Tennis Ball Mathematical Modeling Pdf

A model is a simplified representation of part of the real world. In this chapter we discuss models that can be described mathematically • Models are based on theory. In research models help to test theory by making predictions that can be compared with observations • Models also allow the implications of research results to be explored by making predictions for new situations • Each model is built for a specific purpose. A model that is useful for one job may be inappropriate for another task on a similar topic • Models vary in scope from the simple, which you can put together and use very quickly, to the complex that may take much of your project time to develop and use • Computing tools designed for the job can make modelling feasible for students who are not specialists 4.4 Mathematical models

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• A model is a simplified

representation of part of the

real world. In this chapter

we discuss models that can

be described mathematically

• Models are based on theory.

In research models help to

test theory by making

predictions that can be

compared with observations

• Models also allow the

implications of research

results to be explored by

making predictions for

new situations

• Each model is built for a

specific purpose. A model

that is useful for one job

may be inappropriate for

another task on a similar

topic

• Models vary in scope from

the simple, which you can

put together and use very

quickly, to the complex that

may take much of your

project time to develop

and use

• Computing tools designed

for the job can make

modelling feasible for

students who are not

specialists

4.4 Mathematical models

Catherine Wangari Muthuri

Introduction

Modelling can mean many things in research, and models of

one sort or another play a crucial role in much research.

Experience shows that the role and use of models is rarely

explained in research methods courses. The result is that

many students have only a vague idea about what models can

and should be doing for them. Modelling is often regarded as

the domain of specialists who sit hunched over computers, not

of agricultural researchers who want to solve real problems in

the field. The result is that much research is less effective than

it might be. The aim of this chapter is to start to fill that gap.

The chapter is divided into three major parts. The first

shows you how models are a natural part of the research

process. This is to help you develop your ideas from the general

'models are everywhere' to the main focus of the chapter,

which is concerned with mathematical or simulation models.

The second part discusses your options if you plan to do some

mathematical modelling. Finally, details of the steps you need

to follow to construct, use and test simple models are

described, using examples where modelling tools have been

applied in research studies in Kenya. Research findings can be

enriched by the use of simulation models and this is an

attempt to encourage you not to shy away from using

modelling tools just because you don't like maths!

Model types

Models are everywhere

You may not be aware of them, but you are using models all

the time. Models are not restricted to science. A religious belief,

philosophy or stereotypes are models. Indeed we use models

unconsciously in all decision making from deciding when to

change lane when driving to choosing careers. They come as

physical models in all shapes and sizes from dolls,

miniaturised cars and aeroplanes and globes, or as visual

representations in maps or pictures. They may be presented

as verbal or mental models, or in more abstract arithmetic or

algebraic form, in nearly all we learn. A model is just a

simplified representation of part of the real world.

GEAR: 4.4 Mathematical models 231

Physical models have been used for centuries in research. Engineers use models of boats to

study their stability and resistance to movement through the water. In biological research one

species is often said to 'model' another; in the early stages of medical research monkeys and

mice are used to model man, because they represent some aspects of human physiology well.

The images we carry in our minds, i.e., mental models, are simplified representations of complex

systems. We use them constantly to interpret the world around us and we usually do not realise

that we are doing so.

None of these models involve the complete similarity of real world and model, but similarity

in key features. A model is useful if it behaves in a realistic way for your problem. The scale

model of a ship may be useful for investigating its stability in the water, but it will be useless

for determining the profitability of operating the ship. Different models of the same

phenomenon are useful for different things. Take a 1-ha farm as an example. A map of the farm

(a visual scale model) might be useful when the farmer is planning the location of different

crops. Physical models of the landscape, built up from clay and painted, can be used to examine

the interaction of the farm with neighbouring farms and other land areas. Numerical input-

output models help in making investment decisions. Detailed numerical topological models can

be used to understand water flow and erosion on the farm. Each of these is a 'model of the farm'

and each is useful for its own purpose, but inadequate for other purposes.

Models in the research process

Research involves developing a theory of the real world and testing it with observation, then

perhaps using it to explain and predict further phenomena. Models are representations of the

theory and hence a fundamental part of the research process. Whether the model needs to be

formalised and described mathematically depends on whether the predictions of the theory can

be worked out without formalisation.

