Quote of the Day - Arizona State Universityleml.asu.edu/jingle/Web_Pages/EcoMod_Website/Lecture...Quote of the Day "If you knew what you were doing, you'd probably be bored." --- Anonymous

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

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AN INTRODUCTION TO SOME BASIC MATHEMATICAL

CONCEPTS IN SYSTEMS MODELING I. DIFFERENCE EQUATIONS II. ORDINARY DIFFERENTIAL EQUATIONS III. RELATIONSHIP BETWEEN DIFFERENCE AND DIFFERENTIAL EQUATIONS IV. MATRIX ALGEBRA V. PARTIAL DIFFERENTIAL EQUATIONS

Quote of the Day "If you knew what you were doing, you'd probably be bored." --- Anonymous

I. DIFFERENCE EQUATIONS The mathematical expression of a recursion relation is called a difference equation. Difference equations are suitable for describing discrete systems. For example, they can be used for modeling populations whose generations do not overlap (i.e., adults die and are totally replaced by their progeny at fixed time intervals). A basic understanding of difference equations is important to comprehend the mathematical structure and numerical operations of simulation models. For an excellent description of difference equations and biological examples, see Edelstein-Keshet (1988). A brief introduction to some basic concepts of difference equations is sufficient as far as the purpose of this simulation modeling course goes. For simplicity, let’s begin with a few examples of difference equations in biology. 1. Some Examples (1) Cell Division (linear, first-order difference equation) Suppose a population of cells divides synchronously, with each cell producing r daughter cells. Then, the number of cells in generation n+1 can be written in terms of the number of cells in generation n:


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P

n+1= rP

n (1) where P is the number of cells, n is the generation number, and r (>0) is the number of daughter cells resulting from one parental cell. Note that equation (1) is recursive. Thus, if we know P0, we have:

P1= rP

0

P2= rP

1= r

2P0

P3= rP

2= r

3P0

......

The general analytical solution to equation (1) is:

Pn= rP

n-1= r

n

P0 (2)

Equation (2) allows one to calculate the number of cells in generation n directly from the initial cell population size. From equation (2), it is obvious that: when 0 < r < 1, P decreases; when r = 1, P remains constant; when r > 1, P increases. The increment between two consecutive generations is, therefore, P

n+1- P

n= rP

n- P

n= (r -1)P

n (3) or DP = (r -1)P

n The above two equations describe the rate of change (note that here Dn =1). Rewriting difference equations in terms of the rate of change as in the above example, when possible, helps represent it in a systems (or Forrester) diagram. For instance, a Forrester diagram for equation (1) may look like the following:


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P

Rate of Change

r STELLA equations: P(t) = P(t - dt) + (Rate_of_Change) * dt INIT P = 2 INFLOWS: Rate_of_Change = (r-1)*P r = 2

(2) Density-Independent Population Growth (linear, first-order difference equation) A general difference equation for population dynamics, without consideration of immigration and emigration, can be written as: N

t+1= N

t+ bN

t- dN

t (4) where N is the population size, t is time, and b and d are per capita birth and death rates, respectively, which are usually functions of population size itself and other biotic and abiotic factors. However, in the case of density-independent population growth, b and d are constants. Thus, equation (4) can be simplified into: N

t+1= (1+ r )Nt (5)

where r = b - d r is usually called the intrinsic growth rate. The analytical solution to equation (5) emerges easily after an attempt of calculating N over a few time steps using the recursive equation. As we did with the first example, here we have:

N1= (1 + r)N

0

N2= (1 + r)N

1= (1+ r )

2N0

......


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Thus, the general analytical solution is:

Nt= (1 + r)

t

N0 (6)

To facilitate representing equation (4) in a systems diagram, we can rewrite it in terms of the rate of change: N

t+1- N

t= DN = bN

t- dN

t 2. Linear Difference Equations The two examples above-mentioned are linear, first-order difference equations. These equations take the general form Xn +1 = f (Xn ) (7) or a

0Xn +1

+ a1Xn= b

n (8) where X is the variable of interest, f is a certain functional form, n = 0, 1, 2, ..., N, and a and b are constants. As seen above, linear, first-order difference equations can be easily solved analytically without use of computers. An mth-order, linear difference equation typically takes the form a

0Xn+ a

1Xn-1

+ ... + amXn -m

= bn (9)

or equivalently,

N

Total Birth Rate Total Death Rate

bd


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a0Xn +m

+ a1Xn +m-1

+ ... + amXn= b

n (9’) In equation (9), m is the order of the difference equation, which is the number of previous intervals that directly influence the value of X at a given interval. In general, the number of basic solutions to a difference equation is determined by its order. That is, a first-order equation has one solution, a second-order equation has two, and an mth-order equation has m basic solutions. For example, the following equation N

t+1= aN

t+ bN

t-1- cN

t-1 (10) is a linear, second-order difference equation which, like a system of two coupled first-order equations, has two basic solutions. In general, an mth-order difference equation may be rewritten into a system of m coupled first-order difference equations. For example, a second-order difference equation X

n +2= a

11Xn+1

+ a12a21Xn+ a

22(X

n+1- a

11Xn) (11)

may be expressed equivalently as:

Xn +1

= a11Xn+ a

12Yn

Yn+1

= a21Xn+ a

22Yn

(12)

