California State University, Northridge
STATIC AND DYNAMIC MODELING PROCEDURES
FOR FOOD WEB SYSTEMS
A thesis submitted in partial satisfaction of the
requirements for the degree of Master of Science in
Biology
by
Michael E. Manoogian
May, 1983
The Thesis of Michael E. Manoogian is approved:
~Chair
California State University, Northridge
ii
TABLE OF CONTENTS
Abstract......................................................
vii
I
Introduction .......................................... .
II
Methods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
III
Model Formulation Procedures...........................
7
IV
Static Solution Procedures.............................
20
V
Dynamic Solution Procedures............................
56
VI
Discuss ion.............................................
100
VII
Literature Cited.......................................
112
Appendix A- Matrix Algebra...................................
113
iii
LIST OF TABLES
Table
Page
Values of the Main Diagonal Terms for the
Stiffness Matrix .................................... .
18
Values of the Off Diagonal Terms for the
Stiffness Matrix .................................... .
19
IV. 1
Population Key for Subtropical Food Web ............. .
52
IV.2
Connectivity Key for Subtropical Food Web ........... .
53
IV.3
Stiffness Values for Links for Several Food
Web Mode 1s .......................................... .
54
IV .4
System Link Values for Subtropical Food Web ......... .
55
v. 1
Population Size Amplitudes of Predator and Prey
Populations for Times From 0 to 10 years ............ .
68
V.2
Peak Population Sizes at One Year Intervals for
a Damped, Single Degree of Freedom Food Web ......... .
85
Natural Frequencies of Several Food Web Systems ..... .
99
I I I. 1
III.2
V.3
iv
LIST OF FIGURES
Figure
II.l
Page
Articulated Spring Analogy for Static Food
Web Mode 1s . • . • . . . . . . • . • • • • . • • . . • . . . . . • . . • • • • • . . . . . • . . .
I I. 2
Articulated Spring - Mass Analogy for Dynamic
Food Web Models ...................................... .
III.l
5
6
Population Level Formulation of Tropical
Food Web..............................................
10
I I I. 2
Final Formulation of Tropical Food Web ............... .
11
I I I. 3
Node 1 Showing Connecting Links ...................... .
14
IV. 1
Two Element Food Chain ............................... .
31
IV.2
Three Species Food Web ............................... .
32
IV.3
Result of an Analysis of the Tropical Food Web ....... .
34
IV .4
Undisturbed Food Web ................................. .
35
IV.5
Food Web with One-Third Reduction in Species
Population B......................................... .
36
IV.6
Food Web Analysis After Change ....................... .
37
IV.?
Statically Determinate, Two Degree of Freedom
Food Chain ........................................... .
39
IV.B
Once Indeterminate, Two Degree of Freedom
Food Web. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
IV.9
40
Statically Unstable, Two Degree of Freedom
Food Chain............................................
41
IV.10
Food Web with Link 4 Removed..........................
43
IV.11
Food Web with Link Between Species A and
Source, Populations Removed...........................
44
IV.l2
Food Web with One Species Removed.....................
45
IV.l3
Determinate Food Chain with a Species Removed.........
46
IV.l4
Indeterminate Food Web with One Link Removed..........
47
IV.15
Determinate Food Web with a Link Removed..............
48
v
LIST OF FIGURES
(Continued)
Figure
Page
IV.l6
Subtropical Food Web.................................
51
V.l
Single Degree of Freedom Food Web and Graph
of Response..........................................
58
V.2
Undisturbed and Disturbed Food Web Systems .......... .
82
V.3
Two Degree of Freedom Food Chain with Induced
Oscillations ........................................ .
93
VI.l
Stab 1e Cyc 1e with Refuge ............................ .
101
VI.2
Damped Cycle ...................................... .
102
VI.3
Stab 1e Cyc 1e ..................
•••
104
VI .4
Unstab 1e Cycle .........
o • • • • • • • • • • • • • • • • • • • • • • • • • • • o.
105
vi
o ••••• o
•••••••• o ••
0
o
ABSTRACT
STATIC AND DYNAMIC MODELING
PROCEDURES FOR FOOD WEB SYSTEMS
BY
MICHAEL E. MANOOGIAN
MASTER OF SCIENCE IN BIOLOGY
Traditional modeling procedures in biology have largely been
confined to the development of descriptive models and single degree
of freedom mathematical models.
The emergence of the computer has
enabled the user to model systems with several degrees of freedom in
order to determine the behavior of large systems.
Large system
modeling may be faciliated by using efficient, mechanical formulation
and solution procedures.
Modeling procedures were developed using matrix set-up and
solution procedures based upon static and dynamic structural analysis
techniques.
Methods presented enable the biologist to develop static
and dynamic models of food web structures and the effects of changes
in the food web, and to determine the static and dynamic stability
of a food web.
Models were based upon the assembly of systems of
vii
linear equations or linerized quasi-linear equations.
Modeling procedures were used to analyze several contrived and
actual food web systems.
It was concluded that modeling procedures
were capable of being used in ecological prediction studies, and
could provide an explanation for four and ten year population cycles
noted among some arctic species.
viii
I.
INTRODUCTION
System modeling techniques in biology have largely been limited to
the development of models which are descriptive in form.
Mathematical
modeling techniques have also been used to some extent to model biological systems.
Recently, matrix methods have been proposed for use
in modeling population age structures, food webs, and several other
systems (Maynard-Smith, 1974; May, 1976, 1981).
Though such methods
may be useful in the development of system models, matrix assembly
procedures have generally been overlooked or not analytically based.
Matrix methods have been proposed (Maynard-Smith, 1974; May, l973a,
l973b, 1976, 1981) which rely upon a matrix formulation method known
as the flexibility method, a matrix assembly technique often used in the
development of structural models in engineering.
In recent years, the
direct stiffness approach, another matrix formulation method, has
largely supplanted the flexibility approach.
The principal advantage
of the direct stiffness method, however, lies in the simplicity and
mechanical nature of matrix formulation procedures.
This allows for
the ready adapation of such methods to computer formulation and
solution procedures (McGuire and Gallagher, 1979; Bathe and Wilson,
1976).
The dynamic, time dependent behavior of predator-prey systems has
typically been treated in graphical form (MacArthur and Connell, 1966;
Rosenweig and MacArthur, 1963) or in numercially descriptive form
(Clark, 1972).
These models have been limited to tracing the changes
in population sizes of one and two population systems and long term
censusing.
The static food web represents the configuration of a food
2
web at a particualr point in time.
Food webs constructed in Paine
(1966) are examples of static systems.
System stability has been treated in several forms, but generally
in small problem form.
Models have been constructed in non-linear
problems, randomly constructed systems, to analyze the role of system
complexity, species diversity, or system complexity.
Stability studies
have been centered about Liaponov stability formulations, phase plane
and trajectory plots, limit cycles, and intuitive arguments (Anderson
and May, 1978; Halfon, 1979; Lawlor, 1980; Lawlor and Maynard-Smith,
1976; May, 1973a, l973b, 1976, 1981; Maynard-Smith, 1974; McNaughton,
1978; Murdoch and Oaten, 1975; Roberts, 1974; Siljak, 1974, 1975).
