation in STELLA  predator(t) = predator(t - dt) + (birth_rate - death_rate) * dt INIT predator = 10 INFLOWS: birth_rate = 0.01*predator*prey OUTFLOWS: death_rate = 0.2*predator prey(t) = prey(t - dt) + (prey_births - prey_deaths) * dt INIT prey = 50 INFLOWS: prey_births = per_capita_birth_rate*prey OUTFLOWS: prey_deaths = 0.03*predator*prey per_capita_birth_rate = GRAPH(prey) (0.00, 0.685), (50.0, 0.41), (100, 0.295), (150, 0.215), (200, 0.15), (250, 0.11), (300, 0.07), (350, 0.04), (400, 0.02), (450, 0.01), (500, 0.00)   (From the same model but D-I prey birth rate) (III) Oscillations Generated by Driving Functions Driving variables (or functions) that change periodically (e.g., temperature, light intensity) may introduce oscillations into a system of any order. Oscillations forced by driving variables are called externally generated oscillations. Externally Generated Oscillation  stock(t) = stock(t - dt) + (inflow - outflow) * dt INIT stock = 5 INFLOWS: inflow = driving_variable OUTFLOWS: outflow = stock/time_in_stoch driving_variable = SINWAVE(4,20)+3 time_in_stoch = 1 {time unit}  A general equation for modeling periodic driving functions:  EMBED Equation.2  where Y is a driving variable (e.g., temperature, light intensity), m is the mean value of the function, A is the amplitude of the peak above the mean, (t -  EMBED Equation.2 ) shifts the peak by  EMBED Equation.2  physical units, and the  EMBED Equation.2  is the angular frequency per physical unit. For example, suppose we have a time series: Mean daily air temperatures: 40 degrees F Amplitude: 25 degrees F Period: 365 days Position of the 1st peak: July 30 (Julian day 211) The equation becomes:  EMBED Equation.2  (IV) Chain Structure and Time Delay See modules.   Lecture Notes for Ecological Modeling --- Jianguo (Jingle) Wu  Page  PAGE 1 lsc 405 - Simulation Modeling Lecture Notes J. 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wXfnQxg[`V^SCc_cۓaUOzfWpu)O I_gh꼵)Dd#y<  C AB3йW#H8# T3йW#H D,X,1PxMLA,XPlYQCED  !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXZ[\]^_`abcdefghijmrpqtsxuvwy}z{|~Root Entry FH _MoData Y^SWordDocumentObjectPool.،_MH _M_922000245F،_M،_MOle PIC LPICT   !"#$%&'()*+,-./03479:;<=>?@ABCDEFGHIJKLMNOPSTWYZ[\]^_`abcdefgjmopqrstuvwxyz{|}~LXT1dxpr  1"1 currentpoint ",Times .+ N, Symbol) =) N +0 ( $e (*rt/MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 1568 div 544 3 -1 roll exch div scale currentpoint translate 64 -972 translate /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Times-Italic f1 (N) 13 1356 sh (N) 720 1356 sh (e) 1107 1356 sh 224 ns (rt) 1292 1184 sh 384 /Symbol f1 (=) 389 1356 sh 224 /Times-Roman f1 (0) 982 1452 sh end MTsave restore d=MATH1 N=N 0 e rt1 N=N 0 e rtL WObjInfo Equation Native M_922000698 F،_M،_MOle PIC  LPICT ObjInfo 1Equation Native 2.bdxpr  .b".b currentpoint ",Times .