In modeling biological and ecological phenomena, systems of difference equations are needed when two or more state variables are considered. Solving higher-order difference equations or systems of difference equations is not a trivial matter even when they are linear, and numerical solutions using computers are desirable in many cases. This is especially true for modeling real-world systems. 3. Nonlinear Difference Equations Let’s still consider the previous example on population growth. However, we now assume that birth rate decreases linearly with population size due to increasing crowding effect. Mathematically, this idea can be expressed using the following difference equation:

b(N) = b *K - N

t

K

Ê Ë

ˆ ¯ = b * 1 -

Nt

K

Ê Ë

ˆ ¯ (13)

where b(N) is the per capita birth rate which is a function of N, K is the carrying capacity, and b* is a constant related to birth rate. By examining equation (13), b* is numerically equal to b(N) when N is 0 (biologically makes no sense!). In practice, b* may be defined as the maximum birth rate found over a range of population densities.


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Substituting equation (13) into the general population equation (eq. 4) results in:

Nt+1 = N

t+ b * 1 -

Nt

K

Ê Ë

ˆ ¯ Nt

- dNt (14)

or

Nt+1 = N

t+ b * N

t-Nt

2

K

Ê Ë Á ˆ

¯ - dN

t

When N0 is known, Nt can be calculated recursively using the above equation as follows:

N1

= N0

+ b* 1 - N0

K

Ê Ë

ˆ ¯ - dN

0

N2

= N1

+ b* 1 -N1

K

Ê Ë

ˆ ¯ - dN

1

......

Note that equation (8) contains N

t

2 and, thus, is no longer a linear difference equation. Nonlinear difference equations usually are difficult or impossible to be solved analytically, and numerical solutions with the aid of computers ought to be sought. Equation (14) can be rewritten as

DN = b * 1-Nt

K

Ê Ë

ˆ ¯ Nt

- dNt

A corresponding systems diagram may look like the one below. The result of a sample run is also presented.

N

Total Birth Rate Total Death Rate

b dK N(t) = N(t - dt) + (Total_Birth_Rate - Total_Death_Rate) * dt INIT N = 10 INFLOWS: Total_Birth_Rate = b*(1-N/K)*N OUTFLOWS: Total_Death_Rate = d*N b = 0.1 d = 0.05 K = 100

11:57 AM 2/20/97

0.00 25.00 50.00 75.00 100.00

Time

1:

1 :

1 :

2 :

2 :

2 :

3 :

3 :

3 :

10.00

30.00

50.00

0.50

1.50

2.50

0.00

1.50

3.00

1: N 2: Total Birth Rate 3: Total Death Rate

1

1

1

1

2

2

22

3

3

3

3

Graph 1 (Untitled Graph)


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In general, nonlinear difference equations take the following form Xn = f (Xn,Xn -1,...,Xn-m ) (15) where X

n is the value of X at interval n, and f is a recursion function that contains nonlinear

terms of X (e.g., powers, quadratics, exponentials, reciprocals, etc.). As in the case of linear difference equations, m is the order of the difference equation, indicating the number of previous intervals directly influencing the value of X at the present interval, and determines the number of basic solutions to the difference equation. Similarly, an mth-order nonlinear difference equation may be rewritten into a system of m coupled first-order difference equations. Nonlinear difference equations can rarely be solved analytically, and therefore are mostly solved numerically with computers. In fact, one of the most appealing properties of difference equations is that they readily yield to numerical exploration using computers. This property is not shared with the continuous differential equations we shall discuss later. As wee have seen with the simple examples, solutions to difference equations can be obtained through sequential arithmetic operations, which perfectly fits the way the digital computer works. In practice, finding appropriate difference equations to solve difficult differential equations numerically becomes a key element in many modeling exercises. Thus, it is important to understand the basic properties of difference equations. 4. Some Assumptions of Using Difference Equations

∑ When used for dynamic systems, difference equations assume that time is discrete or discontinuous.

∑ This implies that no events or processes occur between time increments. ∑ This also necessarily makes the rate of change in the values of the state variables, ∆Xi/∆t,

constant over an interval (i.e., the rate of change is essentially an average over the period between t and t+∆t).

∑ In addition, using difference equation models assumes that the systems of study is discrete in nature, or can be reasonably approximated as such.

∑ In theory, it is apparent that most if not all continuous systems can be approximated adequately using discrete mathematics like difference equations if ∆t is small enough.

II. ORDINARY DIFFERENTIAL EQUATIONS (ODES) 1. Derivatives What is a derivative? Simply and intuitively, a derivative is the rate of change in the value of a variable. Mathematically, the first derivative of a function, y = f(x), is defined as


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dy

dx= lim

DxÆ 0

Dy

Dx= lim

DxÆ 0

yx+Dx - yx

Dx

= limDxÆ0

f (x + Dx) - f (x)

Dx

or simply,

dy

dx= lim

DxÆ 0

yx+Dx - yx

Dx (16)

where limDxÆ0

means “when ∆x approaches 0”. In words, dydx

is essentially equal to DyDx

when ∆x

becomes sufficiently small. Derivatives can be denoted in different ways, e.g.,

limDxÆ0

yx+Dx - yx

Dx=dy

dx= f ' (x) = y'