Predictive models have generally not been used in biology to
determine the responses of biological systems to perturbations, due to
difficulties in procedures which are used to model systems, or lack of
confidence in the precision of such models (Levins, 1974).
However,
the role of a predictive model is to aid the biologist in the interpretation of the behavior of a system.
A model is merely an idealization
of nature for it cannot treat random external perturbations that are not
part of the model.
Models, however, may be useful tools in analyzing
a system in association with the knowledge and the experience of a
biologist, and may provide a reasonable approximation of nature.
Modeling procedures presented in the thesis are intended to
1) provide an analytically based, mechanical method of formulating
food web system matrices; 2) provide several different methods of
determining static system models; 3) provide several different methods
of determining the dynamic behavior of a food web system; 4) provide
methods for verifying the static and dynamic stability of a food web
3
system; and 5) provide methods to analyze the effects of changes in the
composition of the food web.
Methods were adapted from static and
dynamic structural analysis procedures employing 1ineraized systems of
equations (Bathe and Wilson, 1976, Clough and Panzien, 1975; Craig,
1981; McGuire and Gallagher, 1979; Meirovitch, 1975; Paz, 1980).
II.
METHODS
Procedures developed in this study to model food web systems employ
techniques which are frequently used in static and dynamic structural
analysis of systems in engineering.
Static models were treated as
systems of articulated springs with the points of articulation being
defined as nodes (Figure II.l).
Each node was defined as a particular
species population and for purposes of the model, it was allowed one
degree of freedom.
Each spring represents a nutrient dependence on a
prey by a predator.
Dynamic models which were developed relied upon the notion of a
system of masses and springs with a mass respresenting a species population and a spring representing a nutrient pathway.
Each population
was defined as a node and allowed one degree of freedom (Figure II.2).
For all of these methods, a general formulation method and a
general solution method was provided.
In addition, at least one con-
trived food web or an actual example from Paine {1966) was formulated
or solved to illustrate each method.
No experimental work was conducted
as this paper merely presents theoretical modeling procedures.
However,
the results of some of the modeling procedures were employed in order
to explain the results of other research (Rosenweig and MacArthur,
1963).
All small problems were solved using hand solution techniques.
Large problems may be solved using an appropriate set of matrix manipulation subroutines such as IMSL.
All large problems presented in this
paper were analyzed using SAPIV, an engineering structural analysis
computer program (Bathe and Wilson, 1973).
4
5
Figure II.l
Articulated Spring Analogy for
Static Food Web Models
Predator
Species~
A ,-r-'r'f\
( \
( (
(
\~- j
f _r'~
_,
,..;
-
/-
<-v \J
\~Links
~
I
\
< r )
Prey Species
6
Figure II.2.
Articulated Spring- Mass Analogy
for Dynamic Food Web Models
Links
Q
(Popul~~ion)
/
I\_\
/
II
I
D - Mass
f\-\
\
\_L.
{1".
(
~
•-,"1.4-
,-"
I
~
("'
9
9
_j_
_L
~
~
t'
(Species)
\-Link (nutrient
pathways)
\-
\
~
'
2
--L..
9
7
III.
MODEL FORMULATION PROCEDURES
II I. 1. ASSUMPTIONS
The intention behind the development of this model is to enhance
the biologists understanding of a food web, since all possible variations cannot be represented.
Static and dynamic models presented here rely upon the following
set of assumptions:
1.
Predators take prey at the same frequency at which they are
encountered and such encounters are random.
Thus, predators
selected prey upon the basis of opportunity.
2.
Interactions between predator and prey may be predicted by
a linear equation.
3.
Since predator-prey interactions may be predicted on the
basis of sets of linear equations, portions of the food web
may be isolated and analyzed separately.
4.
In the absence of prey, the predator does not exist.
5.
In the absence of a predator population, a prey population
may increase unbounded.
This condition may be approached by
an r-selected prey population until it reaches its upper
bound.
Dynamic models presented rely upon an additional set of assumptions which are as follows:
1.
Prey species may seek refuge and avoid predation, thus only a
portion of a prey population can be eliminated.
2.
A reduction in the size of the prey population will cause a
portion of the predator species to die out until the remaining
number of prey can adequately support them.
'
8
3.
Nodal oscillations represent changes in population
size about an equilibrium point.
9
III.2.
FOOD WEB MODEL STRUCTURE
The development of static and dynamic models of the type proposed here, require a simple, mechanical formulation procedure.
An
actual food web, in order to provide the information necessary to
formulate the models, should identify the prey and predator populations,
the portion of diet of the predator that the prey species comprises and
the proper order of consumer for each predator population in the system.
For the purposes of the models, the order of the consumer is defined
as the order of its highest order prey plus one.
is defined as an fundamental source population.
A producer population
For isolated food web
sections, the lowest order population is defined as an ordinary source
population if it is a consumer.
The first step in the development of the model should involve the
mapping of the food web.
Symbolically, circles will be used to
represent species populations, or nodes, and lines between circles
will represent nutrient pathways and will be assigned the term
"link".
The food web is mapped with level 0 representing inorganic
materials level 1 representing producer populations, and levels 2,
3, 4 and so on are the first, second, and third, etc. level consumers.
The tropical food web from Paine (1966) is used to demonstrate the
formulation of the food web (Figure III.l).
The second step involves the formulation of the nutrient pathways.
Once the prey populations of each consumer population are identi-
fied, lines representing links are drawn between them (Figure III.2).
Variables KK represent the portion of the diet of the predator
species that the prey species comprises.
Subscript K represents
10
Figure III.l. Population Level Formulation of Tropical Food Web
1.
Thais biseralis
2. Acanthina brevidentata
0
0
3. Carrion
0
4. Mise
0
5. Bivalues
0
Level 3
6. Barnacles
0
Level 2
Level 1
Ordinary Source Populations
11
Figure III.2.
Final Formulation of Tropical Food Web
Thais biseralis
Acanthina brevidentata
Carrion
Ordinary Source Populations
12
the number used to label the link.
The third step involves the assignment of values to all of the KK
variables.
Each value represents the portion of the total amount of
the nutrient material of the predator for which the prey species
comprises.
The assignment of this value may be based upon whatever
units, such as numbers of organisms, calories, or tissue weight, that
the biologist chooses.
The sum of the connecting links to each pre-
dator should be 1.0, as this value represents the sum of the portions
of the nutrients consumed by the predator.
Figure 111.2 shows the
final formulation of the model of the tropical food web.
of the links are obtained from Paine (1966).
Values
Values for links between
ordinary source populations and level 2 are assigned the value of
1.0 since the source population obtains 100 percent of its nutrients
from unidentified sources.
13
III.3.
STATIC POPULATION (STIFFNESS) MATRIX FORMULATION
The formulation of the stiffness matrix is based upon the
assumption that a food web, at any instant in time is in static
equilibrium.
The procedure for the matrix formulation essentially
involves the perturbation of one node and the fixing of all other
nodes.