+N +T +c, Symbol (=) N + 0 (,e (31(2T +c"1 ( :T +c + =) N +0 ([e/MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 3136 div 1472 3 -1 roll exch div scale currentpoint translate 64 -2601 translate /thick 0 def /th { dup setlinewidth /thick exch def } def 8 th 1534 2882 moveto 242 0 rlineto stroke /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Times-Italic f1 (N) 13 3295 sh (N) 957 3295 sh (e) 1344 3295 sh (N) 2485 3295 sh (e) 2872 3295 sh 224 ns (T) 273 3391 sh (T) 1545 3114 sh (T) 1801 2939 sh 160 ns (c) 387 3447 sh (c) 1659 3170 sh (c) 1915 2995 sh 384 /Symbol f1 (=) 626 3295 sh (=) 2154 3295 sh 224 /Times-Roman f1 (0) 1219 3391 sh (1) 1599 2796 sh (0) 2747 3391 sh end MTsave restore dMATH N T c  =N 0 e 1T c T c =N 0 e N T c  =N 0 e 1T c T c =N 0 eL0_922000785F،_M،_MOle 5PIC 6LPICT 8mdxpr  m"m currentpoint ",Times .+N +nT +c, Symbol (=) N +0 (/e (71(5T +c"5 ( =nT +c + =) N +0 (be (hn/MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 3488 div 896 3 -1 roll exch div scale currentpoint translate 64 -979 translate /thick 0 def /th { dup setlinewidth /thick exch def } def 8 th 1648 1270 moveto 242 0 rlineto stroke /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Times-Italic f1 (N) 13 1683 sh (N) 1071 1683 sh (e) 1458 1683 sh (N) 2713 1683 sh (e) 3100 1683 sh 224 ns (nT) 275 1779 sh (T) 1659 1502 sh (nT) 1917 1327 sh (n) 3285 1511 sh 160 ns (c) 501 1835 sh (c) 1773 1558 sh (c) 2143 1383 sh 384 /Symbol f1 (=) 740 1683 sh (=) 2382 1683 sh 224 /Times-Roman f1 (0) 1333 1779 sh (1) 1713 1184 sh (0) 2975 1779 sh end MTsave restore dMATH N nT c  =N 0 e 1T c nT c =N 0 e n N nT c  =N 0 e 1T c nT c =N 0 e nLXTObjInfoQEquation Native R_922002206 "F،_M،_MOle UPIC VLPICT XObjInfohEquation Native iM O1dxpr  1"1 currentpoint ",Times .+ N, Symbol) =) N +0 ( $e (*rt/MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 1568 div 544 3 -1 roll exch div scale currentpoint translate 64 -972 translate /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Times-Italic f1 (N) 13 1356 sh (N) 720 1356 sh (e) 1107 1356 sh 224 ns (rt) 1292 1184 sh 384 /Symbol f1 (=) 389 1356 sh 224 /Times-Roman f1 (0) 982 1452 sh end MTsave restore d=MATH1 N=N 0 e rt1 N=N 0 e rtL(< <Cdxpr%  C"C currentpoint ",Times .+ 2)N_922002235F،_M،_MOle kPIC  lLPICT n +0, Symbol ( =) N + 0 ( /e (4rT +d (ln) 2) =) rT + d (8T +d (8=(00).)69(A!r"4(81 or )T +d (8Z=) 0).)69)  )T +c (8 H/MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 4448 div 2144 3 -1 roll exch div scale currentpoint translate 64 -4222 translate /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 741 5916 moveto 711 0 rlineto stroke /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Times-Roman f1 (2) -11 4590 sh (2) 340 5182 sh (0) 764 5774 sh (69) 1052 5774 sh (0) 3155 6015 sh (69) 3443 6015 sh 224 ns (0) 476 4686 sh (0) 1315 4686 sh 384 /Times-Italic f1 (N) 214 4590 sh (N) 1053 4590 sh (e) 1440 4590 sh (rT) 939 5182 sh (T) -12 6015 sh (r) 1017 6308 sh (T) 2411 6015 sh (T) 3937 6015 sh 224 ns (rT) 1625 4408 sh (d) 1267 5279 sh (d) 167 6112 sh (d) 2590 6112 sh (c) 4112 6111 sh 160 ns (d) 1829 4465 sh 384 /Symbol f1 (=) 722 4590 sh (=) 630 5182 sh (=) 423 6015 sh (=) 2846 6015 sh 384 /Times-Roman f1 (ln) -7 5182 sh (.) 956 5774 sh (.) 3347 6015 sh 384 /Times-Roman f1 ( or ) 1516 6015 sh ( ) 3841 6015 sh ( ) 4238 6015 sh end MTsave restore dMATH& 2N 0 =N 0 e rT d ln2=rT d T d =0.69r   or   T d =0.69 T c   2N 0 =NObjInfo!Equation Native  _922009865($F،_M،_MOle  0 e rT d ln2=rT d T d =0.69r   or   T d =0.69 T c  L X0^dxpr  ^"^PIC #&LPICT ObjInfo%'Equation Native a currentpoint ",Times .