Geometrically, the derivative of y is the slope at a point x, that is, change in y (i.e., ∆y) divided by an infinitely small change in x (∆x). The notation, dy, is called the differential of the function, y = f(x). The relationship between the derivative and the differential can be easily obtained as follows:

dy

dx= f ' (x)

dy = f ' (x)dx = y' dx (17) The differential of y can be approximated by Dy = y' Dx

y

x

(x+!x, y+!y)

(x, y)

!x

!y

x x+!x

y=f(x)

}}

y

dy

dx= lim

DxÆ 0

f (x + Dx) - f (x)

Dx


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or equivalently, yx+Dx = yx + y' Dx (18) In general, higher-order derivatives may be defined as:

f(n )(x) = lim

DxÆ0

f (n)(x + Dx) - f (n ) (x)

Dx (19)

Note that higher-order derivatives are denoted in several different ways:

f(n )(x) = y

(n)=dny

dxn =

dn

dxn f (x)

2. Differential Equations Equations that relate the rate of change of state variables to these variables themselves and other variables in continuous systems are called differential equations. In other words, differential equations are those that contain one or more differentials. The order of a differential equation is determined by that of the highest-order differential is contains. As with difference equations, an mth-order differential equations may be rewritten into a system of m coupled first-order equations. Also, as in the case of difference equations, most linear differential equations can be solved analytically using established mathematical techniques, whereas complex nonlinear differential equations usually have to be solved numerically using the computer because of the difficulty or impossibility of achieving analytical solutions. There are many familiar differential equation models in biology and ecology. For example, the density-independent population growth we discussed earlier can be expressed as

dN

dt= rN (20)

where N is the population size, and r is the intrinsic increase

Some common formulas for differentiation: d(un) = nun -1du

d(lnu) =du

u

d(eu) = e

udu

d(sinu) = cosudu

d(cosu) = - sinudu

Some common formulas for integration: undu = 1

n+1un+1

Ú + c

du

|u |Ú = lnu + c

eudu = eu + cÚ

sinudu = -cosu + cÚcosudu = sinu + cÚ


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rate. The equation means that the rate of change in population size N is proportional to population size itself. The analytical solution to equation (20) is its integral, i.e.,

N = N0ert

(21) The differential equation for the logistic growth usually takes the form

dN

dt= rN(1 - N / K) (22)

which describes the rate of change in population size as a simple, yet nonlinear function of population size itself. Note that both per capita birth and death rates are affected by crowding effect as prescribed by the equation. Note that the above is a nonlinear differential equation because it contains a term with N2. The integral equation for the above differential equation can be readily (and fortunately) obtained analytically, which is N =

K

1 + K

N0

-1Ê Ë Á ˆ

¯ ˜ e-rt

(23)

The following pairs of figures illustrates the relationship between integrals and corresponding derivatives. INTEGRAL DERIVATIVE

y = x2 + 1 dy/dx = 2x


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y = x3 dy/dx = 3x2

y = erx dy/dx = rerx 3. More ODE Models in Ecology (1) The equilibrium model of island biogeography

dS(t)

dt= I - E

where S is the number of species, I is the immigration rate, and E is the extinction rate. The equation simply mean that the rate of change in the number of species on an island is determined by the difference between immigration rate and extinction rate. An equilibrium is reached when the two rates are equal to each other.


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(2) Lotka-Volterra competition models:

For 2 species:

dN1

dt= r

2N1(K1- N

1- a

12N2

K1

)

dN2

dt= r

2N2(K2- N

2-a

21N1

K2

)

For n species:

�

dNi

dt= riNi(

Ki - Ni - a ijN j

i! j

n

Â

Ki

)

where Ni denote different competing species, Ki is the carrying capacity for species i, ri is the per capita growth rate of species i, and

�

a ij are the competition coefficients, measures of the average per individual impact of species j on the population of species i. (3) Lotka-Volterra predator-prey models: (1 predator & 1 prey)

dN

dt= aN - bPN

= (a - bP)N

dP

dt= cPN - dP

= (cN - d)P

where N is the prey population, P the predator population, and a, b, c, and d are here constants which are related to prey fecundity, the probability of prey being eaten, the probability of predator capturing prey, and predator mortality, respectively. (4) Compartment models of ecosystem processes The following system of difference equations may be used to describe the dynamics of a variety of systems with n compartments:

dx1

dt= k

11x

1+ k

12x

2+L + k

1nxn

dx2

dt= k

21x

1+ k

22x

2+L + k

2nxn

M

dxn

dt= k

n1x

1+ k

n2x

2+L+ k

nnxn


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or briefly,

dxi

dt= k

i1x

1+ k

i 2x

2+L + k

inxn, i = 1,2,K,n

or even more concisely,

dxi

dt= kijx j

j=1

n

Â , i, j = 1,2,K,n

where dxidt

are the rates of change in the values of state variables xi with respect to time, and kij

are coefficients or the instantaneous rates of flow from compartment j to compartment i. As an example, k12x2 is the rate of flow from compartment 2 to compartment 1, whereas k21x1 is the rate of flow from compartment 1 to compartment 2. So, k12x2 and k21x1 represent two opposite flows between the two same compartments. Given the above form of differential equations, it holds that kij ≥ 0 (where i

�

!j). Note that the term kii is negative and represents the total rate of loss of energy or materials from compartment i to all recipient compartments. Here is a specific example of energy flow in an ecosystem (the Silver Springs ecosystem model; see H. T. Odum. 1957. Trophic structure and productivity of Silver Springs, Florida. Ecological Monographs 27:55-112, and H. T. Odum. 1994. Ecological and General Systems: An Introduction to Systems Ecology. Fig. 7-6, pp.99). The major processes in the system may be depicted by the following diagram.