Models can be used in two steps of research:

1 In generating hypotheses or predictions, that will suggest the observations of the real world

that need to be made.

2 In assessing the extent to which our theory (as captured in the model) explains the real-

world observations.

If the model and observations agree then there is nothing in the data to suggest the theory

is not a good description of the real world. But of course we might have collected data that

does not test the theory in ways that are interesting! An important part of research design is

planning observations that do discriminate between models which are fit and unfit for their

intended purpose.

If the model and observations do not agree then you can:

• Question the model structure and assumptions, revise and retest it

• Question the data: perhaps it is not really relevant to the model you have chosen

• Abandon the line of research.

Note that we should not persist with a model for too long when the 'real world' evidence is

that it should be rejected.

Mathematical models

This chapter is about the mathematical models that are used in agricultural research. If the

relationships and rules that make up the model are sufficiently well specified, then they can

be written down mathematically and produce numerical results. In very many models the

basic mathematical relationships and rules are simple (such statements as 'volume = mass /

density 'or 'yield is zero until after flowering'). Complex patterns of results often emerge

because of the many interacting components, rather than because there are some complex

mathematical ideas embedded in the model. This is important. It means you do not have to

GEAR: 4.4 Mathematical models 232

be a mathematician, or even very good at using mathematics, to make effective use of

models in your research.

A mathematical model is a set of equations that represent interconnections in a system, and

can be worked out either by hand or by using a computer. The equations are written in terms

of mathematical objects that correspond directly to physical quantities. If these objects change

as part of the phenomenon they are generally called variables while if they are fixed they are

generally called parameters.

Typically a model will consist of formulae that link some responses or quantities of interest

with inputs, or the things that affect them. For example, a simple model of soil moisture changes

is illustrated in Figure 1.

The soil moisture (W) at time t is Wt. Rainfall is Rt, uptake by plants is Utand drainage is Dt.

The model can be written mathematically as:

Wt+1 = Wt+ Rt– Ut– Dt

If we know the initial conditions (W0) and

the values of R, U and D then we can calculate

W at any time. The model is simplistic. It

ignores soil evaporation. That will not be a

problem if the model is being built for

applications in which soil evaporation can be

ignored, but would be a major deficiency in

other cases. The model also requires inputs

that might be hard to measure (U and D). For

some purposes you might be able to predict U,

by adding another part (some more

components) to the model. For some purposes

you could take:

Ut = c.Pt

where Ptis the potential evapotranspiration

and c is a 'constant'. This model might well be

useful for studying the effect of day-to-day

changes in P on W. However it is still too

simplistic for longer-term studies, as c will

probably not be constant, but will change as the

crop grows and matures. The value of c may

also depend on W, with the plants able to take

up less water when the soil is drier. It is easy to see how this process can quickly lead to models of

ever-increasing complexity, even though each step involves simple and realistic relationships.

Part of the skill in modelling is in choosing the components to model, including the things

which will be necessary but not putting in everything you can think of.

Conceptual and empirical models

Models can either be empirical (data-driven) or theoretical (theory-driven or conceptual). An

empirical model is based mainly on data. It may be used in statistical analysis of study results

and to predict within domains of 'similar' conditions to the empirical base. It does not explain

a system. For example, a fertilizer response curve is an empirical model. It can be developed

from observations on the yield of crops with different amounts of fertilizer, and used to predict

the yield at any fertilizer level. However, it does not explain why the yield response is the way

it is. An empirical model consists of one or more functions that capture the trend of the data.

Although you cannot use an empirical model to explain a system, you can use such a model to

GEAR: 4.4 Mathematical models 233

Figure 1. Simple model of soil moisture

predict behaviour. We use data to suggest the model, to estimate its parameters, and to test

the model. An empirical model is not built on general laws and is a condensed representation

of data. However many statistical or empirical models are built on elements of an underlying

theory, for example, we construct the input variables in a regression model based on a

theoretical understanding of factors that should determine the response.