When the perturbation, in the form of a nodal displacement,
has the value of one, then the value of each link is equal to the
value to be placed into the stiffness matrix.
The first step in the formulation procedure involves the determination of the number of degrees of freedom.
Since each species pop-
ulation in the food web represents a node which is allowed one degree
of freedom, the number of degrees of freedom equals the number of
populations in the food web.
This is analogous to a system of articu-
lated springs in which the points of articulation may translate or
move in a linear manner.
If the system of springs is represented
in a Cartesian coordinate system, the movement of any one node may
be described in relation to only one axis if it is defined as having
only one degree of freedom (McGuire and Gallagher, 1979).
The nature
of the food web is such that the number of columns must equal the number
of rows, or it is definded as a square matrix.
As an example, the
tropical food web in Figure III.3 has six species and therefore six
degrees of freedom.
As a result, the stiffness matrix has six rows
and columns and corresponds to the number of degrees of freedom.
generalized form of the stiffness matrix for the tropical food web
is shown on the following page.
The
14
Figure III.3.
Node 1 Showing Connecting Links
1.
2.
3.
0
15
Kll
Kl 2 K13
K14
Kl 5 Kl6
K21
K22
K24
K25
~1
~2 ~3 ~4 ~5 ~6
~1
~2 ~3 ~4 ~5 ~6
K51
K52
~1
~2 ~3 ~4 ~5 ~6
K23
K53
K54
K55
K26
K56
The next step involves the assignment of values to each position
in the stiffness matrix.
It may be appropriate to number each pop-
ulation and link which composes the food web.
Figures III.l and III.2.
One scheme is shown in
The terms on the main diagonal of the stiff-
ness matrix are determined by summing the values of the links connecting it to prey populations as shown in Figure III.3.
This node is
assigned the first row in the static matrix.
The value of K11 is
Using the corresponding values
merely the sum of K1 , K2 , K3 , and K4 .
from Figure III.2, the value of K is 1.0.
The values of K22 through
K66 are obtained in a similar manner and are complied and shown in
Tab 1e I I I. 1 .
Off diagonal positions in the stiffness matrix include all
elements of the matrix with mixed value.
These nodes denote the
effect of the perturbation of any one population on any other population in the food web.
Again, node 1 is isolated as in Figure III.3
in order to illustrate the formualtion of the off-diagonal terms K12
through K16 . By convention, all off-diagonal terms are negative.
Since no link exists between populations 1 and 2, the value of K12
16
is zero.
The values of K13 through K16 are -K1 , -K2 , -K3 , and -K4 .
Using the values from Figure III.2, the values of K13 through K16 are
-.11, -.09, -.69, and -.11, respectively.
Using a similar procedure
the other off diagonal terms are compiled and shown in Table III.2.
At this point, a set of simple checks should be carried out to
determine if the matrix was properly constructed.
In order for the
matrix to be correct in form, 1) all main diagonal terms must be
positive; 2) all off diagonal terms must be negative; 3) the order
of the matrix must equal the number of degrees of freedom, or nodes;
4) the matrix should be symmetric about its diagonal , or K.. must
lJ
equa 1 Kji' where subscripts i and j denote the row and column in
which the stiffness term is positioned in the matrix.
The final form of the stiffness matrix for the tropical food
web is shown below:
1.0
0
-.11
-.09
-.69
-.11
0
l.O
0
0
-.25
-.75
-.11
0
1.11
0
0
0
-.09
0
0
1.09
0
0
-.69 -.25
0
0
1.69
0
-.11
0
0
0
1.11
-.75
17
III.4.
DYNAMIC POPULATION (MASS) MATRIX FORMULATION
The formulation of the mass matrix is a much simplier procedure
than the development of the stiffness matrix.
Again, the size of
matrix is dictated by the number of degrees of freedom in the food
web.
Thus for the tropical food web example, used in previous
sections, the dynamic matrix will be of order six.
Since the dynamic
model is analogous to a discretized spring-mass system, the population
matrix will contain the value 1.0 in each position on the main diagonal
and 0.0 in each off-diagonal position.
The value of 1.0 is selected
since the populations in the food web are assumed to be in full
complement at the instant that sampling was conducted.
Any number
could be used to represent the equilibrium complements of the populations in the matrix, however.
In dynamic models, population size
variations are measured about the equilibrium point and are treated
in modeling changes by increasing or decreasing the value of the
equilibrium by a fraction of 1.0.
The final form for the mass
matrix is shown below:
1.0
0
0
0
0
0
0
1.0
0
0
0
0
0
0
1.0
0
0
0
0
0
0
1.0
0
0
0
0
0
0
1.0
0
0
0
0
0
0
1.0
18
Table III.l.
Values of the Main Diagonal Terms for the
Stiffness Matrix
K..
lJ
FOR~,1ULATION
Kll
Kl + K2 + K3 + K4
1.0
K22
Ks + K6
1.0
K33
Kl + K7
1.11
K44
K + K
8
2
1.09
K55
K3 + K5 + Kg
1.69
K66
K4 + K6 + KlO
1.11
VALUES
19
Table III.2.
K ..
lJ
Values of the Off Diagonal Terms for the
Stiffness Matrix
FORMULATION
K12' K21
VALUE
0
K13k K31
-Kl
-.11
K14' K41
-K2
-.09
K15' K51
-K3
-.69
Kl6' K61
-K4
-.11
K23' K32
0
K24' K42
0
K25' K52
-K5
-.25
K26' K62
-K6
-.75
K31' K43
0
K35' K53
0
K36' K63
0
K45' K54
0
K46' K64
0
K56' K65
0
IV.
STATIC SOLUTION PROCEDURES
The intent behind the development of the static spring analogy is
to provide information about the nutrient flow from source populations
to consumer populations at the top of the food web.
To accomodate
this, Hooke•s Law (McGuire and Gallagher, 1979), a basic relationship
from structural mechanics is used.
The general relation is shown
below:
Variables R, K, and x refer to the force vector, stiffness matrix
and the displacement vector.
In the case of a static model, forces
and displacements are merely devices through which link values are
determined.
The general principle involves the application of a force
of 1.0 on each node representing a highest order consumers as each
was assumed to exert a predation pressure of 1.0 on the food web
system.
This produces displacement of the nodes in the food web to
a new equilibrium position.
On the basis of the relative displace-
ments of the nodes, the new system link values can be calculated.
Two basic solution procedures are useful in calculating the system
link values.
The first method involves the use of Cramer•s Rule
for inverting the stiffness matrix.
The second method uses the Gauss
elimination procedure which is a numerical matrix inversion procedure
used for larger matrices and in computer applications (Bathe and
Wilson, 1976).
20
21
IV.l.
SMALL PROBLEM SOLUTION PROCEDURE
Problems with two or three degrees of freedom are often more
conveniently solved using an appropriate hand solution technique.
The method shown below utilizes Cramer's rule and is a suitable
techinque.
The solution technique can be reduced to a stepwise
procedure as follows:
1.
After assembling the stiffness matrix, it may be included
in the general static equation shown below:
2.