The dynamics of energy in the ecosystem can be described using the following system of nonlinear differential equations:

dX1/dt = F

10- T

21X1X2- M

51X1- L

01X1- P

01X1

dX2/ dt = F

20+ T

21X1X2- T

32X2X3- M

52X2- P

02X2

dX3/dt = T

32X2X3- T

43X3X4- M

53X3- P

03X3

dX4/ dt = T

43X3X4- M

54X4- P

04X4

dX5/dt = M

51X1+M

52X2+ M

53X3+ M

54X4- P

05X5


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where X1, X2, X3, X4, and X5 are the state variables with the unit of kilocalories per square meter, denoting producers, herbivores, carnivores, top carnivores, and decomposers, respectively; Fij are the external forcing inputs, Tij the trophic level feeding rates, Mij the natural mortality rates, Pij the respiration rates, and L01 the rate for downstream losses. Note that, in this example, the rate of loss from each compartment is not expressed as the total rate of loss. With the values for all the parameters plugged in, the system of differential equations becomes: dX1/dt = 20810 - .0039X1X2 - 1.01X1 - .73X1 - 3.50X1 dX2/dt = 486 + .0039X1X2 - .0272X2X3 - 5.13X2 - 8.86X2 dX3/dt = .0272X2X3 - .0382X3X4 - 74X3 - 5.10X3 dX4/dt = .0382X3X4 - .676X4 - 1.466X4 dX5/dt = 1.01X1 + 5.13X2 + .74X3 + .676X4 - 188.6X5 About Compartment Models Compartment models refer to models that describe the flow of physical material between physical or biological storage pools (i.e., compartments). In the form of difference equations, differential equations, or matrices, compartment models represent the rates of change of energy or materials in compartments as the sum of all flow rates into the compartment minus the sum of all the flow rates out of the compartment. Each compartment is a state variable, with which one or more rates of flow between compartments are associated. The compartment from which the flow is coming is called the donor compartment, and the one to which the flow goes is called the recipient compartment. Compartment modeling provides a very general conceptualization that applies to many biological and ecological problems. Indeed, this methodology has been dominant in modeling flows of energy and material in ecosystems (see Shugart and O’Neill 1979 for a review of early work on compartment models in ecology). Systems models usually capitalize on the compartment conceptualization. STELLA, as a simulation package, is well suited for building compartment models.


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III. RELATIONSHIP BETWEEN DIFFERENTIAL AND

DIFFERENCE EQUATIONS 1. A Comparison between FDEs and ODEs As mentioned earlier, difference equations (discrete mathematics) can be used to approximate differential equations (continuous mathematics). Differential equations may be perceived as the continuous time version of difference equations. To convert differential equations to difference equations, the following guidelines should help. For simplicity, take the differential equation of logistic growth as an example:

(i) Approximate the differential on the left-hand side of the equation with its equivalent difference form. That is, use Nt+Dt

- Nt

Dt to replace dN

dt.

(ii) Replace all N on the right-hand side of the equation with Nt. (iii) Rearrange the equation so that Nt+Dt

is the only term on the left-hand side of the equation.

It is extremely important to realize that the accuracy of this approximation depends on the size of ∆t. There is a tradeoff in choosing an appropriate ∆t. Accuracy, computing time, and other related issues must be considered simultaneously. The rule of thumb for choosing ∆t will be discussed later. The following figure illustrate the effects of the size of ∆t on the accuracy of predictions from the exponential population growth model (Hall and Day 1977).

Here are some simple examples: Differential equations Difference equations dN

dt= r N

t+Dt= N

t+ rDt

dN

dt= rN N

t+Dt= N

t+ rN

tDt

dN

dt= rN(1 - N / K) N

t+Dt= N

t+ rN

t(1-

Nt

K)Dt


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IV. SOME BASIC CONCEPTS IN MATRIX ALGEBRA The concepts and techniques of matrix algebra are important to users of mathematics. Applications of matrix algebra are found in such fields as economics, physics, sociology, engineering, biology, and ecology. Matrix algebra has been used to express ideas, solve mathematical problems, and model real-world activities and processes. In particular, methods in matrix algebra are often used to simplify the representation of, and to derive analytical solutions to, systems of equations that describe the dynamics of systems under study. Although a detailed account of matrix algebra is not necessary here, a brief introduction to some notations in matrix algebra and their relationship to systems of difference and differential equations is very useful to anyone who is interested in systems modeling. 1. Definitions (1) A matrix is simply a two-dimensional array of mn symbols (numbers, variables, functions, or notions of other types), with m rows and n columns. The dimension of a matrix is the number of its rows and columns. For instance, the dimension of a matrix with m rows and n columns is (m x n). The notation of matrices uses parentheses (...) or [...]. Here are some examples of matrices:

