Lecture Notes for Ecological Modeling Jianguo (Jingle) Wu

Lecture Notes for Ecological Modeling --- Jianguo (Jingle) Wu

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 II. BUILDING BLOCKS FOR SYSTEMS MODELS X

(I) FIRST-ORDER SYSTEMS

o First-order: One state variable (or 1 stock) dX dt o Linear: No non-linear combinations of the state X(t) = X(t - dt) + (dX_dt) * dt variable of any sort in the algebraic equation of rates. INIT X = 0 INFLOWS: 1. Linear Change dX_dt = 2 o One state variable. o One rate variable. o The rate of change is a constant, not dependent on the state variable. Thus, there is no feedback loop.

1: X 2: dX dt (1) Linear Growth 1: 30.00 2: 3.00 o The rate of change is positive, and thus represented

as an inflow. 1

1: 15.00 2 2 2 2 o Examples: 2: 2.00 o Increase in water level in a 1

tank with a constant inflow of 1 water into the tank. 1: 0.00 2: 1.00 1 o Plant biomass sometimes is 0.00 3.00 6.00 9.00 12.00 found to increase linearly Graph 1 (Linear Growth) Time 11:40 PM 4/7/97 with certain soil nutrients.

(2) Linear Decline o The rate of change is negative, and thus Linear Decline in STELLA represented as an outflow. X o 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 dX dt of time. X(t) = X(t - dt) + (- dX_dt) * dt  Decrease in crop yield with declining INIT X = 40 annual precipitation sometimes exhibits OUTFLOWS: dX_dt = 2 linear pattern.

(3) Linear Growth and Decline

o Combination of the linear growth and decline module.

o One state variable. 1: X 2: dX dt 1: 45.00 o Either two uniflows (one 2: 3.00

inflow and one outflow) or 1 one biflow (compare the 1

structural diagrams below). 1: 30.00 2 2 2 2 2: 2.00 o The uniflow version gives 1 more details, whereas the biflow version is more 1

1: 15.00 concise. The choice of the 2: 1.00 0.00 3.00 6.00 9.00 12.00

two forms is dependent on Graph 1 (Linear Growth) Time 12:07 AM 4/8/97 the degree of details is needed. o Linear growth occurs Linear Growth and Decline in STELLA when the total rate of (1) Two-uniflow model: (2) One-biflow model: change (inflow minus X Y outflow) is positive, and linear decline does otherwise.

dX dt in dX dt out dY dt

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 (1) Exponential Growth N  Also known as compounding process  One state variable.  One rate variable.

 One auxiliary variable. dN dt  The rate of change is equal to a constant

proportion of the state variable. r  Because the rate variable depends on the state N(t) = N(t - dt) + (dN_dt) * dt variable itself, they form a positive feedback INIT N = 2 loop, in which the state variable and the rate of INFLOWS: change reinforce each other, generating an dN_dt = r*N r = 0.1 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. o 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:

Lecture Notes for Ecological Modeling --- Jianguo (Jingle) Wu rt

. Page 4 N = N0e Lecture Notes for Ecological Modeling --- Jianguo (Jingle) Wu

When t is equal to Tc, we have

1: N 2: dN dt 1: 300.00 2: 30.00

1: 150.00 2: 15.00 2 1

2 1 1: 0.00 1 2 2: 0.00 1 2 0.00 12.50 25.00 37.50 50.00 Graph 1: Page 1 (Exponenti… Time 11:26 AM 4/8/97

Lecture Notes for Ecological Modeling --- Jianguo (Jingle) Wu 1 Tc NPage 7 = N eTc = N e T c 0 0 Lecture Notes for Ecological Modeling --- Jianguo (Jingle) Wu

In general, if t = nTc,

Lecture Notes for Ecological Modeling --- Jianguo (Jingle) Wu 1 Page 9 nTc Tc n N nT c = N0e = N0e Lecture Notes for Ecological Modeling --- Jianguo (Jingle) Wu

o 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 rt

, we have: Page 11 N = N0e Lecture Notes for Ecological Modeling --- Jianguo (Jingle) Wu rt