A unit force is applied to the node or nodes representing
the
highest order consumer(s) such that the force vector
contains the value 1 in the appropriate position(s).
For
example, the force vector for the tropical food web is shown
below:
R
=
11
1
0
0
0
olT
Positions 1 and 2 contain the value
the highest order consumers.
since both represent
All other positions contain
zeros since those populations represented by these nodes
are not highest order consumers.
3.
The stiffness matrix is inverted using Cramer's Rule.
4.
The vector of displacements is determined using the following
equation:
The superscript, -1, denotes an inverted matrix.
5.
Each link value is determined based upon the relative
displacements at both ends using the following
formula~
22
R
=
K (x. - x.)
J
1
Variable K refers to the link value of a particular link.
scripts i and
link.
nodes.
j
Sub-
refer to the number of the node at either end of the
Variables xiand xjare the new equilibrium positions of the
Each new link value equals the percentage of the diet of the
top level predator that the prey species composes.
This may be a
direct contribution, as a prey species may be fed upon by a predator,
or an indirect contribution, as a prey may be fed upon by a higher
order consumer which may subsequently be fed upon by another predator.
23
Example IV.l.
Calculate the displacement vector and system link values
for the food web system shown below using small problem
Assume K1 = K2 = 1 and Species 8
exerts a predation pressure of 1.0 the system.
solution techniques.
1.0
Species 8
Species A
1.
Determine the stiffness matrix and force vector.
[ KJ
=[ Kl +
-K2
K2
-K2
!Rl =
2.
K2
2
l-1
-1 ]
1
gl
Invert stiffness matrix using Cramer•s Rule.
=
3.
]= r
1
1
[~ ~]
Calculate the displacement vector.
{::}
=
{::} =
[:
{! J
:]
g}
24
4.
Determine the system link values
Rl
=
K (X.
J
= 1
=
1
X;)
0)
1.0
R2 = 1 (2- 1)
R2 = 1.0
Species 8 obtains all of its nutrients from feeding on species A and
indirectly from source populations of species A.
25
IV.2.
GAUSS ELIMINATION SOLUTION -- LARGE PROBLEM SOLUTION PROCEDURE
A numerical method which can conveniently facilitate the calculation of nodal displacements is the Gauss elimination procedure.
This method can be used to solve sets of equations which are positive,
definitive, symmetric, and banded. The solution procedure involves a
numerical triangularization of a static matrix by successively
reducing the below diagonal stiffness elements to zero.
This procedure
can be illustrated using the static equations shown below:
2
-1
-1
0
0
ul
2
-1
u2
-1
2
u3
0
=
1
(2)
0
Using the set of equations (2), the initial step of the procedure
involves the subtraction of a multiple (-l/2) of the first equation
from the second equation and a multiple (O) of the first equation
from the third equation.
2
-1
0
1.5
0
-1
The result is shown below:
0
ul
-1
u2
2
u3
0
=
1
(3)
0
The second step uses the set of equations (3) and involves the
subtraction of a multiple (-2/3) of the second equation from the
third.
The result is shown below:
2
-1
0
1.5
0
0
0
ul
-1
u2
1.333
u3
0
=
1
.667
( 4)
26
Using the result from the set of equations (4), the elements of
the displacement vector can be determined by a back substitution
<
procedure.
In formal form, the reduction of the static matrix into the
upper triangular configuration can be represented by the equation:
-1
-1
L
L
1-n
-1
L
2
K
=S
(5)
1
Variables S and L are the final triangularized static matrix and
the Gauss multipliers, respectively.
The equation can be represented
in the form:
(6)
Equation (6) can be represented as follows for L = L1 L2 •.. Ln:
K
=L
(7)
S
Since S is an upper-triangular matrix, it can be shown that
s = o s·
where D is the diagonal matrix of S and s• is LT.
(B)
As a result, the
following equation can be established:
K
= L D LT
(9)
The solution to K U = R can be determined using the equations:
L V
=R
D LT = V
(10)
(11)
The load vector R is reduced to obtain V and U is obtained by
back substitution using the following equations:
27
v
=
-1
L
n-1
LT U =
o- 1 v
-1
L
2
-1
R
L
1
( 12)
( 13)
28
Example IV.2.
Calculate the displacement vector and link values for
the system shown below using Gauss Elimination procedure.
1.0
Species B
Species A
1.
Determine static matrix and set up force vector.
[2 -1]
-1
1
2.
Set up the static equations.
3.
Triangular the static matrix by multiplying the first equation
by -1/2 and subtract it from the seocnd equation to obtain the
following set of equations:
29
4.
Determine the displacement vector by back substitution.
1;2
x2 =
1
=
o
2x 1 - x2
Solution of these equations results in the following displacement
vector:
C:l
5.
=
1
{:}
K
Determine system linl values.
l
R.
=
Rl
= 1.0 = Link value #1
R2
=
R2
= 1.0 = Link value #2
1
K
K
K(l-0)
Kl
(2-l)
30
IV.3.
INTERPRETATION OF THE MODEL
The general intent of the static model is to provide the biologist
with a notion of the proportion of each prey population which directly
or indirectly through higher order consumers, contributes to the diet
of the highest order consumer(s).
Using the food chain in Figure IV.l,
link values may be found to be 1.0 as both populations modeled in the
chain feed exclusively on the next lowest species in the chain.
Either
solution method may be sued to determine that the system link value
for both links is 1.0.
This indicates that species A derives 100 per-
cent of its diet from species B and that species B obtains 100 percent
of its diet from undesignated source populations.
food web is shown in Figure IV.2.
A more complex
The stiffness matrix and vector of
forces are shown below:
{~ 1 =r~-75
l
-.75
0
1.25
-.25
-.75]
-.25
1.00
Using either solution method, the link values may be calculated to
be 0.75 for links 1 and 3, and 0.25 for links 2 and 4.
Thus species
band its source populations account for 25 percent of the diet of
species C.
Likewise, species A and its source populations account
for 75 percent of the diet of species C.
Link values may often be determined, for small problems, without
the aid of hand calculations or computer methods.
Food webs which
contain many species and/or complicated link structures, however,
may be difficult to model without the use of such methods.
Computers
may often facilitate a more convenient set of solution techniques.
31
Figure IV. 1.
Two Element Food Chain
2.
Species B
1. Species A
Source Populations
32
Figure IV.2.
Three Species Food Web
3. Species C
R3 --
1. Species A
K
4
= .25,
R4 - .25
2. Species B
33
The tropical food web is an example of such a system.
Using matrix
formulation techniques from section III.3 to obtain the final stiffness matrix determined for the tropical food web in the same section,
the vector of forces from the same section, and solution techniques
from section IV.3, an analysis of the tropical food web was carried
out.
The results are shown in Figure IV.3.
Link values, R..
lJ
indicate the predation pressure induced upon the populations due to
the top predators.
One of the primary uses of the static model is in the evaluation
of the effects of changes in the food web structure.
Should a
species, or several species undergo a substantial change in population size, the static model may be used to determine the shift in
nutrient flow through the food web.