A conceptual, theoretical or 'process based' model includes a set of general laws or

theoretical principles. If all the governing physical laws were well known and could be described

by equations of mathematical physics, the model would be physically based. However, all

existing theoretical models simplify the physical system and often include obviously empirical

components. Thus the distinction between conceptual and empirical models is not clear-cut.

And again, it is the modellers job to use something appropriate for the task, rather than to

assume that one approach has more intrinsic value than another.

Roles of models

Models play several roles including:

•Exploring the implications of theory. It may not be possible to see the implications of theories

that involve several interacting components without calculating what happens in different

conditions. Used in this way, models provide insights and add creativity

•Prediction or forecasting tools help users make sensible educated guesses about future

behaviour. These can be used in planning, scenario analysis and impact analysis

•Explaining observations and generating hypotheses

•Training so that learners can carry out 'virtual experiments', exploring the result of making

changes.

In research models can help answer such questions as:

'Can I construct a theory that explains my observations?'

'Is my hypothesis credible?'

'What new phenomenon does my theory help to explain?

Used for prediction, models can answer such questions as:

'Given the model, what will happen in the future?'

'Given the model, what's going on between places where I have data?'

'What is the likelihood of a given event?'

How to model

You have three options if you decide to use simulation models in your work. You can use an

already existing developed model, modify an existing model or develop a new model altogether.

Using an already developed model

Hundreds of models relevant to agricultural research have been developed and described and

are available to you. A few have to be purchased. Many are available free to researchers and can

be down loaded from web sites or obtained from the authors. The advantages of using a model

that someone else has developed include:

•Time saving. Some of the hard work has already been done

•Recognition. Some models have been widely used and described. Their value is already

recognised so you will find it easy to justify their use

•Support. You will find documentation, examples and maybe technical assistance in using

the models.

However there are also disadvantages, compared to the alternatives of developing your own

models. These include:

• You may not find a model that actually describes the phenomena in which you are interested

at the right level of simplification

GEAR: 4.4 Mathematical models 234

• The available models may require inputs that are not available to you

• You may not fully understand how the model is constructed (the theory on which it is based)

• The model may not run on any computer available to you, or in the way you need for your

research.

If you are considering using a model, then select it by:

1 Determining exactly what you want to do with it. You will only be able to decide if candidate

models are suitable when your task is clear.

2 Searching literature and the Internet for references to models that tackle your problems, and

asking experts in the field.

3 Evaluating each possible model against your requirements. If you end up with more than

one candidate then choose the simplest.

Modifying an already existing model

You may well find that no available model meets your requirements but that some come close.

Therefore it may be desirable to modify a model. Modifying it may mean anything from

changing the way input files are handled to adding to or changing some of the underlying theory.

Often modification will mean adding a description of further components and processes to

address a specific situation.

If you plan to adapt or modify an existing model, all the points above about selecting it apply.

In addition you will have to be able to:

• Get access to the original computer code and description of the theory behind it

• Understand them fully

• Know how to modify it for your needs.

The computing skills you need will probably be more than those you need to just run an

existing model.

Some models are much easier to adapt than others. If they were originally designed and

produced with adaptation in mind then the task may be straightforward. If they were not built

to be adapted the task of modification may be all but impossible.

Adapting a model takes longer than using an already existing model. You need to go though

all the steps in the modelling process that are discussed later in this chapter. This implies that

the exercise becomes a major component of your research. It therefore demands that you have

sufficient skills and are familiar with the language of the packages and software used.

Developing a new model

The third option is to develop your own model. Situations that necessitate developing models

include those when:

• The outputs generated and inputs required are not catered for in the existing models

• Exiting models are too clumsy or complicated, or have a poor track record

• You are working in an area where no existing models can be found.

Given the novelty of most research, the last is likely to be the case. Building and using your own

models could be:

• Something that takes a few hours, if you are simply looking at a few interacting components

and are familiar with a suitable computing environment

• Something that takes most of your 3 years as a PhD student!

More likely it will be somewhere between the two. The steps in developing a model are

outlined below. The most critical are the first ones: defining useful and realistic objectives .

You will probably be most successful if you start with simple objectives. Reduce the problem to

its simplest objectives, and work on the simplest model that will meet those. This might be a

model with no more than two interacting components and simple rules describing them. Yet

even these models can give insights into your theory and observations that are not apparent

GEAR: 4.4 Mathematical models 235

until the model is formalised.