1 4

8 9

Ê Ë Á ˆ

¯

a11

a12

a13

a21

a22

a23

a31

a32

a33

Ê

Ë

Á Á

ˆ

¯

˜ ˜ x11

x12

x13

x14

x15

x21

x22

x23

x24

x25

x31

x32

x33

x34

x35

Ê

Ë

Á Á

ˆ

¯

˜ ˜

A (2

�

¥2) matrix A (3

�

¥3) matrix A (3

�

¥5) matrix In general, an (m x n) matrix may be written as

Y = {yij} =

y11

y12

L y1n

y21

y22

L y2 n

M M O M

ym1 ym2 L ymn

Ê

Ë

Á Á Á

ˆ

¯

˜ ˜ ˜

where yij is called the element or entry which is found in the ith row and the jth column of the matrix. Note that determinant and matrix are two different concepts. An (m

�

¥ n) determinant is a number, whereas an (m

�

¥ n ) matrix is not a number, but a table consisting of mn elements arranged in a particular order. For example,


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2 4

1 3= 2 or in general,

a11

a12

a21

a22

= a11a22- a

12a21

However, the above operations do not apply to matrices:

2 4

1 3

Ê Ë Á ˆ

¯ ! 2 or in general,

a11

a12

a21

a22

È

Î Í ˘

˚ ˙ ! a11a22 - a12a21

(2) A vector is a special kind of matrix, i.e., a (1

�

¥ m) or an (m

�

¥ 1) matrix. The former is called a row vector, and the latter a column vector. A vector can be visualized as arrows or points in Euclidean n-space. A scalar, on the other hand, is an ordinary number, which can be perceived as a (1

�

¥ 1) matrix. A row and column vector can be expressed, respectively, as

V = v1,v2,L,v

n[ ] V =

v1

v2

M

vn

È

Î

Í

Í

Í

˘

˚

˙

˙

˙

2. Some Basic Rules of Matrix Operations (1) The Transpose of a Matrix The transpose of a matrix A, denoted by

�

¢ A , is obtained simply by switching the rows and columns of the original matrix. In the case of vectors, the transpose of a row vector is a column vector, and vice versa. For example,

if A =1 4

2 5

3 6

Ê

Ë

Á Á

ˆ

¯ ˜ ˜ , its transpose is then A' =

1 4

2 5

3 6

Ê

Ë

Á Á

ˆ

¯ ˜ ˜

¢

=1 2 3

4 5 6

Ê Ë Á ˆ

¯

(2) Addition and Subtraction of Matrices The addition (or subtraction) of two or more matrices is simply the addition (or subtraction) of corresponding elements in these matrices. These operations require all matrices must be of the same dimension. For example,

if A =1 4

2 5

3 6

Ê

Ë

Á Á

ˆ

¯ ˜ ˜ , and B =

0 0

2 2

0 0

Ê

Ë

Á Á

ˆ

¯ ˜ ˜ , then A + B = B + A =

1 4

4 7

3 6

Ê

Ë

Á Á

ˆ

¯ ˜ ˜ .

In general, we have


Page 18

X + Y = Y + X =

x11

+ y11

x12

+ y12

L x1n + y

1n

x21

+ y21

x22

+ y22

L x2n + y

2n

M M O M

xm1 + ym1 xm2 + ym2 L xmn + ymn

Ê

Ë

Á Á Á

ˆ

¯

˜ ˜ ˜

where

X =

x11

x12

L x1n

x21

x22

L x2n

M M O M

xm1

xm2

L xmn

Ê

Ë

Á Á Á

ˆ

¯

˜ ˜ ˜

, and Y =

y11

y12

L y1n

y21

y22

L y2 n

M M O M

ym1 ym2 L ymn

Ê

Ë

Á Á Á

ˆ

¯

˜ ˜ ˜

(3) Matrix Multiplication: (i) The product between a matrix X and a scalar k is a matrix in which each new element is the element in X multiplied by k. For example,

2 3 -2

1 0

Ê Ë Á ˆ

¯ =

3 -2

1 0

Ê Ë Á ˆ

¯ 2 =

6 -4

2 0

Ê Ë Á ˆ

¯

In general, we have

kX = Xk =

kx11

kx12

L kx1n

kx21

kx22

L kx2n

M M O M

kxm1

kxm2

L kxmn

Ê

Ë

Á Á Á

ˆ

¯

˜ ˜ ˜

(ii) The product of two matrices A and B is a matrix C (where C = AB) in which element cij is the summation of the respective products of corresponding elements in the ith row of A and the jth column of B. Thus, the operation of matrix multiplication requires that the number of columns of matrix A must be equal to the number of rows of matrix B. Graphically, this can be depicted as follows:

=

q

m q

n

m

n

The rule of matrix multiplication may look complex, and it is best understood through examples.


Page 19

Example 1 --- Multiplication of two vectors:

1 2 3 4[ ]

1

2

3

4

È

Î

Í Í Í

˘

˚

˙ ˙ ˙

= 30 ,

1

2

3

4

È

Î

Í Í Í Í

˘

˚

˙ ˙ ˙ ˙

1 2 3 4[ ] =

1 2 3 4

2 4 6 8

3 6 9 12

4 8 12 16

È

Î

Í Í Í Í

˘

˚

˙ ˙ ˙ ˙

In general, we have

u1

u2

L um[ ]

v1

v2

M

vm

È

Î

Í Í Í Í

˘

˚

˙ ˙ ˙ ˙

= u1v

1+ u

2v

2+ Lu

mvm[ ] ,

and

v1

v2

M

vm

È

Î

Í Í Í Í

˘

˚

˙ ˙ ˙ ˙

u1

u2

L un[ ] =

v1u

1v

1u

2L v

1u

n

v2u

1v

2u

2L v

2u

n

M M O M

vmu

1vmu

2L v

mu

n

È

Î

Í Í Í Í

˘

˚

˙ ˙ ˙ ˙

Example 2 --- Multiplication of a matrix by a vector:

