1. Yield Gap Analysis and Crop Modeling Workshop
Nairobi, Kenya
RESEARCH PROGRAMS ON
Climate Change,
Agriculture and
Food Security
SYSTEMS ANALYSIS IN AGRICULTURE
Integrated Systems
for the Humid
Tropics
Roots,
Tubers
and Bananas
International Potato Center
Sub-program: Production Systems and Environment

2. SYSTEMS ANALYSIS IN AGRICULTURE

8. 1. Collection of elements
2. Connected
3. Forming a unit

9. A particular attribute of most agricultural systems
is their complexity. Therefore, when studying
complex systems we should follow Albert
Einstein’s rule: Make things as simple as possible,
BUT NOT SIMPLER THAN THAT

14. Mathematics is used to synthesize and
understand the behavior of a system:
•Reductionist knowledge of the parts of a system (known as mathematical
models)
•Mean of articulating our ideas and formalizing them in an abstract way

20. Stephen W. Hawking
Theoretical Physicist
Cambridge University

21. Methodology
Define objectives
Analysis of the system
Synthesis
5000
Verification
y = 1.0657x - 195.55
R2 = 0.9925
Validation
Sensitivity analyses
500
0
-500
Scenario analyses
-1000
Y -1500
-2000
Documentation
-2500
8.0
5.0
-3000
2.0
-1.0
-3500
1 2 3
4 5 6
7
-4.0
8
9 10 11
12 13 14
15 16 17
X1
18 19 20
21
-7.0
-10.0
X2
Simulados
4000
3000
2000
2000
2500
3000 3500 4000
Observados
4500
5000

22. Methodology
Defining Objectives
Problem to be Addressed
Define objectives
Analysis of the system
Synthesis
Verification
Defining Effective Measurements
Analysis of the System
Determine Components of the
System
Defining model Variables
Synthesis
Validation
Sensitivity analyses
Scenario analyses
Documentation
Defining working hypotheses
Abstraction of components
Developing the Mathematical
Algorithm
Programming

23. Define objectives
Irradiance
Day t hour h
Analysis of the system
Respiration
Day t
Synthesis
Verification
Validation
Sensitivity analyses
Scenario analyses
Documentation
Biomass
Day t-1
GPP
Day t
NPP
Day t

24. Linear Regression (Observed vs. Simulated).
5000
y = 1.0657x - 195.55
R2 = 0.9925
Define objectives
Simulados
4000
3000
Analysis of the system
2000
2000
Synthesis
2500
3000 3500 4000
Observados
4500
5000
Ho (1) : o = 0
Ha (1) : o  0
Verification
Validation
Ho (2) : 1 = 1
Ha (2) : 1  1
Residual Analysis (Observed vs. Simulated).
Residuales (y-ye)
Scenario analyses
150
100
Sensitivity analyses
200
150
100
Residuales (y-ye)
200
50
0
-50
50
0
-50
-100
-100
-150
-150
-200
Documentation
-200
Observaciones
Observaciones
ei = yi – yei

25. Define objectives
Analysis of the system
Synthesis
Running the model to generate
desired information
Find estimated values of input
and state variables that maximize
(or minimize) ouput variables
Verification
Validation
Sensitivity analyses
Scenario analyses
Documentation
What
Happens if

28. Basic concepts required
to model systems
dynamics

29. Hierarchy of Yield Drivers and Associated Yield Levels
Crop Traits
Germplasm
Production Situation
Defining factors
Potential yield (Yp)
CO2
Radiation
Limiting factors
Attainable yield
Climate
Temperatu
re
Yield increasing measures
Reducing factors
Water
Actual yield (Ya)
Yield protecting measures
Nutrients
Soils
Weeds
Pests
Dry Matter Yield, Mg/Ha
Diseases
Modified by R. Quiroz from Penning de Vries & Rabbinge, 1995

30. Growth and development
Growth. The increase of weight or volume of the total plant
or various plant organs.
Development. The passing through consecutive
phenological phases. Characterized by the order and rate
of appearance of vegetative and reproductive plant organs.