For example, Figure IV.4 shows
a simple three species food web formulation in its undisturbed form.
Figure IV.5 shows the food web formulation after the population
of species B was reduced by one third.
To model this change, the
value of K was reduced from 0.75 to 0.50.
The resulting stiffness
matrix is shown in the solution formulation below:
{l [1. ~0
~
=
1
-.50
0
1.25
-.25
-.50]
-.25
-.75
Using either solution procedure, the result may be calculated and is
shown in Figure IV.6.
Species A and its source populations account
for 67 percent of the diet of species C.
Likeswise, species B
accounts for 33 percent of the diet of species C.
34
Figure IV.3.
1.
Result of an Analysis of the Tropical Food Web
2.
Thais biseralis
Acanthina brevidentata
• 75
3. Carrion
R8 = .09
R9 = .94
R
10
=
.86
35
Figure IV.4.
Undisturbed Food Web
Species C
= • 75
Species A
Species B
K
2
= 1.0
36
Figure IV.5.
Food Web with One-Third Reduction in Species
Population B
Species C
.50
Species A
K = 1. 0
1
Species B
37
Figure IV.6.
Food Web Analysis After Change
Species C
Species A
Species B
IV.4.
STATIC STABILITY
The arrangement and complexity of links in a food web may affect
its stability.
In a mathematical sense, stability exists when the
number of equations in a set of simultaneous equations equals or
exceeds the number of unknowns to be determined.
Should the number
of unknowns exceed the number of equations, the system is mathematically unstable.
A determinate system is a system in which the
number of equations equals the number of unknowns.
An indeterminate
or redundant system is one in which the number of equations exceeds
the number of unknowns from the number of equations.
For example,
a system is described an once indeterminate or once redundant if
it has one more equation than unknown.
With respect to a food web, each link represents one equation
and each species represents one unknown.
For example, the system
in Figure IV.? represents a two degree of freedom food chain.
Since
the system contains two links and two species, it is considered a
determinate food chain.
with three links.
Figure IV·.8 shows a two species food web
Since it has one more link than species, it is
considered a once indeterminate food web.
An unstable food web is
represented by Figure IV.9 since it has two species and only one
1 ink.
Two sources may exist in an indeterminate or determinate food
web on the basis of the assumptions stated at the beginning of
Chapter III.
One exists when a prey species is not constrained by
a predator species.
Should a predator species be removed from
38
39
Figure IV.?.
Statically Determinate, Two Degree of Freedom
Food Chain
Species B
Species A
40
Figure IV.B.
Once Indeterminate, Two Degree of Freedom
Food Web
Species B
Species A
Q
41
Figure IV.9.
Statically Unstable, Two Degree of Freedom
Food Chain
Species B
Species A
-I
'
42
the food web, the growth of the population of the prey species
will be unbounded.
In reality, this condition will not occur as the
control mechanisms of the prey species will eventually intercede and
cause a leveling off of the population size.
Such a system is shown
in Figure IV.lO.
The other instance exists when a prey population is deleted from
the food web.
If the prey was the sole source of food for a predator
population, the predator would become extinct from the food web.
a configuration is shown in Figure IV.ll.
Such
The result would include
the extinction of species A and C from the food web.
Species C
would become extinct since species A became extinct as it was the sole
prey species.
The role of food web redundancy lies in the degree of resiliency to
changes in the food web structure.
A large degree of redundancy may
allow the food web to be less sensitive to the loss of a species or a
link.
If a species is lost from a highly interconnected food web
structure, the result may simply be changes in the nutrient flow
pattern through the food web rather than the loss of one or several
species that would occur in a more simply connected food web.
For
example, the elimination of a single species from the system in
Figure IV.l2, will not cause the extinction of the other species
from the food web.
The removal of a species from a determinate
food web, as in Figure IV.l3, will result in the extinction of the
other species.
Likewise, the removal of one link from the food web
in Figure IV.14 will not affect the species complement of the food web
while the removal of a link from the determinate food web in
Figure IV.l5 will cause the extinction of the species in the food web.
43
Figure IV.lO.
Food Web with Link 4 Removed
Species C
\
''
\
\
'\
K4
\
\
'
Species A
\
9
Species B
44
Figure IV.ll.
Food Web with Link Between Species A
and Source, Populations Removed
Species B
Species A
45
Figure IV.l2.
Food Web with One Species Removed
Species C
Species A
a)
Species B
Undisturbed Food Web
Species C
,
,
I
I
,,
I
I
Species A
b)
Species B
Food chain after Species A was deleted
46
Figure IV.l3.
Determinate Food Chain with a Species Removed
Species C
Species B
Species A
47
Figure IV.l4.
Indeterminate Food Web with One Link Removed
Species C
Species B
Species A
Link Removed
48
Figure IV.15.
Determinate Food Web with a Link Removed
Species C
Species B
Species A
-Link removed
I
.....L.
50
IV.5.
STATIC LARGE MODEL EXAMPLE USING REAL DATA
As a demonstration of the static food web modeling system
proposed in this thesis, a subtropical food web (Paine, 1966) was
analyzed (Figure IV.16).
Four cases were analyzed based upon the
conditions shown below:
Model A--All link stiffnesses set to 1.0.
Model B--Link stiffnesses represent the fractional number of
prey consumed by each predator.
Model C--Model B with the stiffness of link 6 reduced by half.
Model 0--Model B with the stiffness of links 12 to 16 reduced
by half.
Table IV.3
analysis.
shows the link stiffness values used for each
Table IV.4
shows the link values that resulted from
the analysis of each model.
Model A demonstrates the imprecision
of a model in which all link stiffnesses are assumed to be 1.0 as
species which are not common in the diet of the keystone predator,
Heliaster, are modeled as being common.
The key example is
Muricathus which comprises only a minimal portion of the diet of
Heliaster but is modeled as being the most common constituent.
In
constrast, Model B demonstrates the role of a more precise method
which assigns link stiffnesses based upon the prey selection
frequency of each predator.
In comparison to Model A, Muricanthus
was modeled as comprising only .001 of the diet of Heliaster.
Models C and D demonstrate the effects of changes in the link
stiffnesses in a food web.
Model C shows the effect of a reduction
of the value of link 6 from 0.36 to 0.18, representing a one half
50
reduction in the number of bivalves consumed by Muricanthus.
The
result indicated no perceptable change with respect to Model B
due to the relatively weak and indirect connection between Heliaster
and bivalves consumed by Muricanthus.
Thus the food web was essen-
tially unaltered by the reduction of the bivalve consumption by
Murjcanthus.
The analysis of Model D demonstrates the effect of a one-half
reduction in the size of the barnacle population on the food web.
The result is a shift to bivalves as a more frequent direct and
indirect nutrient source and a shift away from barnacles.
The analysis of the static food webs presented above involved
the use of SAPIV, a structural analysis program.
Essentially, the
program assembles a stiffness matrix and calculates forces, which
are analogous to link values, in the manner demonstrated by the large
problem solution procedures (Bathe and Wilson, 1973).