3 -2

1 0

Ê Ë Á ˆ

¯ 2

3

Ê Ë Á ˆ

¯ =

0

2

Ê Ë Á ˆ

¯ ,

or in general,

a1 1

L a1n

M O M

am1

L amn

Ê

Ë

Á Á Á

ˆ

¯

˜ ˜ ˜

x1

M

xn

Ê

Ë

Á Á Á

ˆ

¯

˜ ˜ ˜

=a

1 1x

1+ L+ a

1nxn

M

am1x

1+ L+ a

mnxn

Ê

Ë

Á Á Á

ˆ

¯

˜ ˜ ˜

Example 3 --- Multiplication of two matrices:

-1 2 1

0 -1 2

È

Î Í ˘

˚ ˙

1 1

0 -2

1 3

È

Î

Í Í Í

˘

˚

˙ ˙ ˙

=0 -2

2 8

È

Î Í ˘

˚ ˙


Page 20

In general, multiplication of an (m

�

¥ q) matrix by a (q

�

¥ n) matrix results in a matrix C, which has a dimension of (m

�

¥ n). The element of C can be expressed mathematically as

cij = ai1b1 j + ai2b2 jL + aiqbqj = aikbkjk=1

q

Â

3. Matrix Notations of Systems of Equations With the information presented above, we now can conveniently represent systems of equations using matrix notations. For example, the system of equations

x1+ x

2+ x

3+ x

4= 0

x1+ 2x

2+ 3x

3+ 4x

4= 1

5x1+ 6x

2+ 7x

3+ 8x

4= 2

9x2+10x

3+11x

4= 3

can be equivalently rewritten as

1 1 1 1

1 2 3 4

5 6 7 8

0 9 10 11

È

Î

Í Í Í Í

˘

˚

˙ ˙ ˙ ˙

x1

x2

x3

x4

È

Î

Í Í Í Í

˘

˚

˙ ˙ ˙ ˙

=

0

1

2

3

È

Î

Í Í Í Í

˘

˚

˙ ˙ ˙ ˙

Matrix notations are also frequently used for differential and difference equations. For example, the following system of ODEs (see previous sections on ODEs)

dx1

dt= k

11x

1+ k

12x

2+L + k

1nxn

dx2

dt= k

21x

1+ k

22x

2+L + k

2nxn

M

dxn

dt= k

n1x

1+ k

n2x

2+L+ k

nnxn

is equivalent to the matrix form dX

dt= KX ,

where X is the state variable vector, and K is the coefficient matrix:


Page 21

X =

x1

x2

M

xn

È

Î

Í

Í

Í

˘

˚

˙

˙

˙

,

K =

k11

k12

L k1n

k21

k22

L k2 n

M M O M

kn1

kn2

L knn

Ê

Ë

Á Á Á

ˆ

¯

˜ ˜ ˜

When the above system of differential equations is linear, techniques in matrix algebra can be used to derive solutions analytically. However, if the system is nonlinear, which is the case for most real-world problems, computer simulation must be adopted to obtain numerical solutions. Usually, systems of difference equations also can be expressed in matrix form, and vice versa. For example, the Silver Spring ecosystem model can be represented in the following form of FDEs:

X1,t +Dt

= X1,t+ F

10- T

21X1,tX2,t-M

51X1,t- L

01X1,t- P

01X1,t

X2,t +Dt

= X2, t+ F

20+ T

21X1,tX2,t- T

32X2,tX3,t- M

52X2,t- P

02X2,t

X3,t +Dt

= X3,t+ T

32X2,tX3,t- T

43X3,tX4, t-M

53X3,t- P

03X3,t

X4,t +Dt

= X4,t+ T

43X3,tX4,t- M

54X4,t- P

04X4,t

X5,t +Dt

= X5,t+ M

51X1, t+ M

52X2, t+M

53X3,t+ M

54X4, t- M

05X5,t

This FDE system can easily obtained from the ODE system by replacing dXdt

with Xi ,t +D t

- Xi,t

Dt

on the left-hand side of the equations, and, correspondingly, substituting Xi ,t

for Xi on the right-

hand side of the equations. In a little more compact fashion, the above system can be rewritten as

X1,t +Dt = F10 + (1- T21X2,t -M51 - L01 - P01)X1,t

X2,t +Dt = F20 + (1 + T21X1,t - T32X3,t - M52 - P02)X2,t

X3, t+Dt = (1 + T32X2,t - T43X4,t -M53 - P03 )X3,t

X4,t+Dt = (1 + T43X3,t -M54 - P04)X4,t

X5,t +Dt

= M51X1, t + M52

X2, t +M53X3,t + M54

X4,t + (1- M05)X

5, t

where X

i ,t and X

i ,t +D t are the values of the state variable Xi at time intervals t and t+∆t,

respectively. All variables and parameters are defined as previously. A matrix form that is equivalent to the above system of FDEs is X