31. 20
0
10
Bacteria, number
30
40
Let us say we put a single bacteria in a culture that divides itself
every half minute; in 15 min there will be 45
0
5
10
15
Time,min
Most living organism present growth patterns similar
to this figure. That is, it follows an exponential
increase in number or weight.

32. Let’s assume we have a culture that divides itself every
unit of time (t). If we record the weight and we say that
the first cell had a weight w0, then when divided into
two the weight is 2w0, son on and so forth, we will have:
Time, t
Weight, w
1
w0
2
2w0
3
3w0
4
4w0
5
5w0
The shape of the growth response, as a function of
time, might be generically described by an exponential
function:
W(t) = w0 *e k*t

33. The growth rate at any time is:
30
10
20
dy
dx
0
Bacteria, number
40
dw/dt = k* W0 *Exp (k*t)
0
5
10
Time,min
15

34. We can calculate now the relative growth
rate (RGR), defined as the rate of growth
divided by the weight:
RGR =
dw/dt
RGR = k
W (t)
k* W0 *Exp (k*t)
=
W0 *Exp (k*t)

35. Now we have a little problem, plants and
other biological systems do not grow
indefinitely; as the organisms get bigger,
their growth rate slows until it reaches its
mature size, when RGR becomes zero
Therefore we need to modify our equation
for RGR. There are different ways and we
will use an arbitrary but convenient way

36. RGR*=
dw/dt
W
*(1 – g*W)
=
k (1 – g*W)
Where: g=1/Wmax
Putting this in words, when W is close to W0 RGR
is close to k but as W approaches Wmax RGR also
approaches zero

37. 0.000
0.010 0.020 0.030
RGR
0
0
50
Time,days
50
100
Weight
150
100
150
0
10
20
Weight
30
40

38. Now, let us say we have a plant growing without
restriction (water, climate, pest control, etc.)
Irradiance
Day t
Biomass
Day t-1
Respiration
Day t
GPP
Day t
W (t)= W0 *e k*t
Where: W(t) – weight at any time t
W0 – weight at t=0
k – growth constant
NPP
Day t

39. Conceptual representation of a
horizontal surface at the top of the
canopy
G
B
R NIR

40. A. Effect of temperature on the metabolic reaction rate
Optimal t°
Emergency Rate
Reaction Rate
%
B. Effect of so
potato plan
Temperature ( °C )
Respiration/photosynthesis rates
(gCO2 cm -2 hoja min -1
Total
D. Relationship b
solar energy u
M (gcm -2)
C. Effect of temperature on photosynthesis and
respiration in potato

41. Thermal time and growth
Growth and development of crops are strongly dependent on
temperature.
Each species requires a specific temperature range for
development to occur. They are named cardinal temperatures:
• Base temperature, Tb
• Optimum temperature, To
• Maximum temperature, Tm
Thermal time are commonly calculated as a Growing Degree
Days (GDDs), Growing Degree Units (GDUs), or heat units
(HUs). Different methods exist for calculating heat units.

42. Growing degree days calculation
Classical approach
ET = TX-Tb
Effective temperature
40
Where
Tx Mean temperature
30
20
10
0
-20
-10
0
10
20
30
40
50
40
50
Temperature
Tx < To, ET = TX (1-((Tx-To)/(To-Tb))2
Effective temperature
Alternative Approach
To
20
10
Tx > To, ET = TX (-((Tx-Tm)/(Tm-To))2
Tb
Tm
0
ET Effective temperature
ADD Acummulated degree days
-20
-10
0
10
20
Temperature
30

44. Potato phenology
Phase 0
between planting and emergence
Phase 1
between emergence and tuber initiation
Phase 2
between tuber initiation and the moment
when 90% of assimilates are partitioned to
the tubers
Phase 3
until the end of crop growth

45. Potato phenology
Patacamaya, La Paz
17°16' S 68°55' W 3800 m.a.s.l.
100%
Canopy Cover
80%
60%
40%
20%
GDD
Luk'y
Waycha
Alpha
Ajanhuiri
Gendarme
Phase 1
350 - 450 GDD
Phase 2
800 – 1000 GGD
Phase 3
1200 – 1400 GDD
1400
1200
1000
800
600
400
200
0
0%