, we have: Page 12 N = N0e Lecture Notes for Ecological Modeling --- Jianguo (Jingle) Wu

rTd 2 N 0 = N0e

lnPage 14 2 = rTd 0.69 T = or T = 0.69 T d r d c Lecture Notes for Ecological Modeling --- Jianguo (Jingle) Wu

o 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: o Savings increase in a bank account due to interest income o Population growth when resources are not limiting. o Drug addiction.

o Arms race (proliferation). 1: Super N 2: N 1: 100.00  Superexponential (or 2: supraexponential) growth: o In many positive feedback

systems, doubling time decreases, 1: 51.00 2 2: rather than remain constant, as the state variable increases. This 2 1 means that the rate of change is a 2 1 2 1: nonlinear function of the state 2: 2.00 0.00 5.00 10.00 15.00 20.00 variable (see Figure). For Graph 1: Page 1 (Superexponen… Time 1:36 PM 4/8/97 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

PagedN 16 = (rN)N = rN2 dt Lecture Notes for Ecological Modeling --- Jianguo (Jingle) Wu

o 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. o The superexponential growth has the Exponential Decay in STELLA same feedback structure as the N exponential.

(2) Exponential Declay  Also known as draining process dN dt  One state variable.  One rate variable. r  The rate of change is equal to a constant N(t) = N(t - dt) + (- dN_dt) * dt proportion of the state variable, but in contrast INIT N = 100 with exponential growth it is negative. OUTFLOWS:  The rate variable depends on the state variable dN_dt = r*N itself, and form a feedback loop between them. r = 0.1 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.

-1 o For exponential decay, Tc is the time required for the state variable to become e (=

0Page 18 . 368 Lecture Notes for Ecological Modeling --- Jianguo (Jingle) Wu

0Page 19 . 368 Lecture Notes for Ecological Modeling --- Jianguo (Jingle) Wu

) 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 o Mathematically, relaxation time can be derived as follows:

1: N 2: dN dt 1: 100.00 2: 10.00 1

1: 50.00 2: 5.00

1 2 1: 0.00 1 2 2: 0.00 0.00 12.50 25.00 37.50 50.00 Graph 1: Page 1 (Exponentia… Time 11:45 AM 4/10/97

Lecture Notes for Ecological Modeling --- Jianguo (Jingle) Wu -rt

N = N0e Lecture Notes for Ecological Modeling --- Jianguo (Jingle) Wu -rt

N = N0e Lecture Notes for Ecological Modeling --- Jianguo (Jingle) Wu

For a time period from t=0 to t=Tc,

Lecture Notes for Ecological Modeling --- Jianguo (Jingle) Wu 1 Page 24 - Tc Tc -1 N T c = N0e = e N0 = 0.368 N0 Lecture Notes for Ecological Modeling --- Jianguo (Jingle) Wu

In general, when t = nTc, we have

Lecture Notes for Ecological Modeling --- Jianguo (Jingle) Wu 1 Page 26 - nTc Tc - n n N nT c = N0e = e N0 = 0.368 N0 Lecture Notes for Ecological Modeling --- Jianguo (Jingle) Wu

 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:

1Page 28 - rT N = N e h 2 0 0 Lecture Notes for Ecological Modeling --- Jianguo (Jingle) Wu

1Page 29 - rT N = N e h 2 0 0 Lecture Notes for Ecological Modeling --- Jianguo (Jingle) Wu

1 ln = -rT, or ln2 = rT 2 h h Lecture Notes for Ecological Modeling --- Jianguo (Jingle) Wu

, thus,

0.69 T = or T = 0.69 T h r h c Lecture Notes for Ecological Modeling --- Jianguo (Jingle) Wu

o 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: o Draining water from a tank o Population-death rate process o Redioactive decay process

(3) Exponential Growth and Decay Combined o Simply a combination of exponential growth and exponential decay. o The behavior of the first-order linear system exhibits three different patterns: 1) exponential growth when growth constant is larger than decay constant; 2) exponential decay when growth constant is smaller than decay constant; and 3) remaining unchanged (unstable equilibrium) when the two constants are equal.