Steps in modelling

The steps involved in the modelling process

are summarised in the flowchart (Figure 2).

However, developing any useful model will be

an iterative process – you will certainly have

to return to early steps, for example, if you are

looking again at the interactions in your

model when it does not seem to give sensible

predictions.

The model-building process can be as

enlightening as the model itself, because it

reveals what you know and what you don't

know about the connections and causalities

in the system you are studying. Thus

modelling can suggest what might be fruitful

paths for you to study and also help you to

pursue those paths.

Formulate a clear problem statement and characterise the results expected

As with all other aspects of research, what you do depends on what you want to find out. Setting

realistic and detailed objectives for your modelling will determine the whole nature of the task.

It will help you decide on the following important characteristics of models:

1 Will the model need to be deterministic or stochastic?

In deterministic models the future state of the system is completely determined (in principle)

by previous behaviour. In stochastic models the system is subject to unpredictable, random

changes. These models involve probability and statistics. If you are interested in risks, your

model will have to use stochastic components.

2 What timescale is appropriate?

The timescale of the processes in question determines the timescale of the models. Depending

on the time taken for the processes under question to reach an equilibrium or to be felt, useful

decisions on what to include/exclude in the model can be reached. For example, when looking

forward 100 years, you need to ignore daily/monthly or seasonal variations of the parameters

in question. Such variations can be ignored in a long-term model but could be important in a

short-time model. Examples of scales and typical times are:

• Metabolic (enzyme-catalyzed reactions; seconds to minutes)

• Epigenetic (short-term regulation of enzyme concentration; minutes to hours)

• Developmental (hours to years)

• Evolutionary (months to years).

3 Does the model need to be spatial?

All agriculture takes place in a spatial context, but only some problems require you to

specifically describe spatial interactions. Think of the problem of modelling small farms. If you

want to describe economic inputs and outputs of the farm you need to know that there are

crops, animals and trees, but it may not matter where on the farm they are. If you want to

model nutrient flow between tree and crop plots, then their location matters and the model

you use will have to be explicit about that. Many of the management decisions made by small-

GEAR: 4.4 Mathematical models 236

Figure 2. A summary of the steps involved

in the modelling process

scale farmers living in heterogeneous environments make use of spatial variability on their

farms, such as growing different crops on different patches of land, abandoning part of their

land, or focusing their efforts only on those patches with the highest returns to investment of

labour or inputs. Most of the current models in agriculture do not handle spatial variability well,

if at all. There is a clear need to develop existing models further, or to construct new ones, in

order to address this limitation. Unfortunately, the structure of many existing models does not

facilitate transformation to spatially explicit versions, as their linear nature restricts them to

being run in sequence many times, in order to simulate each patch of land in turn. This makes

it difficult to simulate simultaneous interactions between patches of land (e.g., soil, or water

flow down a gradient). In circumstances where spatial variability is a key factor affecting the

study it is advisable for you to explore using a model that takes this into consideration.

Determine the key variables in the system and their interconnections

In this step you need to determine the key variables in your study that will be represented by

variables in the computer model. Key variables are the few most important, significant factors

that affect the system and their relationships. The cause- and-effect connections in the real

system will be represented by interconnections in the computer model. Adding more and more

interconnections makes the model complex, though by design, models should be a

simplification of the system under study. A determinant of model usefulness is therefore the

ability of the modeller to leave out unimportant factors and capture the interactions among the

important factors. Note that a model is:

• Too complex when there are too many assumptions and relations to be understood

• Too simple when it excludes factors known to be important.

Constructing a model

Building a model is an interactive, trial and error process. A model is usually built up in steps

of increasing complexity until it is capable of describing the aspects of the system of interest.

Note: It will never 'reproduce reality'.

The appropriate tools you need to construct a model depend on the complexity of the model.

The simplest tools may be paper and pencil. Others may use spreadsheets, while the more

complex models may require dedicated modelling software that uses its own language. The

simplest mathematical model takes the form of equations show how the magnitude of one

variable can be calculated from the others and spreadsheets like Excel are adequate for the task.