Figure 1V.16, Subtropical Food Web
--~-----.,..,..-,"" / 7
,."
I I
I
/
I
,.
r
,.'
./,
I I
I
I
I
I
I
1
I
I
I
//
/
,
I
,'
I(
I
I
\
I
I
t
\
\
\
\
t
I
I
II
I
I
I
I
I
/
I
I
I
,'
I
,' I
/
/
--------_Greater
than or equa)
to .10
--------Less than .Jo
/I
/1
/
.,,"
,.,.,.
I t"
I
,,
,."
I
I
I
/1
//
~~~_,.-"{/
- uY
/1
I
t
\
\
\
l
l
\
''
'
'
\
t
I
I
I
l
l
I
0'
\
'
\
\
\
\
\
\
\
\
\
\
\
\
\
\
®
<.n
._.
52
Table IV.l.
Population Key for Subtropical Food Web
NODE NO.
POPULATION
1
Co 11 umbe 11 i dae
2
Bivalves
3
Herb. Gastropods
4
Barnacles
5
Chi tons
6
Brachiopods
7
Morula
8
Cantharus
9
Acanthina a.ngelica
10
Hexaplex
11
Acanthina tubercul ata
12
Muricanthus
13
Heliaster
53
Table IV.2.
LINK NO.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
Connectivity Key for Subtropical Food Web
PREY NODE NO.
1
1
1
2
2
2
2
3
3
3
3
4
4
4
4
4
5
6
7
7
7
8
9
10
11
11
12
PREDATOR NODE NO.
13
10
12
13
10
12
11
10
13
12
11
7
8
12
13
9
13
13
13
12
10
12
11
13
12
13
13
54
Table IV.3.
LINK NO.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
Stiffness Values for Links for Several Food Web Models
MODEL A
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
MODEL B
.03
.04
. 16
. 31
.77
.36
.14
.16
.08
.24
.28
.28
1.0
. 31
. 51
1.0
.01
.05
.01
.001
.02
.03
.57
.001
.01
.01
.001
MODEL C
.03
.04
. 15
. 31
.77
.18
.14
.16
.08
.24
.28
.28
1.0
. 31
. 51
1.0
. 01
.05
.01
.001
.02
.03
.57
.001
.01
.01
.001
MODEL D
.03
.04
.16
. 31
.77
.36
. 14
.16
.08
.24
.28
.28
0.5
.155
.255
0.5
.01
.05
.01
.001
.02
.03
.57
.001
. 01
.01
.001
55
Table IV .4.
LINK NO.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
System Link Values for Subtropical Food Web
MODEL A
.094
.023
.037
.094
.023
.037
.027
.023
.094
.037
.027
.086
.029
.036
.093
.052
.095
.095
.071
.014
.001
.029
.052
.071
.039
.067
.228
MODEL B
.030
0
.001
.308
.001
0
.002
0
.080
.001
.004
.006
0
0
.505
.004
.010
.050
.007
0
0
0
.004
.001
0
.01
.001
MODEL C
.030
0
.001
.308
.001
0
.002
0
.080
.001
.004
.006
0
0
.505
.004
.010
.050
.007
0
0
0
.004
.001
0
.01
.001
MODEL D
.04
0
.011
.411
.001
0
.002
.001
.106
.001
.006
.008
0
0
.338
.005
.013
.066
.009
0
0
0
.005
.001
0
.013
.001
V.
DYNAMIC SOLUTION PROCEDURES
V.l.
CONSERVATIVE SYSTEMS
The oscillatory behavior of any species, predator or prey, will
affect all species in its food web as it does in the dynamic behavior
of a structure.
A system in which energy dissipation is demonstrated
by a long term, net reduction in the amplitude of population size
oscillations is known as a non-conservative system.
For a dynamic
system, nodal displacements are assumed to demonstrate oscillations
of population sizes in a dynamic model.
The simplest dynamic system is a single degree of freedom, or
one population food web system.
The general equation of oscillation for
a single degree of freedom system is shown below:
mx
+
ex
+ kx
= f
(t)
Varipbles m, c, k, and fare the mass, damping, stiffness, and inducing
terms.
Variables
x, x,
displacement terms.
and x are the acceleration, velocity, and
The equation for a freely oscillating system is
shown below:
mx
+ kx
=0
This equation may be divided by m to obtain the following equation:
x + w2x
Variable,
=0
w, is the natural frequency of the food web and is obtained
using the relation shown below:
w2 = k/m
Since w is expressed in terms of radians per unit time, the time for
the system to travel 1 radian is 1/2rr times unit time.
56
The period
57
of the system is the amount of time for one cycle to be completed and
is obtained using the relation shown below:
T = l/w
The amplitude of the system is the maximum displacement with respect
to the equilibrium point (Figure V.l).
The response of the population may be predicted using the
following realtion:
x(t)
=
A sin wt + 8 cos wt
Where constants A and 8 are constants that are determined from the
initial conditions of the system.
In terms of a biological system, the relationships shown above
will be used to predict the oscillation in the size of populations
about an equilibrium point.
For example, a particular species in a
study site may have a maximum population size of 110 and a minimum of
90.
The equilibrium point will be assumed to be 100.
In terms of
our formulation, the equilibrium point will be defined as 1.0 and the
maximum and minimum points as 1.1 and 0.9. These values indicate the
fraction of the equilibrium population size that exists at a particular
time.
Initial conditions will then be assumed to be the difference
between the peak amplitude and the equilibrium point.
For the system
described above, the initial displacement, x(O), will be defined as
0.1 and the initial velocity, x(O), will be 0.
For k=l.O and
m=l.O, the equation of oscillation for the system shown in Figure V.l
is shown below:
1 X+ 1 X
=
0
58
Figure V.l.
Single Degree of Freedom Food Web and
Graph of Response
Species A
1.1
1.0
0.9
t
59
The natural frequency for the system is 1 cycle per unit time since
w equa)s w equals 1.
The response of the system may be determined
from the initial conditions at t=O and the following relations:
x(O)
= 0.1
=A sin wt + B cos wt
x(O)
= 0.0 = A w cos wt - B w sin wt
After solving for constants A and B, the response equation for
the populations about its equilibrium is shown below:
x(t)
= 0.1 cos lt
The equilibrium population size may be multiplied by the function
shown above to obtain the amplitude of population size oscillation
about the equilibrium point.
Thus the population that was modeled
oscillates in size between 110 and 90 about an equilibrium of 100
and will continue to do so if left undisturbed.
Figure V.l shows
the result in graphic form.
As a result population system may be represented using a freely
oscillating spring-mass system, a multiple population system may
be represented using a system of springs and masses.
The general
equation for a multiple population system is shown below:
[M]
{xf
+ [C]
{x}
+ [K]
{x} =
F (t)
Vaiables [M], [C], and [K] are the dynamic population matrix, the
population damping matrix, and the population stiffness matrix.
Variables (x), {x), and (x) are the acceleration, velocity, and
displacement vectors.
Variable F(t) is a forcing function.