Exponential Growth/Decline in STELLA Exponential Collapse in STELLA

N level

inflow outflow outflow

growth constant decay constant M collapse constant N(t) = N(t - dt) + (inflow - outflow) * dt INIT N = 150 level(t) = level(t - dt) + (- outflow) * dt INFLOWS: INIT level = 99 inflow = growth_constant*N OUTFLOWS: OUTFLOWS: outflow = decay_constant*N outflow = collapse_constant*(M-level) decay_constant = 0.1 collapse_constant = 0.15 growth_constant = 0.1 M = 100

1: N 2: N 3: N 3. 1: 300.00 Exponential Collapse

1: 150.00 3 3 3 3

1 2 1 2 1: 0.00 1 0.00 12.50 25.00 37.50 50.00 Graph 1: Page 1 (Exponenti… Time 1:37 PM 4/10/97

 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:

dNPage 38 = -r(M - N) dt Lecture Notes for Ecological Modeling --- Jianguo (Jingle) Wu

dNPage 39 = -r(M - N) dt Lecture Notes for Ecological Modeling --- Jianguo (Jingle) Wu

where N (< M) is the state variable, and r and M are two constants.

 Examples: o Population shrinking when smaller than MVP (minimum viable population). o Change in the interactive force between molecules when water is heated up and boils. o Decline of species diversity with increasing habitat fragmentation. o 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:

dNPage 41 = r(N - M) dt Lecture Notes for Ecological Modeling --- Jianguo (Jingle) Wu

dNPage 42 = r(N - M) dt Lecture Notes for Ecological Modeling --- Jianguo (Jingle) Wu

Exponential Growth/Collapse in STELLA N

r M level(t) = level(t - dt) + (- outflow)*dt INIT level = 99 OUTFLOWS: outflow = collapse_constant*(M-level) collapse_constant = 0.15 M = 100

 The simple structure exhibits 3 different behavioral patterns: o Exponential growth when N > M; o Exponential collapse when N < M; and o Remaining constant when N = M.

NdN0MStateExponentialRate(unstableExponential ofVariable Change equilibriumgrowthdt decay point)

1: N 2: N 3: N 1: 250.00

1: 125.00 1 1 2 3 2 2 2 3

1: 0.00 3 3 0.00 12.50 25.00 37.50 50.00 Graph 1: Page 1 (Exponentia… Time 3:06 PM 4/10/97

5. Goal-Seeking Behavior Simple Goal-Seeking in STELLA

(1) Simple Goal-Seeking level  A simple goal-seeking model may consist of 1 state variable, 1 biflow, and 2 constants.  The state variable and the flow form a Inflow negative feedback.

 A simple goal-seeking model may take constant the form: goal discrepancy

level(t) = level(t - dt) + (Inflow) * dt INIT level = 2

INFLOWS: Inflow = constant*discrepancy constant = 0.1 discrepancy = goal-level goal = 100

dLPage 46 = c(G - L) dt Lecture Notes for Ecological Modeling --- Jianguo (Jingle) Wu

dLPage 47 = c(G - L) dt Lecture Notes for Ecological Modeling --- Jianguo (Jingle) Wu

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!

1: level 2: level 3: level (2) S-Shaped Growth 1: 200.00

 Also known as logistic growth 2 or sigmoidal growth.

 It is a first-order nonlinear 2 2 2 1: 100.00 1 1 1 3 1 3 system (see the Rate-Level 3 graphs).  The state variable and the flow 3 form a negative feedback that 1: 0.00 ultimately gives rise to the goal 0.00 12.50 25.00 37.50 50.00 seeking behavior (see the Graph 1: Page 1 (Simple goa… Time 4:11 PM 4/10/97 Forrester diagram).  It may take different forms (uniflow or biflow versions). [For example, in level population regulation, crowding may Inflow affect only the per capita birth rate, or only the per capita death rate, or both.]