t+Dt=AX

t

where


Page 22

Xt+ Dt =

X1,t+ Dt

X2,t+ Dt

X3,t+ Dt

X4, t+ Dt

X5,t+ Dt

È

Î

Í

Í

Í

Í

˘

˚

˙

˙

˙

˙

, Xt

=

X1,t

X2, t

X3,t

X4, t

X5,t

È

Î

Í

Í

Í

Í

˘

˚

˙

˙

˙

˙

and A is the ugly matrix

A =

F10

X1,t

+1 - T21X2, t

- M51

- L01

- P01

0 0 0 0

F20

X2,t

+1+ T21X1,t

- T32X3, t

- M52

- P02

0 0 0 0

0 0 1 + T32X2, t

- T43X4, t

- M53

- P03

0 0

0 0 0 1 + T43X3,t

- M54

- P04

0

M51

M52

M53

M54

1- M05

È

Î

Í

Í

Í

Í

Í Í

˘

˚

˙

˙

˙

˙

˙ ˙

Note that to facilitate the conversion, the first two FDEs can be rewritten as:

X1,t +Dt = F10 + (1- T21X2,t -M51 - L01 - P01)X1,t

= (F10

X1, t

+ 1- T21X2, t -M51 - L01 - P01)X1,t

X2,t +Dt = F20 + (1 + T21X1,t - T32X3, t - M52 - P02)X2, t

= (F20

X2,t

+1+ T21X1,t - T32X3, t - M52 - P02)X2,t

Leslie matrix models One of the renowned matrix models in biology is the Leslie matrix model of age-structured population dynamics (in recognition of P. H. Leslie’s important role in its development). Leslie matrix models of age-structured population dynamics usually take the form:

N1,t+1

N2,t +1

M

Nm ,t +1

È

Î

Í

Í

Í

˘

˚

˙

˙

˙

=

f1

f2

f3

L fm

p1

0 0 L 0

0 p2

0 L 0

M O O L M

0 0 0 pm-10

È

Î

Í

Í

Í

Í

˘

˚

˙

˙

˙

˙

N1, t

N2,t

M

Nm ,t

È

Î

Í

Í

Í

˘

˚

˙

˙

˙

or simply Nt+1 = P Nt


Page 23

where Ni,t and Ni, t+1 are the number of individuals of age class i at times t and t+1, respectively, fi are the fertility coefficients (giving the number of age class 1 individuals at time t+1 per age class i individual at time t), pi are the survival probabilities of members of age class i from time t to time t+1, and m is the number of age classes in the population. The matrix, P, is called the population projection matrix, and also referred to as a Leslie matrix. The Leslie matrix model can be easily translated into a difference equation model, and simulation package such as STELLA can be readily used to simulate the dynamics of age-structured populations. In particular, the FDE form of the matrix model can be written as

N1,t+1

= f1N

1,t + f2N

2, t + f3N

3, t +L + fmNm,t

N2,t +1

= p1N

1,t

N3, t+1

= p2N

2, t

M

Nm ,t +1= pm-1

Nm,t

V. PARTIAL DIFFERENCE EQUATIONS (PDES)

So far we have discussed only ordinary differential equations (ODEs), which contain only defferentials with respect to single independent variables. ODEs are appropriate and have been widely used for modeling many biological and ecological processes. However, when processes have to be considered explicitly over a continuous space, partial differential equations provide an appropriate (and difficult) mathematical framework. This is because in this case the rate of change in any state variable must be expressed with respect to time and space (thus more than one independent variables!). PDEs are in general beyond the scope of this course, but the following examples should help understand the basic concepts and applicability of these equations as a mathematical framework. Such an understanding, even though preliminary, is very helpful for choosing appropriate simulation modeling approaches for different problems. 1. Fluid Dynamics Let’s imagine a situation in which a pollutant flows in a river. Suppose that we want to model the temporal dynamics of the concentration of the pollutant (C) at different points along the river (assuming it is only one-dimensional for simplicity). Thus, the state variable (C) varies over two independent variables: time (t) and space (x). There are four fundamental processes affecting fluids and solutes in such systems: 1) Advection: the flow of media and the solute from point to point. 2) Molecular diffusion: the movement of mass due to random motion of individual molecules.


Page 24

3) Turbulent diffusion: the fluxes due to turbulence which depend on the size of one’s observational scale (i.e., ∆x). The larger the ∆x, the larger the eddies and fluxes involved. By assuming that the time scale is long enough, the average effect of turbulent diffusion can be treated as a component of the advection term.

4) Reaction: Any processes other than advection and diffusion that change the concentration of

a solute inside the spatial interval ∆x. These may be chemical interactions (e.g., the substance going in or out of solution), or biological uptake and excretion (e.g., the uptake of nitrogen by plants). These processes can be treated mathematically as an ordinary differential equation.

Now let’s focus on advection and molecular diffusion. In a segment of the spatial dimension (∆x), we have an inflow (Fin) and an outflow (Fout).