More complex computer simulations use special software that allows the building and testing

of a model. There are software products available that make building and running some types

of models very easy even if you know nothing about computer programming. Investigate such

software as STELLA and ModelMaker before trying to write your own code in lower-level

computing languages. They make the job of developing and running your own models very

much simpler!

The development of the simple soil water model outlined in Figure 1 is shown here to give

you an idea of what is involved. The model represented in Figure 1 is drawn in STELLA. In Figure

3a. STELLA uses four main types of building blocks:

Stocks. These are stores of 'stuff', represented by rectangles. They may describe water, money,

people, biomass,… whatever you are modelling.

Flows. These are the movements of material into and out of stocks, represented by broad

arrows. The arrow can be thought of as a pipe, with a tap on it to regulate the flow. Sources and

sinks of the material are represented by 'clouds'.

Converters. These are represented by circles. They hold values of constants and formulae used

to convert one type of material to another.

Connectors. These narrow arrows show the logical connections between components in the

GEAR: 4.4 Mathematical models 237

model. The equations describing the model

must be consistent with these connections.

The stock of soil water (W) has an inflow of

rain (R) and outflows of uptake (U) and drainage

(D). The actual values of these are read from

data files. The model is completed by filling in

a formula or other details in each location

marked by '?'. The model can then be run.

In Figure 3b the uptake is now calculated as

c.P, where P is the potential evapotranspiration

(PeT), also read from a file. It should be clear

from this that modifying the model requires

little more than adding components to the

diagram. The real challenge of course is

deciding how to model uptake, not changing

the computer code – this is why software such

as STELLA is so important. The final step

(Figure 3c) shown here displays two more

changes that the modeller thought would

help. The drainage is now calculated (because

there was no measured data available) and the

uptake now depends on both the crop biomass

and the soil water. The latter involves keeping

track of the biomass growth, a second stock in

the model. Many physiologists would be

uncomfortable with a single 'type' of biomass,

and start differentiating it into, say, roots,

stems, leaf and grain. Then you need to add

components that describe what the

partitioning depends on. Similarly the soil

scientist would like to have several soil layers,

each with different hydraulic properties. The

model can quickly become complex. The value

of software such as STELLA is that it allows

you, as researcher, to think about what

constitutes a sensible model for you, rather

than worrying about computer codes.

Sensitivity analysis, validation,

verification and calibration

Sensitivity analysis

Through sensitivity analysis, you can gain a

good overview of the most sensitive

components of the model. Sensitivity analysis

attempts to provide a measure of the

sensitivity of other parameters or forcing

functions, or sub-models to the stated

variables of greatest interest in the model. It

helps you to systematically explore the

response of the model to changes in one or

GEAR: 4.4 Mathematical models 238

Figure 3a. Simple soil water

model in STELLA

Figure 3b. Simple soil water model with

uptake modelled as c.PeT

Figure 3c. Simple soil water model with

uptake depending on both crop biomass

and soil water

more parameters, to see how sensitive the overall model outcome is to a change in value. This

sensitivity is always dependent on the context of the setting of other parameters, so you should

be careful about the conclusions you draw. Some parameters only matter in particular types of

circumstance. Others, however, seem to always matter, or to matter hardly at all. This type of

model analysis is used to see which parameters should get priority in a measurement

programme. You must be provided with affordable techniques for sensitivity analysis if you are

to understand which relationships are meaningful in complicated models. This is equally true

whether you are using an already developed model, modifying a model or developing one.

Validation, verification and calibration

In general, verification focuses on the internal consistency of a model, while validation is

concerned with the correspondence between the model and the reality. Calibration checks that

the data generated by the simulation matches real (observed) data, it can also be considered as

tuning of existing parameters. These steps can be among the most conceptually difficult. No

model is universally 'valid' in the sense that it will give 'correct' predictions in all circumstances.

There will always be discrepancies between observed and predicted values. These discrepancies

can be made smaller by calibration and by making adjustments to the model. However this

does not necessarily increase the usefulness of the model in either: explaining your observations

of the real world, or making predictions about behaviour in the real world.

Simulate a variety of scenarios to generate non-obvious discussion

Simulation models have been used widely in Kenya to address various problems. Three

examples are given to help you see how they can be used.