Models presented in this section behave according to the
equation shown below:
[M]
{x}
+ [K] { xJ
=
o
60
The equation shown above is a general equation of oscillation for a
freely vibrating conservative system with no forcing function.
Non-
conservative systems and forced systems will be treated in later
sections.
The solution for the free oscillation of a system is shown
below:
Variables t, w, and i refer to time, natural frequency, and the mode
number. A dynamic system oscillates in a combination of deformation
shapes which are called modes such that n populations will haven degrees
of freedom, n mode shapes, and n natural frequencies. Constants A
and B are determined from initial conditions for the system since
the solution represents an eigenvalue solution (Craig, 1981;
Mer i o v itch , 19 75) .
The general solution for the responses of a multiple degree
of freedom system are shown below:
x1 (t) = (A.1
(A.
J
<!> ••
1J
<j> ••
11
sin w1. t +B.1
<P •.
11
cos w.1 t) +
sin w. t +B. cp .. cos wJ. t) ...
J
J 1J
x2 (t) = (A; cpji sin Wi t + B; cpji cos wi t) +
(A. cp .. sin w. t +B. cp .. cos w. t) ...
J JJ
J
J JJ
J
For a multiple degree of freedom system, the solution procedure may
be performed in the stepwise form that follows:
1.
Set up the mass static and stiffness dynamic matrices
as in section III and assemble into the following equation:
[M]
{x}
+ [K]
{xJ = o
61
2.
Assume a harmonic solution of the form shown below with u.
1
and U. being defined as an arbitrary variable and constant:
1
u1. (t) = U.1 sin w.1 t + u.J cos w.1 t
3.
Set up an algebraic eigenvalue problem on the basis of the
assumed solution by multiplying the mass matrix by -w 2 :
2
[ [K] - w [M] ] {::} = 0
4.
Since this represents a set of linear differential
equations, the responses correspond to the values of w.1
that satisfy the equation shown below for non-trivial
solutions.
The number of values will equal the number of
degrees of freedom, or species, in the food web.
[ [K] - w2 [M] ] = 0
5.
The determinant of the matrix shown in step 4 is known as
the· characteristic equation which is a polynomial of the
form shown below:
An+ al An-1 + a2 An-2 + a3 An-3 + ... +an
where Ai 1 s equal the square of the frequencies wi.
6.
Solve for the roots of the polynomial equation.
7.
Since many values of w satisfy the matrix in step 4, it may
be thought of an an eigenvalue problem.
Each of the
eigenvalues would possess a corresponding eigenvector which
is a matrix with the number of columns equal to the number
of species in the food web.
62
8.
Eigenvectors consititute the columns of the matrix.
The latter has an order equal to the number of species in
the food web as shown below:
411
~ln
4nl
~nn
[4] =
9.
These values and the natural frequencies are substituted
into the general equation of oscillation.
The general
solution for a two degree of freedom is shown below
(Craig, 1981):
x1 (t) = A1 411 sin w1 t + 81 411 cos w1 t +
A2 412 sin w2 t + 82 412 cos w t
x2 (t) = A1 421 sin w1 t + 81
~
21 cos w1 t
+
A2 ¢ 22 sin w2 t + 82 4 22 cos w2 t
Constants A1 , A2 , 81 , and B2 , are determined from the initial
conditions of the system.
63
Example V.l.
Determine the natural frequencies and the response
Assume K1 = K2= 1.0,
M1 = M2 = 1.0, population sizes of species A and Bare
of the food web system shown below.
1000 and 100.
Assume an initial displacement x1 (0) of
All other initial conditions, x (o),
2
species A of 100.
x1 (o), and x2 (0) are 0. Assume that the unit of time
is 1 year.
K
3
= 1.0
Species B
Species A
1.
Set up stiffness and mass matrices.
Ml
[M] = [ 0
64
2.
Assume a harmonic solution of the form shown below:
Ui(t) = Ul
3.
Wit+ U2 sin Wit
COS
Since-w2mx = kx, the dynamic population matrix is multiplied by
-w2 and the algebraic eigenvalue problem is set up as shown
below:
2.0
-1.0
[ [
4.
-1.0]
2.0
For a non-trival solution the following must be true:
2.0
[[
-1.0
-1.0]
2.0
combine matrices and set w2
2-A
[
5.
-1
-1
]
A
=0
2-A
Solve the determinant to obtain the polynomial shown below:
2 - 4A + 3
A
6.
=
=0
The roots of the equation in step 5 are:
The natural frequencies are:
w1 =
JA;
=
~
w2
Jf2
=
rJ3 = 1. 732 cycles/year
=
= 1 cycle/year
._
65
7.
Roots Al and A2 are substituted into the algebraic eigenvalue
expression.
Root 1 :
2-Al
-1 ]
[ -1
= [
2- Al
l
2-1
-1
-1
2- lJ
Determince matrix of cofactors
Mode shape 1 is shown below:
I<P1l = {1.0}
1.0
Root 2:
2-A 2
[
-1.0]
2-A.2
-1.0
-1.0]
=
[2-3
-1.0 2-3
Matrix of cofactors:
[cp]
=[-1 1]
1
-1
The second mode shape is shown below:
8.
The matrix of eigenvalues is shown below:
[~] [::~
=
:::]
[:
-~]
66
9.
Substitute eigenvalues and frequencies into the general response
equation as shown below:
x1 (t) = A1 (t)
cost+ B1 (1) sin t
= A2 (1) cos 1.732 t + B(l) sin 1.732t
At + = 0, constants A1 , A2 , B1 , and B2 , and B2 are determined
using the specified initial conditions. It will be assumed that
species A oscillates, so the initial conditions will be assumed
Based upon these assumptions the following expressions result.
x1 (O) = .1 - A1 + A2
x2 (0)
= 0 = A1
- A2
x1 (o) = o = s1
+ 1.732 s2
x2 (o) = o = s1
- 1.732 s2
81
= 82
=
0
Solve for A1
then
0.5 = A + A
1
1
0. 5
and
= A1
67
The response equations are:
x1 (t) = .05 cost+ .05 cos 1.732 t
x2 (t)
= .05 cost- .05 cos 1.732 t
Table V.1 shows amplitudes at 1 year intervals for times from
0 to 10 years for both predator and prey species.
68
Table V.l.
Population Size Amplitudes of Predator and Prey
Populations for Times From 0 to 10 years
TIMES (YEARS)
0
POPULATION
POPULATION 2
1100
100
1044
106
2
1001
110
3
1066
103
4
1095
101
5
1023
108
6
1011
109
7
1085
101
8
1081
102
9
1006
109
10
1029
107
Q •
69
V.2.
NORMAL MODE METHOD
Coefficient matrices [M] and [K] often contain non zero terms
in the off diagonal positions, thus producing a set of coupled
equations.
The normal mode method, or mode superposition method as
it is often called, uses the property of orthogonality of natural
modes to diagonalize the coefficient matrices (Craig, 1981).
result is a set of uncoupled equations.
The
The normal mode solution
procedure incorporates the general solution procedure representd in
Section V.l through step 8.
At this point the following procedure
is used:
1.