+ dN+dt" , Symbol(=) ()rN) ))N) =) rN ( X2/MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 3008 div 896 3 -1 roll exch div scale currentpoint translate 64 41 translate /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 0 404 moveto 519 0 rlineto stroke /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Times-Italic f1 (dN) 27 262 sh (dt) 106 796 sh (rN) 1064 503 sh (N) 1652 503 sh (rN) 2337 503 sh 384 /Symbol f1 (=) 627 503 sh (=) 2028 503 sh 384 /Times-Roman f1 (\() 927 503 sh (\)) 1500 503 sh 224 /Times-Roman f1 (2) 2780 331 sh end MTsave restore dQMATHE  dNdt=(rN)N=rN 2E dNdt=(rN)N=rN 2LD_922179184*F،_M،_MOle PIC ),LPICT dd dxpr  "  currentpoint ",Times .+ 0).)368/MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 928 div 384 3 -1 roll exch div scale currentpoint translate -3813 -986 translate /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Times-Roman f1 (0) 3868 1306 sh (368) 4156 1306 sh 384 /Times-Roman f1 (.) 4060 1306 sh end MTsave restore d$MATH 0.368aObjInfo+-Equation Native 4_922179603F0F،_M،_MOle  0.368LNX$T5dxpr  5"5 currentpoint ",Times .+ N, Symbol) =) N +0 ( $e (*-)rt/MTsave save def 40 dict begin currentpoint 3 -1 rolPIC /2LPICT ObjInfo13Equation Native Pl sub neg 3 1 roll sub 1696 div 544 3 -1 roll exch div scale currentpoint translate 64 60 translate /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Times-Italic f1 (N) 13 324 sh (N) 720 324 sh (e) 1107 324 sh 224 ns (rt) 1439 152 sh 384 /Symbol f1 (=) 389 324 sh 224 ns (-) 1295 152 sh 224 /Times-Roman f1 (0) 982 420 sh end MTsave restore d@MATH4~ N=N 0 e -rt/n4 N=N 0 e -rt_9221799336F،_Mp_MOle PIC 58LPICT :L 0:dxpr  " currentpoint ",Times .+N +T +c, Symbol (=) N + 0 (,e ( 1-(81(6T +c"6 ( >T +c + =) e (Y-)1 +N +0 (q=) 0).)368) )N +0/MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 5312 div 896 3 -1 roll exch div scale currentpoint translate 64 -602 translate /thick 0 def /th { dup setlinewidth /thick exch def } def 8 th 1683 893 moveto 242 0 rlineto stroke /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Times-Italic f1 (N) 13 1306 sh (N) 957 1306 sh (e) 1344 1306 sh (e) 2608 1306 sh (N) 3051 1306 sh (N) 4835 1306 sh 224 ns (T) 273 1402 sh (T) 1694 1125 sh (T) 1950 950 sh 160 ns (c) 387 1458 sh (c) 1808 1181 sh (c) 2064 1006 sh 384 /Symbol f1 (=) 626 1306 sh (=) 2303 1306 sh (=) 3559 1306 sh 224 ns (-) 1532 950 sh (-) 2796 1134 sh 224 /Times-Roman f1 (0) 1219 1402 sh (1) 1748 807 sh (1) 2918 1134 sh (0) 3313 1402 sh (0) 5097 1402 sh 384 ns (0) 3868 1306 sh (368) 4156 1306 sh 384 /Times-Roman f1 (.) 4060 1306 sh 384 /Times-Roman f1 ( ) 4739 1306 sh end MTsave restore dMATH N T c  =N 0 e -1T c T c =e -1 N 0 =0.368 N 0ObjInfo79Equation Native _9221799534@<Fp_Mp_MOle  N T c  =N 0 e -1T c T c =e -1 N 0 =0.368 N 0Ld 0PIC ;>LPICT JObjInfo=?Equation Native       !"%'()*+,-./0123456789:=@BCDEFGHIJKLMNOPQRSTUVWZ[^`abcdefghijklmnopqrstuvyz}Jdxpr  " currentpoint ",Times .