Soil fertility management in western Kenya: Dynamic simulation of productivity,

profitability and sustainability at different resource endowment levels

A farm economic-ecological simulation model was designed to assess the long-term impact of

existing soil management strategies, on-farm productivity, profitability and sustainability. The

authors developed a model that links biophysical and economic processes at the farm scale.

The model, which runs in time units of 1 year, describes soil management practices, nutrient

availability, plant and livestock productivity, and farm economics. It concluded that low land

and capital resources constrain the adoption of sustainable soil management practices on the

majority of farms in the study area. Previously it had been assumed that low-input organic

methods were suitable for the poorest farmers. For more details, see Shepherd and Soule (1998).

Modelling leaf phenology effects on growth and water use in an agroforestry system

containing maize in the semi-arid Central Kenya using WaNuLCAS

The three tree species under study were Grevillea robusta (evergreen), Alnus acuminata

(semidecidous) and Paulownia fortunei (deciduous). The inputs included climate data, soil data,

calendar of events, crop and tree parameters, agroforestry zones and layers, and leafing

phenologies. The scenario outputs included soil water balance, tree and crop biomass and

stem diameter. WaNuLCAS model simulations demonstrated that altering leaf phenology

from evergreen through semi-deciduous to deciduous decreased tree water uptake and

interception losses but increased crop water uptake, and drainage rates in all the species. It

was therefore concluded that deciduous tree species would compete less with crops and be

more advantageous in increasing stream flow than evergreen trees. Phenology had not

previously been a major consideration in determining tree selection For more details, see

Muthuri (2004).

Modelling the benefits of soil water conservation using PARCH; A case study from a semi-

arid region of Kenya.

GEAR: 4.4 Mathematical models 239

The PARCH model was used to simulate maize grain yield under three soil/water conservation

scenarios: 1. a typical situation where 30% of rainfall above a 15 mm threshold is lost as runoff,

2. runoff control, where all rainfall infiltrates, and 3. runoff harvesting, which results in 60%

extra 'rainfall' for rains above 15 mm. The study showed that runoff control and runoff

harvesting produced significant maize yield increases in both the short and the long rains.

Previously runoff control was justified more for erosion benefits than increased crop production.

For more details, see Stephens and Hess (1999).

Conclusions

The success of models developed by physicists and chemists has led to the rapid development of

modern technology, the conquest of many diseases resulting in increased life expectancy, and the

improvement of human lives on earth. But, no matter how successful a model has been, scientists

realise there may be aspects of the world that the model fails to explain, or worse, predicts

incorrectly. Nevertheless, creating and using models is one of the most powerful tools ever

developed. But, there is a need to revise and improve models as new information is discovered.

Further resource material and references

There are many books, journals and articles on models. Most tend to be specialised and specific

to certain models or application of models in specific areas of specialisations. To understand

some basics on what models are, and how you can build a model, three books are listed below

particularly useful.

Appendix 1. The Craft of Research. Paul L. Woomer.

Appendix 11. ICRAF. 2003. Genstat Discovery Edition and Other Resources. World Agroforestry Centre

(ICRAF), Nairobi, Kenya.

Anon. 2003. Why the analysis of wide range of physical phenomena leads to consistent and successful

results when applying the BSM concept and models? http://www.helical-structures.org/Applications/

why_successful.htm [accessed June 2009]

Ford, A. 1999. Modelling the Environment. An introduction to system dynamics modelling of Environmental

Systems. Island Press, California, USA. 401 pp.

Jorgensen, S.E. 1994. Fundamentals of ecological modelling. Elsevier, London, UK. 628 pp.

Matthews, B.R. and Stephens, W. 2002. Crop-Soil Simulation Models: Applications in Developing Countries. CAB

International, Wallingford, UK.

Muthuri, C.W. 2004. Impact of Agroforestry on crop performance and water resources in semi-arid central Kenya.

PhD Thesis. Jomo Kenyatta University of Agriculture and Technology (JKUAT). 289 pp.