~ Rate Value level(t) = level(t - dt) + (Inflow) * dt INIT level = 300 INFLOWS: Inflow = Rate_Value Rate_Value = GRAPH(level)... Rate-Level Graph

The behavior of this simple structure has three patterns: 1) Increasing and approaching the goal if simulation starts with a value of the level smaller than the goal; 2) Decreasing and approaching the goal if simulation starts with a value of the level larger than the goal; 3) Remaining unchanged if simulation starts with the value of the level equal to the goal.

1: level 2: level 3: level 1: 300.00

2 2 3 2 3 1 2 3 1: 150.00 1

1: 0.00 1 0.00 12.50 25.00 37.50 50.00 Graph 1: Page 1 (Simple sig… Time 5:08 PM 4/10/97

A familiar example of sigmoidal growth is the logistic equation:

PagedN 50 = rN(1 - N/ K) dt Lecture Notes for Ecological Modeling --- Jianguo (Jingle) Wu

This is a first-order nonlinear differential equation. The following is an equivalent simple systems model. N N(t) = N(t - dt) + 1: N v. dN dt 5.00 (dN_dt) * dt INIT N = 2

dN dt INFLOWS: 2.50 dN_dt = r*N*(K- r N)/K K = 100

0.00 K r = 0.2 0.00 50.00 100.00 Graph 1: Page 2 (Untitled Graph) N 10:46 PM 4/10/97

1: N 2: dN dt

1: 100.00 1 1: N 2: N 3: N 2: 5.00 1: 200.00 2

1: 50.00 2 2: 2.50 2 1: 100.00 3 3 2 3 1 2 3

1 2 1: 0.00 1 2: 0.00 1 0.00 12.50 25.00 37.50 50.00 1: 0.00 0.00 12.50 25.00 37.50 50.00 Graph 1: Page 1 (Simple goal-seeking) Time 10:46 PM 4/10/97 Graph 1: Page 3 (Untitled Graph) Time 11:12 PM 4/10/97 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!

0maxNegativeRateLevel,PositiveGoaldL L feedbackdt region

0NegativeRateLevel,PositiveGoalmaxdL L feedbackdt region

(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 6. Overshoot and Collapse population

 A simple structure that generates overshoot and collapse behavior birth rate death rate includes 2 stocks (e.g., population per capita birth rate ~ and food resource) and a few rates per capita death rate associated with them (see the per capita consumption rate Forrester diagram).  In this example, population grows

exponentially when food is not food limited. More and more food consumption rate resource is consumed by the growing population, resulting in less food(t) = food(t - dt) + (- consumption_rate) * dt INIT food = 1000 {tones} and less food resource. At some OUTFLOWS: point, death rate will rise above consumption_rate = per_capita_consumption_rate*population birth rate, and the population population(t) = population(t - dt) + (birth_rate - death_rate) * dt eventually collapse due to the lack INIT population = 100 of food. 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)

1: population 2: food 1: 400.00 2: 1050.00 2

1: 200.00 2 2: 650.00 1

2 1: 0.00 1 2: 250.00 2 0.00 12.50 25.00 37.50 50.00

Graph 1: Page 1 (Overshoot/c… Time 12:13 AM 4/11/97

Oscillation in STELLA 1: consumption rate 2: death rate 3: per capita death rate 7. Oscillations 1: 40.00 Stock 1 2: 60.00 inflow 1 outflow 1 3: 0.40 3 3  Two stocks and four 1 rates associated. 2  The first stock promotes 1: 20.00 productivity2: 1 30.00 1 the inflow to the second 3: 0.20 2