+N +nT +c, Symbol (=) N +0 (/e ( 5-(<1(:T +c": ( BnT +c + =) e (`-)n +N +0 (y=) 0).)368 (n +N + 0/MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 5664 div 896 3 -1 roll exch div scale currentpoint translate 64 -1638 translate /thick 0 def /th { dup setlinewidth /thick exch def } def 8 th 1797 1929 moveto 242 0 rlineto stroke /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Times-Italic f1 (N) 13 2342 sh (N) 1071 2342 sh (e) 1458 2342 sh (e) 2836 2342 sh (N) 3318 2342 sh (N) 5183 2342 sh 224 ns (nT) 275 2438 sh (T) 1808 2161 sh (nT) 2066 1986 sh (n) 3168 2170 sh (n) 5003 2170 sh 160 ns (c) 501 2494 sh (c) 1922 2217 sh (c) 2292 2042 sh 384 /Symbol f1 (=) 740 2342 sh (=) 2531 2342 sh (=) 3826 2342 sh 224 ns (-) 1646 1986 sh (-) 3024 2170 sh 224 /Times-Roman f1 (0) 1333 2438 sh (1) 1862 1843 sh (0) 3580 2438 sh (0) 5445 2438 sh 384 ns (0) 4135 2342 sh (368) 4423 2342 sh 384 /Times-Roman f1 (.) 4327 2342 sh end MTsave restore dMATH N nT c  =N 0 e -1T c nT c =e -n N 0 =0.368 n N 0 N nT c  =N 0 e -1T c nT c =e -n N 0 =0.368 n N 0L 0Hdxpr  H"H currentpoint ",Times .+ 1*2_922180889BFp_Mp_MOle #PIC AD$LPICT &" ( N +0, Symbol (=) N + 0 (2e ( 7-)rT +h/MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 2304 div 896 3 -1 roll exch div scale currentpoint translate 64 42 translate /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 0 403 moveto 235 0 rlineto stroke /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Times-Roman f1 (1) 21 261 sh (2) 21 795 sh 224 ns (0) 574 598 sh (0) 1413 598 sh 384 /Times-Italic f1 (N) 312 502 sh (N) 1151 502 sh (e) 1538 502 sh 224 ns (rT) 1870 320 sh 160 ns (h) 2073 377 sh 384 /Symbol f1 (=) 820 502 sh 224 ns (-) 1726 320 sh end MTsave restore dhMATH\  12N 0 =N 0 e -rT h d\ 12N 0 =N 0 e -rT hL: 0ObjInfoCE;Equation Native <x_922181066:^HFp_Mp_MOle >PIC GJ?LPICT AObjInfoIKXEquation Native Y}dxpr  }"} currentpoint ",Times .+ln( 1*2" , Symbol(=) -)rT + h (7,) or )ln) 2) =) rT + h /MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 4000 div 896 3 -1 roll exch div scale currentpoint translate 64 -985 translate /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 351 1430 moveto 235 0 rlineto stroke /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Times-Roman f1 (ln) -7 1529 sh (,) 1703 1529 sh (ln) 2508 1529 sh 384 /Times-Roman f1 (1) 372 1288 sh (2) 372 1822 sh (2) 2855 1529 sh 384 /Symbol f1 (=) 694 1529 sh (-) 1008 1529 sh (=) 3145 1529 sh 384 /Times-Italic f1 (rT) 1234 1529 sh (rT) 3454 1529 sh 224 ns (h) 1560 1626 sh (h) 3780 1626 sh 384 /Times-Roman f1 ( or ) 1844 1529 sh end MTsave restore dsMATHg^ ln12=-rT h , or  ln2=rT hcug ln12=-rT h , or  ln2=rT hLwx 0dxpr  "_922181091NFp_Mp_MOle \PIC MP]LPICT _ currentpoint ",Times .+T +h, Symbol (=( 0).)69(!r" (0 or )T +h (Z=) 0).)69)  )T +c@/MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 4288 div 896 3 -1 roll exch div scale currentpoint translate 64 -2589 translate /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 728 3034 moveto 711 0 rlineto stroke /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Times-Italic f1 (T) -12 3133 sh (r) 1004 3426 sh (T) 2398 3133 sh (T) 3911 3133 sh 224 ns (h) 165 3230 sh (h) 2575 3230 sh (c) 4086 3229 sh 384 /Symbol f1 (=) 410 3133 sh (=) 2820 3133 sh 384 /Times-Roman f1 (0) 751 2892 sh (69) 1039 2892 sh (0) 3129 3133 sh (69) 3417 3133 sh 384 /Times-Roman f1 (.) 