Van Noordwijk, M. and Lusiana, B. and Ni'matul Khasanah. 2004. WaNuLCAS version 3.01: Background

on a model of water, nutrient and light capture in agroforestry systems. International Centre for

Research in Agroforestry (ICRAF), Bogor, Indonesia, 246 pp

Shepherd, K.D. and Soule, M.J. 1998. Soil fertility management in Western Kenya: dynamic simulation of

productivity, profitability and sustainability at different resource endowment levels. Agriculture,

Ecosystem and Environment 71: 131-145.

GEAR: 4.4 Mathematical models 240

Soto, R. 2003. Introducing System Thinking in High School. The Connector 1(5).

http://www.iseesystems.com/community/connector/Zine/SeptOct03/jake.html [accessed June 2009]

Stephens, W. and Hess, T.M. 1999. Modelling the benefits of soil water conservation using the PARCH

model –a case study from a semi-arid region of Kenya. Journal of Arid Environments 41: 335-344.

Vohnout, K. 2003 Mathematical Modelling For System Analysis In Agricultural Research. Elsevier, New York.

452pp

Internet resources

Ecological models http://ecobas.org/www-server/

CERES crop models http://www-bioclim.grignon.inra.fr/ecobilan/cerca/ceres.html

FALLOW model at http://www.worldagroforestry.org/Sea/Products/AFModels/fallow/Fallowa.htm

FLORES model at http://www.cifor.cgiar.org/acm/methods/models.html An example of model building in

participatory research

PARCHED-THIRST at http://www.ncl.ac.uk/environment/people/publication/13158/

WaNuLCAS model at http://www.worldagroforestry.org/SEA/Products/AFModels/wanulcas/

STELLA software; ISEE Systems at http://www.iseesystems.com

Powersim software; The business simulating company http://www.powersim.com

Vensim PLE. Vantana Systems Inc. http://www.vensim.com

Management Unit of the North sea Mathematical Models (MUMM) (2003)

http://www.mumm.ac.be/EN/Models/Development/Ecosystem/how.php

Model Maker: available from http://www.modelkinetix.com/modelmaker/

GEAR: 4.4 Mathematical models 241

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Models of tree-soil-crop interactions in agroforestry should maintain a balance between dynamic processes and spatial patterns of interactions for common resources. We give an outline and discuss major assumptions underlying the WaNuLCAS model of water, nitrogen and light interactions in agroforestry systems. The model was developed to deal with a wide range of agroforestry systems: hedgerow intercropping on flat or sloping land, fallow-crop mosaics or isolated trees in parklands, with minimum parameter adjustments. Examples are presented for simulation runs of hedgerow intercropping systems at different hedgerow spacings and pruning regimes, a test of the safety-net function of deep tree roots, lateral interactions in crop-fallow mosaics and a first exploration for parkland systems with a circular geometry.

K.D. Vohnout

This book provides a clear picture of the use of applied mathematics as a tool for improving the accuracy of agricultural research. For decades, statistics has been regarded as the fundamental tool of the scientific method. With new breakthroughs in computers and computer software, it has become feasible and necessary to improve the traditional approach in agricultural research by including additional mathematical modeling procedures. The difficulty with the use of mathematics for agricultural scientists is that most courses in applied mathematics have been designed for engineering students. This publication is written by a professional in animal science targeting professionals in the biological, namely agricultural and animal scientists and graduate students in agricultural and animal sciences. The only prerequisite for the reader to understand the topics of this book is an introduction to college algebra, calculus and statistics. This is a manual of procedures for the mathematical modeling of agricultural systems and for the design and analyses of experimental data and experimental tests. It is a step-by-step guide for mathematical modeling of agricultural systems, starting with the statement of the research problem and up to implementing the project and running system experiments.