If we assume that the system is in temporal equilibrium (i.e., the rate of change in time is zero), we have Cout = Cin - ∆x [NetChange(x)] or Cx+∆x = Cx - ∆x [NetChange(x)] Thus, the net change in C can be expressed as an ordinary differential, i.e., [NetChange(x)] = lim (Cx+∆x - Cx ) / ∆x ∆x->0 However, when we consider temporal change in C as we originally proposed, a partial differential equation must be used. Because C is being changed by processes both in time and space, according to the law of conservation of mass the temporal changes in C must equal the spatial changes in C. This basic conservation equation can be expressed as

!C

!t= -

!F

!x

In the case of advection, if the velocity of the water is a constant U over the small spatial interval ∆x, then the flux of C is simply F = UC So,


Page 25

!C

!t= -

!F

!x= -

!(UC)

!x

In the case of molecular diffusion, the flux F through a medium is proportional to the spatial gradient of the concentration over a small ∆x, i.e.,

F = D!C

!x

Thus, the conservation equation becomes

!C

!t= -

!F

!x

=

!D!C

!x

!x

where D is called diffusivity. If D is a constant over x, then the above equation can be simplified to

!C

!t= D

!2C

!x2

Considering advection and diffusion together to find changes in C, we now have

!C

!t= D

!2C

!x2-!(UC)

!x

2. Patch Dynamics The framework of the diffusion-reaction models takes the following general form:

dYi

u

dt= fi

u(Y

u,X

u) + (netexchangewithotherpatches) + (netexchangewithmatrix)

in which Yu is the vector (Y1

u

,Y2

u

,...,Yn

u ) of state variables for a given patch u, Xu the vector (X1u, X2u, ..., Xnu) of parameters accounting for the same patch, and fu the specific functional relationship (Levin, 1976). These models take into account both temporal and spatial heterogeneity for given state variables, such as population densities, with the aid of analytical


Page 26

power of mathematical diffusion theory. They can further be divided into continuous and discrete types according to their different conceptualization and mathematical details. A simple, yet representative, example of the continuous diffusion-reaction model for population dynamics may be in the partial differential equation form

�

!N (x, t)

!t= N f(N) +

�

![D!N(x,t)]

!x

where N(x,t) is the population density relative to the spatial position x, D is the diffusion rate of individuals of the population, and f(N) is the population growth rate. The corresponding (spatially) discrete model may be written as

�

dNi

dt= Ni f(Ni) +

�

[dij (N j - Ni)j !iÂ ]

where dij is the exchange rate of individuals between patch i and patch j and Ni and Nj are population sizes in the two patches, respectively. Levin and Paine (1974) formulated a patch demographic model to predict the distribution pattern of an age- and size-structured patch population in an intertidal landscape. The approach, a nonequilibrium island biogeographic construct, differs importantly from the diffusion-reaction approach. The Levin-Paine patch demographic model considered the distributions of the age and size of patches in the form:

!r

!t+!r

!a+

!

!x(gr) = -m ( t,a ,x)r

where r(t ,a, x ) is the probability density function describing the frequency distribution of patches of age and size x at time t, m(t,a, x) is the mean extinction rate of patches of age a and size x at time t (due to intra-patch succession), and g( t,a ,x) is the mean growth rate of patches of age a and size x at time t (due to patch shrinkage or expansion). A spatially explicit extension of the Levin and Paine model has been developed to understand the pattern and process interactions in a grassland (Wu and Levin 1994, 1997), known as the spatial patch dynamics modeling approach. From the above, it becomes clear that spatial problems can be treated using the compartment modeling framework (ODEs and FDEs) if the spatial heterogeneity or pattern can be regionalized into separate sections. In situations where no homogeneous regions can be reasonably assumed (i.e., there is a continuous gradation of the spatial structure), however, PDEs are more appropriate. Although there are mathematical techniques that can be used to solve rather simple PDEs analytically, they are usually solved numerically with the aid of computers.


Page 27

Classification of Mathematical Problemsa and Their Ease of Solution by Analytical Methodsb (from Hall and Day 1977).

Linear Equations

Nonlinear Equations

Equat ion

One Equation

Several Equations

Many Equations

One Equation

Several Equations

Many Equations

Algebraic Trivial Easy Essentially impossible

Very difficult

Very difficult

Impossible

ODEs Easy Difficult Essentially impossible

Very difficult

Impossible Impossible

PDEs Difficult Essentially impossible

Impossible Impossible Impossible Impossible

a Courtesy of Electronic Associates, Inc. b After Franks (1967).

RECOMMENDED READINGS:

∑ Walters, C. J. 1971. Systems Ecology: The Systems Approach and Mathematical Models in

Ecology. In: Odum, E.P. 1971. Fundamentals of Ecology. 3rd ed. W.B. Saunders Co., Philadelphia. pp.276-292.

∑ Kitching, R. L. 1983. Systems Ecology: An Introduction to Ecological Modelling. University of Queensland Press, St Lucia. [Chapter 5]

∑ Edelstein-Keshet, L. 1988. Mathematical Models in Biology. McGraw-Hill, New York.

∑ Wu, J. and J. L. Vankat. 1991. A System Dynamics model of island biogeography. Bulletin of Mathematical Biology 53:911-940.

∑ Haefner, J. W. 1996. Modeling Biological Systems: Principles and Applications. Chapman & Hall, New York. [Chapter 4]

∑ Hastings, A. 1997. Population Biology: Concepts and Models. Springer, New York.

∑ Wu, J. and S. A. Levin. 1994. A spatial patch dynamic modeling approach to patternand process in an annual grassland. Ecological Monographs 64(4):447-464. [http://leml.asu.edu/jingle/cv.html#PUBLICATIONS]

∑ Wu, J. and S. A. Levin. 1997. A patch-based spatial modeling approach: conceptual framework and simulation scheme. Ecological Modelling 101:325-346. [http://leml.asu.edu/jingle/cv.html#PUBLICATIONS]

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