943 2892 sh (.) 3321 3133 sh 384 /Times-Roman f1 ( or ) 1503 3133 sh ( ) 3815 3133 sh end MTsave restore dMATH}@ T h =0.69r   or   T h =0.69 T cl ObjInfoOQwEquation Native x_922187839LXTFp_Mp_MOle {} T h =0.69r   or   T h =0.69 T cL 0Udxpr  U"U currentpoint ",Times .+ dN+d`E !zQЀkchD11O9I4** o&~+3~߷;LA/zjV Œ{`klmЃ6#=@KP~Ѩ$("TIYEqSq,X5  TӬTӬU'ы$=W,P%XTUUI)PXT>gj<˖wNزVykpRthh8N7_̜M9 xNMUUkc ݎnx5NVGDN92uZ}*Lb/_z -ty/4 sH#ΓmG94EIS/HjU w*{(ÿ˿Wwϥ:t+a#N: ByɨAC ^Loی#xbz{N+[-Vت ]jukZ;Րy`HQGpgML+V߲_2-X.ז+1K/lx6GͷX6'(72id%f6Zes]5fV^3ygdww{)Ug*AC4,<8OL Wx+r[<!m,51yA \1LE&,:c# CQEUOŢ3e.N~gke~rl?1G.>lgd+vҲVd˫yƽ+>YI5~~F4'D"Y! S&WordMicrosoft Word&Word %  2 @@$Use Word 6.0c or later to 2 @$view Macintosh picture. %XS XS XSXSHH_iwwwwwwUUUUUUffff ff ff33ff33 &f&f&f333333@@@LLLYYYffffffs3s3s3̌̌fff333 ٙٙٙ!fff"222#ff$ff%ff30&ffL'ffY(fffb)34ff*ff+ff30,ffL-ffff.Yff/Lff0@ff136ff29ff3ff430ff5fbff6ffDF7ff*89&h:L;s3LΙ?@LAs3BCgDs5E`FLΙG.יHILJKfjL@MN35OfdP˙Q˳0RSfhT UfdV˙WX3YZ[fh\>s]^f`_`aUXbc@defghijklmnopqNrstuvjwLLxyLyeLz3L{L|L}gL~LLLeL.LLL9LͼLͦgL̓qLL`LLLGL͗u1əf/f3gٛəEٗc1ee23gib ff33ff33ff33ffffffffff̙ffffffff33ffDDwwΪ3333ә33ff33ՙff33̙33ffUUww33ff333333ffff癙333333ffffffff""DD̙ffffff3333ffDDDDDD""""""XSXS XSC\  ^ Z  Z  ^  E  HH HH H H H H H H H H H H H HHHHH H H$H&HH#H HHHHHH HHHHHH HHHHHHHHHH H%HH H H H H HHHHHH HHHH HHHH$HHHHHHHHHHH HHHHHHHHH HH H!HHHH"HHHH H H H H HHHHHH HHHHHHH%HH"HHHHH HHHHHH HHHHHH HHH HHHHHHHHHHHHHHHH H H H H H H H H H H H H5HQK5)6=:'3Fru u!s*(Di's&X S&WordMicrosoft Word&Word &  2 @@$Use Word 6.0c or later to 2 @$view Macintosh picture. &F2]P 2FP] F2]PF2]PHH_wwwwwwUUUUUUffff ff ff33ff33 &f&f&f333333@@@LLLYYYffffffs3s3s3̌̌fff333 ٙٙٙ!fff"222#ff$ff%ff30&ffL'ffY(fffb)34ff*ff+ff30,ffL-ffff.Yff/Lff0@ff136ff29ff3ff430ff5fbff6ffDF7ff*89&h:L;s3LΙ?@LAs3BCgDs5E`FLΙG.יHILJKfjL@MN35OfdP˙Q˳0RSfhT UfdV˙WX3YZ[fh\>s]^f`_`aUXbc@defghijklmnopqNrstuvjwLLxyLyeLz3L{L|L}gL~LLLeL.LLL9LͼLͦgL̓qLL`LLLGL͗u1əf/f3gٛəEٗc1ee23gib ff33ff33ff33ffffffffff̙ffffffff33ffDDwwΪ3333ә33ff33ՙff33̙33ffUUww33ff333333ffff癙333333ffffffff""DD̙ffffff3333ffDDDDDD""""""F2]PF2]P F2]P#ttttttttttt(ttttttttttttt^ttttttttttttttttttttttttttttttt[ttttttttttttttt tttttttttttttttttUt tttttttttttt tttttttt ttttttttUtt ttttttttttt tttttttt tttttttUttttttttttttttttttttttt tt ttt*t5t)tttttttttttttttttttttttttttttttttttttttttttttt tttttttttttttttttttttttttttttttttttttttttttttttttttttttt$t.tt2t*t6t#ttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttt#t%t./+3SNORR?%$ $   DZD && S&WordMicrosoft Word&Word <  2 @@$Use Word 6.0c or later to 2 @$view Macintosh picture. <  2 UU!"U """"  " """"  " """"  " """"  ""U """" W "U """" W "U """" NWN "NU """" W "U $2$$p0000>> %52%5%5P _@@@@ 7G27G7GD D>|p  , Geneva .(1:36 PM 4/8/97(S0.00)]5.00)a10.00)d15.00)Z20.00(Time(1:(1:(1:*2:(2:(%2:+52.00" ##(551.00"| ##(/100.00" ##( 1: Super N)2: N W"W###############################################################################"W############################################################################### 2\c(]12(12pw(q22(228?(922}(2(Graph 1: Page 1 (Superexponen tpt 2 ~ (P`D  S&WordMicrosoft Word&Word n  2 @@$Use Word 6.0c or later to 2 @$view Macintosh picture. n  0'FC, Geneva .