46. SOLANUM Conceptual framework
Light
Light
Interception
LUE
(—)
DM
PAR
Kg DM.ha¨¹.d ¨¹
Photosynthetic
Apparatus
T
GC
LAI
Light Reflectance
Tubers
Roots
Stems
Leaves

47. Dry matter accumulation equation
The growth model, based on light interception and
utilization as proposed by Spitters (1987, 1990) and
Kooman (1995), was used to simulate the daily dry
matter accumulation, through the following general
equation:
Wt = flint*PAR*LUE
Where:
•Wt Growth rate at day t (g DM.m-2.d-1)
•flint Fraction of PAR intercepted by the foliage
•PAR Photosynthetically active radiation (MJ.m-2.d-1)
•LUE Light utilization efficiency (g DM.MJ-1 PAR)

48. The main growth processes
Light interception
Light use efficiency
Tuber partitioning

49. Model parameters
Fraction of light intercepted (FLINT)
Growth phase:
FLINT = (MCC * N * f0 * exp (R0*t)) / (N *f0 * exp(R0*T) + 1 – N *f0).
P1 maximum canopy cover, MCC
P2 initial light interception capacity, f0 (m2 pl-1)
P3 initial relative crop growth rate R0 (ºCd-1)
Senescence phase:
Ft = 0.5 – (t - t0.5) / d.
P4 duration of leaves senescence, d (ºCd),
P5 time when light interception was reduced to 50%, t0.5 (ºCd).

50. Fraction of light intercepted (FLINT)
MCC
1.0
0.8
0.6
Canopy cover
0.4
0.2
R0
f0
0.0
0
500
1000
Thermal time
1500
2000

51. Model parameters
Radiation use efficiency
P6 light use efficiency, RUE (gr MJ-1)
Partitioning harvest index function
HI=M/(1+(t_ac/A)b)
P7 asymptotic harvest index, M
P8 initial slope of the harvest index curve, b (ºCd-1),
P9 thermal time at the initial harvest index curve, A (ºCd)
Tuber dry matter
P10 tuber dry matter content (DMcont)

52. Radiation use efficiency - RUE
3000
y = 5.552x
R² = 0.933
Total dry matter (gr. m-2)
2000
1000
0
0
100
200
300
Intercepted PAR (MJ.m-2)
400

53. Asymptotic harvest index
1.0
M
0.8
0.6
Tuberization index
b
0.4
0.2
A
0.0
0
500
1000
Thermal time
1500
2000

54. The soil - plant - atmosphere system
CO2
Temperature
Radiation
Rainfall
Photosynthesis
Respiration
Photorespiration
Transpiration
Dry matter
Atmosphere
Plant
Farmer
practices
Water
Nutrients
Evaporation
Soil

55. Model parameterization
―Minimum data set‖
What to measure?
When to measure?
How to measure?

56. What to measure for estimating
potential production?
Solar radiation
Temperature
Planting date
Emergence date
Harvest date
Canopy cover/LAI/VI
Dry matter by plant organ
Dry matter content of tubers
Atmosphere
Plant

57. When to measure?
Daily meteorological data
Periodic crop growth measurements
Weekly
10 days
15 days

58. How to measure?

59. Meteorological data and
equipment
• Minimum and maximum
air temperature
• Solar incoming radiation
• Rainfall
• Reference evapotranspiration
• Soil temperature

60. Leaf area data acquisition

61. Determining leaf area index (LAI)
from NDVI
NIR - R
NDVI =
NIR + R
Where:
NIR: Near Infrarred
R: Red

62. Relationship between LAI and NDVI
data
simulated

63. Canopy cover data acquisition
Grid method

64. Canopy cover data acquisition
Segmented image method
Post-processing

65. Measuring dry matter
Leaves
Stems
Tubers
Roots

66. Parameter calculation example
La Molina, Peru
Latitude 12º 04’39‖ S
Longitude 76º 56’53‖ W
Altitude 280 m.a.s.l.
June - November 2006

67. Thanks