Field experiments in the semi-arid regions of Kenya have shown that soil water conservation techniques can result in increased maize grain yields. The degree of benefit in a particular season is dependant on the rainfall amount and distribution. However, the results of field experiments are limited to a few years of observations at specific locations and it is therefore difficult to make generalized conclusions about the benefits in years of differing rainfall patterns.The PARCH model has been calibrated for Katumani Composite 'B'—a local variety of maize used in the Machakos district of Kenya—and validated against observed grain and dry matter yields from experimental plots. Historical daily rainfall data were collected for Katumani Research Station, in the semi-arid region of Kenya, during the period 1961 to 1994. Seasonal totals for the short rains (October to February) and long rains (February to August) were calculated and nine years were selected as representative of 'wet', 'average' and 'dry' seasons for the long and short rains, respectively.The PARCH model was then used to simulate the maize grain yield under three soil water conservation scenarios: (i) a typical situation where 30% of rainfall above a 15 mm threshold is lost as runoff, (ii) runoff control, where all rainfall infiltrates, and (iii) runoff harvesting, which results in 60% extra 'rainfall' for rains above 15 mm. The soil was taken to be a sandy clay loam which is typically found in the region. Two planting densities of 4·4 and 8·8 plants m−2were used to simulate normal and high levels of management. Planting dates were determined from the 30% runoff scenario and were fixed for the other scenarios to avoid confounding the results.The results showed that runoff control and runoff harvesting produce significant yield increases in 'average' years in both the long rains and the short rains. However, in 'dry' years there were only small yield increases in the short rains and negligible benefit in the long rains. In 'wet' years there were no significant yield increases due to water conservation in either season.Clearly, these results are a simplification of the real situation where water conservation strategies may allow earlier planting or be accompanied by increased planting densities, both of which may result in yield increases. However, this work demonstrates the usefulness of appropriate crop growth models in evaluating a wider range of crop management strategies under a realistic range of climatic conditions than would be possible in the field.

Keith Duncan Shepherd
M.J. Soule

A farm simulation model was designed to assess the long-term impact of existing soil management strategies, on farm productivity, profitability and sustainability. The model, which runs in time units of 1 year, links soil management practices, nutrient availability, plant and livestock productivity, and farm economics A case study is presented of the application of the model to existing, mixed farm systems in Vihiga district, in the highlands of western Kenya. Three representative farm types were developed using participatory techniques to reflect differences in resource endowments and constraints faced by farmers. The model was used to assess the sustainability of the existing systems for the three farm types as a basis for recommending improved practices for each. A summary model for calculating new sustainability indicators of soil productivity is presented. The low (LRE) and medium (MRE) resource endowment farms, which comprise about 90% of the farms in the area, have declining soil organic matter and low productivity and profitability. In contrast, the high resource endowment category of farms (HRE) have increasing soil organic matter, low soil nutrient losses and are productive and profitable. Crop nutrient yields were 17, 19 and 86 kg N ha−1 year−1 on LRE, MRE and HRE farms, respectively. Soil C, N and P budgets were negative in LRE and MRE but positive in HRE. Farm revenue in LRE and MRE was 2–13% of farm revenue in HRE. It comprised 7% of household income in LRE compared with 25% in MRE and 63% in HRE. It is concluded that low land and capital resources constrain the adoption of ecologically and economically sustainable soil management practices on the majority of farms in the area. Strategies are needed to (i) increase the value of farm output (ii) increase high quality nutrient inputs at low cash and labour costs to the farmer, and (iii) increase off-farm income.

Andrew Ford

Curso de: Ecología (01PT029) Introducción a los sistemas dinámicos utilizados como modelos de simulación en el área de estudios medioambientales. Además de presentar los conceptos básicos, ilustra al lector sobre la mecánica para la construcción de modelos, así como un gran volumen de ejercicios.

Why the analysis of wide range of physical phenomena leads to consistent and successful results when applying the BSM concept and models?

Anon

Anon. 2003. Why the analysis of wide range of physical phenomena leads to consistent and successful results when applying the BSM concept and models? http://www.helical-structures.org/Applications/ why_successful.htm [accessed June 2009]

Crop-Soil Simulation Models: Applications in Developing Countries

B R Matthews
W Stephens

Matthews, B.R. and Stephens, W. 2002. Crop-Soil Simulation Models: Applications in Developing Countries. CAB International, Wallingford, UK.

Impact of Agroforestry on crop performance and water resources in semi-arid central Kenya

C W Muthuri

Muthuri, C.W. 2004. Impact of Agroforestry on crop performance and water resources in semi-arid central Kenya. PhD Thesis. Jomo Kenyatta University of Agriculture and Technology (JKUAT). 289 pp.

Introducing System Thinking in High School