stock, and the second 1 3 stock accelerates the

1: 0.00 depletion of the first Stock 2 2: 0.00 2 3 1 2 3: 0.00 stock. This negative inflow 2 0.00 outflow12.50 2 25.00 37.50 50.00 feedback is essential for Graph 1: Page 2 (overshoot/c…productivityTime 2 12:13 AM 4/11/97 oscillatory behavior. Stock_1(t) = Stock_1(t - dt) + (inflow_1 - outflow_1) * dt  With STELLA, you INIT Stock_1 = 10 must use one of the INFLOWS: Runge-Kutta simulation inflow_1 = Stock_2*productivity_1 algorithms. OUTFLOWS: outflow_1 = 10 Stock_2(t) = Stock_2(t - dt) + (inflow_2 - outflow_2) * dt (1) A Simple General INIT Stock_2 = 15 Structure (cf. Richmond et INFLOWS: al. 1993). inflow_2 = 10 OUTFLOWS: outflow_2 = Stock_1*productivity_2 productivity_1 = 1 productivity_2 = 1

1: Stock 1 2: Stock 2 1: 20.00 2:

2 2 1 1: 10.00 2: 1

1: 2: 0.00 0.00 4.00 8.00 12.00 16.00 Graph 1 (Oscillation) Time 2:21 AM 4/11/97 (2) A Predator-Prey Model

Oscillation in STELLA prey prey births prey deaths

~ per capita birth rate

predator

birth rate death rate

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)

1: prey 2: predator

1: prey 2: predator 1: 90.00 1: 55.00 2: 50.00 2: 40.00 1 1

1 1 1: 45.00 2 2: 25.00 2 2 1: 30.00 1 2: 25.00 2 2

1 1: 0.00 1: 5.00 2: 0.00 0.00 25.00 50.00 75.00 100.00 2: 10.00 0.00 10.00 20.00 30.00 40.00 Graph 1 (Untitled Graph) Time 2:47 AM 4/11/97 Graph 1 (Untitled Graph) Time 2:25 AM 4/11/97 (From the same model but D-I prey birth rate)

(III) OSCILLATIONS GENERATED BY DRIVING FUNCTIONS

o Driving variables (or functions) that change periodically (e.g., temperature, light intensity) may introduce oscillations into a system of any order. o Oscillations forced by driving variables are called externally generated oscillations.

Externally Generated Oscillation

stock 1: stock 2: driving variable

1: 7.00 2 2: 7.00 1

inflow outflow

1 2 driving variable time in stoch 1: 3.50 2: 3.00 2 stock(t) = stock(t - dt) + (inflow - outflow) * dt 1 INIT stock = 5

1: 0.00 INFLOWS: 2: -1.00 1 2 inflow = driving_variable 0.00 15.00 30.00 45.00 60.00 OUTFLOWS: Graph 1 (Oscillation by Driving… Time 8:50 AM 4/11/97 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:

YPage 58 = m + Acos(w(t- t )) Lecture Notes for Ecological Modeling --- Jianguo (Jingle) Wu

YPage 59 = m + Acos(w(t- t )) Lecture Notes for Ecological Modeling --- Jianguo (Jingle) Wu

Lecture Notes for Ecological Modeling --- Jianguo (Jingle) Wu 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 -

tPage 62 Lecture Notes for Ecological Modeling --- Jianguo (Jingle) Wu

) shifts the peak by

tPage 64 Lecture Notes for Ecological Modeling --- Jianguo (Jingle) Wu physical units, and the

wPage 66 Lecture Notes for Ecological Modeling --- Jianguo (Jingle) Wu is the angular frequency per physical unit.

For example, suppose we have a time series: o Mean daily air temperatures: 40 degrees F o Amplitude: 25 degrees F o Period: 365 days o Position of the 1st peak: July 30 (Julian day 211)

The equation becomes:

2 p T = 40 + 25cosж (t- 211)ц и365 ш Lecture Notes for Ecological Modeling --- Jianguo (Jingle) Wu

(IV) Chain Structure and Time Delay

See modules.

SV16 SV17 SV18 DV16

I16 O16 O17 O19

exit fraction 3 transfer fraction 1 transfer fraction 2

Stock 1 Stock 2 Stock 3

inflow transfer 1 transfer 2 exit 3

exit fraction 1

exit 1 exit 2 exit fraction 2