+, N`0:`0:`5?<`5?P9VQn++@dN dt"4B;"6B;p/};4}/}5;}6}"9b"4b"2_PX{p+.rff`$3P_aDOR]f2PB3H9`D]nWa>gLu62PU[ D[@#|$P   S&WordMicrosoft Word&Word    2 @@$Use Word 6.0c or later to 2 @$view Macintosh picture.    2 PP!"P """"  " """"  " """" ll "l """"  ""P """" R "P """" |R| "|P """" JRJ "JP """" R "P $2$$p0000>> %52%5%5P _@@@@ 7G27G7GD D>|p  , Geneva .(`11:45 AM 4/10/97(N0.00)T12.50)_25.00)^37.50)T50.00(Time(1:(|1:(1:*2:(2:($2:+10.00(|150.00(+100.00+ 0.00(75.00($110.00(1: N)2: dN dt R"R#######################################################################################################################################################################################################"R####################################################################################################################################################################################################### 2%V/](.W12+_12+^$12qx+^ 12NiXp(Wj22+_f22&-+^22+^2({Graph 1: Page 1 (Exponentia tpt 2 ~ (P`D"YT S&WordMicrosoft Word&Word   2 @@$Use Word 6.0c or later to 2 @$view Macintosh picture.   0'F, Geneva .+ N`0:`0:`5?<`5?P9Q+)@outflow"4;"6;p/;4/5;6"9"4"2P\t+#decay constant`2@<J`2E<O`7EAO<`7@AJP;^Sv(b]inflow"6P."8P.p1~=6~1~7=~8~";j"6j"4gP^3vK(growth constantff`&RaESb2PAG`Cq_a=K72PY_`IA:aCWQe2P\AbG`.oTf}aFnT|2PAG2D>"#!! S&WordMicrosoft Word&Word   2 @@$Use Word 6.0c or later to 2 @$view Macintosh picture.   2 NN!"N """"  " """"  " """" bb "b """"  ""N """" P "N """" yPy "yN """" GPG "GN """" P "N $2$$p0000>> %52%5%5P _@@@@ 7G27G7GD D>|p  , Geneva .([1:37 PM 4/10/97(L0.00)R12.50)\25.00)[37.50)R50.00(Time(1:(~1:(1:+/0.00(~)150.00()300.00(1: N)2: N)3: N P"P#######################################################################################################################################################################################################"P#######################################################################################################################################################################################################"xP####################################################################################################################################################################################################### 2T[(U12(12 ( 12gn(h12KgUn(Th22+[c22%+\22z+\22sy}(|z32s})\32s0}7)[32s})\3(zGraph 1: Page 1 (Exponenti tpt 2 ~ (P`DCl 8 PIC SV|LPICT ~ObjInfoUWEquation Native Rt" , Symbol(=) -)r)()M) -) N) )/MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 2720 div 896 3 -1 roll exch div scale currentpoint translate 64 41 translate /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 0 404 moveto 519 0 rlineto stroke /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Times-Italic f1 (dN) 27 262 sh (dt) 106 796 sh (r) 1167 503 sh (M) 1484 503 sh (N) 2216 503 sh 384 /Symbol f1 (=) 627 503 sh (-) 941 503 sh (-) 1902 503 sh 384 /Times-Roman f1 (\() 1329 503 sh (\)) 2503 503 sh end MTsave restore dBMATH6 V dNdt=-r(M-N)6 dNdt=-r(M-N)L 0Ndxpr  N"N_922189165ZFp_Mp_MOle PIC Y\LPICT  currentpoint ",Times .+ dN+dt" , Symbol(=) r)()N) -) M) )u/MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 2496 div 896 3 -1 roll exch div scale currentpoint translate 64 41 translate /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 0 404 moveto 519 0 rlineto stroke /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Times-Italic f1 (dN) 27 262 sh (dt) 106 796 sh (r) 936 503 sh (N) 1257 503 sh (M) 1927 503 sh 384 /Symbol f1 (=) 627 503 sh (-) 1617 503 sh 384 /Times-Roman f1 (\() 1098 503 sh (\)) 2272 503 sh end MTsave restore d?MATH3  dNdt=r(N-M)gh3 dNdt=r(N-M)L 0ObjInfo[]Equation Native O_922194297Rp`Fp_Mp_MOle PIC _bLPICT ObjInfoacEquation Native OIJ     !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGH`KNLMPRQTSVUWZXY\[x]^_yabcdefghijklmnopqrstuvw|}~Hdxpr  H"H currentpoint ",Times .+ dL+dt"  , Symbol(=) c)()G) -) L))t/MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 2304 div 896 3 -1 roll exch div scale currentpoint translate 64 41 translate /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 0 404 moveto 456 0 rlineto stroke /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Times-Italic f1 (dL) 27 262 sh (dt) 75 796 sh (c) 870 503 sh (G) 1178 503 sh (L) 1852 503 sh 384 /Symbol f1 (=) 564 503 sh (-) 1539 503 sh 384 /Times-Roman f1 (\() 1043 503 sh (\)) 2076 503 sh end MTsave restore d?MATH3  dLdt=c(G-L) p3 dLdt=c(G-L)L l0_922216076fFp_Mp_MOle PIC ehLPICT _dxpr  _"_ currentpoint ",Times .+ dN+dt" , Symbol(=) rN) ()1)-) N) /)K) )/MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 3040 div 896 3 -1 roll exch div scale currentpoint translate 64 41 translate /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 0 404 moveto 519 0 rlineto stroke /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Times-Italic f1 (dN) 27 262 sh (dt) 106 796 sh (rN) 936 503 sh (N) 2026 503 sh (K) 2546 503 sh 384 /Symbol f1 (=) 627 503 sh (-) 1712 503 sh 384 /Times-Roman f1 (\() 1365 503 sh (/) 2362 503 sh (\)) 2834 503 sh 384 /Times-Roman f1 (1) 1469 503 sh end MTsave restore dHMATH<  dNdt=rN(1-N/K)< dNdt=rN(1-N/K)ObjInfogiEquation Native X_922254645lFp_Mp_MOle  S&WordMicrosoft Word&Word   2 @@$Use Word 6.0c or later to 2 @$view Macintosh picture.   0'FC, Geneva .+$ level`0:`0:`5?<`5?P9VQn+-@outflow"4B;"6B;p/};4}/}5;}6}"9b"4b"2_Pay+(collapse constantPa'y?(0Mff`#7K_ya@NN\l2PB:H@`F;Fa@fNtM2P_e"e<aFRT`2Pb9h? 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def /ns {cf sf} def /sh {moveto show} def 384 /Times-Italic f1 (Y) -16 261 sh (m) 636 261 sh (A) 1316 261 sh (t) 2643 261 sh 384 /Symbol f1 (=) 322 261 sh (+) 997 261 sh (-) 2845 261 sh 384 /Times-Roman f1 (cos\() 1587 261 sh (\() 2511 261 sh (\)\)) 3337 261 sh /f2 {ff matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (w) 2213 261 sh (t) 3117 261 sh end MTsave restore dEMATH9  Y=m+Acos(w(t-t))9 Y=m+Acos(w(t-t))L>>Q dxpr  " currentpoint ", Symbol .+t/MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 _922254412d|rFp_Mp_MOle PIC qtLPICT Q  "$%&'()*+,-./05789:;<=>?@ABCDEFGHIJKLMPRSTUVWYZ[\]_1 roll sub 288 div 288 3 -1 roll exch div scale currentpoint translate -3082 -37 translate /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf 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