Use the matrix of eigenvalues to diagonalize the coefficient matrices and to transform them to the modal
coordinate system.
The transformed coefficient matrices
are known as transformation matrices are shown below:
[MR] = [~JT [M] [~]
[KR] = [~]T [K] [~]
2.
Due to the principle of orthogonality, modal mass and
stiffness matrices contain non-zero terms only on the
main diagonal.
The result is a set of equations of
oscillation in the form shown below:
..
2
n.1 + w.1 n.1
3.
=
o
The solution of each equation from step 2 is shown below:
70
4.
The vector of transformed initital conditions is obtained
by transforming the general initial conditions using the
following relation:
I
(MRi) ni(O)
5.
l
=
[~JT [M] [Xi(O)]
Constants A and B are determined from the modal initial
conditions and the solutions are transformed back into the
general coordinate sytem using the following relation:
x(t)
= [~] ln(t)J
7l
Example V.2.
Determine the response equations for the food web shown
in example V.l.
Use the same conditions as stated in
example V.l.
From example V.l, the mass and stiffness matrices are:
[M]
=
[1.0 0]
0
1.0
-
[
2.0
[K]
=
-1.0
- 1.0]
2.0
Natural frequiences are:
w, = 1.0
w2 = 1. 732
The matrix of eigenvalues is:
1.
Determine the modal matrices
[K] = [~JT [K] [~]
72
2.
The modal set of equations is:
The uncoupled equations are:
2Til
= 2111 = 0
2Ti2 + 6112 = 0
3.
The modal response equations are:
11 1 {t) = A1 cost+ s1 sin t
11 2{t) = A2 cos 1.732 t + B2 sin 1.732 t
4.
The general initial conditions are transformed into modal
initial conditions
MRl Tjl
I
{O)l
.
t~R2 112 (0)
= [~]T [M] x(O)
73
5.
Solve for constants A and B.
~ (0) =
1
= .05- A cos 1.732 t + B sin 1.732 t
~ (0)
I
2
~,
.05- A cost+ B sin t
(t)
=
-~2(t) =
.05 cos t
.05 cos 1.732 t
Transform modal response equations into the general
coordinate system.
x, (t)
= ~,
x2 (t)
=
f"jl (t) - YJ2(t)
x, (t)
=
.05 cos t + .05 cos 1. 732 t
x2 (t)
=
.05 cos t - .05 cos 1 . 732 t
(t) + YJ2(t)
or
74
V.3.
MATRIX ITERATION METHOD
One of the more effective hand computation procedures for
small problems is the method of matrix iteration.
Using the form of
the basic differential equation shown below, eigenvalues and natural
frequencies may be obtained:
[[K] - w2 [M]]
{x} = 0
The previous equation may be written in the following form:
[K]
l xJ
= w2
[M]
lxJ
The general procedure is carried out by assuming a vector of
eigenvalues and performing a series of iterations until the above
equation is satisfied.
have been identified.
At this point, an accurate eigenvector will
The method can be reduced to the stepwise
procedure that follows.
1.
Set up the mass and stiffness matrices.
=[:11
[K] =rll
:2J
[M]
Kl~
K22
K21
2.
Invert the stiffness matrix.
3.
Premultiply both side of the equation by [K]-l to obtain:
1
w2
[KT
1
1
w2
IXI =
1
w2
{x/
[K]
{x}
= [KT 1 [M]
[KT 1 [M]
I
= [D] X}
IXl
{x}
75
4.
Choose an arbitrary starting eigenvector.
A starting
eigenvector with a 1 in each position is generally assumed
for a first mode iteration.
The general equation for
.I
a two degree of freedom system is shown below:
=
5.
[D]
ln
An eigenvector is obtained then divided through such that the
largest eigenvalue is reduced to 1.
The general form is
shown below:
Variables a- 1 and 'Yare the second eigenvalue and the divisor
of the eigenvector.
6.
The new eigenvector is used in the next iteration procedure.
The relation is shown below:
Variables a 2 and a 3 are the eigenvalues calculated in step 5.
The procedure is carried out until the eigenvalues do not
change significantly with each successive procedure.
The
final solution is of the form shown below:
1
w2
The first frequency is found using the following equation:
wl
=
(-1-) 1/2
'Y
76
7.
Higher modes may be determined using the same principle.
The first natural mode is eliminated from the configuration
using the following relation:
8.
The result of the operations from step 7 is used to form
the sweeping matrix.
A1
=
A2
= · -a 1A2
A1
The sweeping matrix [S] developed from these relationships and
is shown below:
9.
The new mass matrix from which the second mode is calculated
is found using the following relation:
[D][sd :: l
'1 { ::} =
After the new mass matrix is calculated, steps 4 through
6 are repeated to obtain the second mode configuration.
10.
The matrix of eigenvalues is then formed as shown below.
[c!>]
=
[~11
~n 1
::: cp~n]
P~n
:::
77
Example V.3.
Determine the response and natural frequencies for the
food web shown in example V.l using the matrix iteration
method.
l.
Set up the mass and stiffness
[M]
=
[K] =
2.
r·o 1.:]
0
-1.0]
r-0
2.0
-1.0
Invert the stiffness matrix
[Kr 1 = I-121 -12 I
3.
4.
matrices.
[~ :1
Determine the system dynamic matrix
[OJ
= [Kr l
[D]
=[
[M]
2/3
1/3]
l/3
2/3
Choose a starting vector and determine the first eigenvector.
Starting vector =
{ 21}
78
1
w2
=
2/3j
{2}1
{5/3}
=
5.
1/31
[2/3
l/3
3/3
Divide the eigenvector by 5/3 to obtain
~
{Al}
w A
6.
= 5/3 {1.6 J
2
The new eigenvector is used in the next iteration procedure
as shown below.
12/3/3] {1.6}
Further iteration will result in the following
~
1 { }
=
The final eigenvector is
=
{~}
The natural frequency is:
wi
7.
(-+f/2
Use the first mode eigenvector to eliminate the first mode.
jl
8.
=
ll[:
~]
C}= alAl
Form the sweeping matrix
=
=
+
a2A2
79
The sweeping matrix is
9.
Form the new dynamic system matrix
2/3
[0(2)]
=
=
[ l/3
1/3J [ 1
2/3
-1
[1 /3 00]
-1/3
Assume a starting vector of
{~} and use steps 4 and 6
until the result converges.
= [1/3
1
-1/3
w2
=
~]
{:l
1/3
-1/3
Divide the new vector by 1/3 to obtain
=
1/3 {_:
l
This turns out to be the second mode solution
w2
=
1.732 cycles/year
The response equations can be determined using either the
normal mode method or general solution procedure.
The
80
response equations are shown below:
x1 (t) = .05 cost+ .05 cos 1.732 t
x2 (t) = .05 cost
.05 cos 1.732 t
Again, the populations at yearly intervals between 0 and
10 years are shown in Table V.l.
V.4.
CHANGES IN THE DYNAMIC MODEL
As in the case of the static model, the primary goal of a
dynamic model is to provide the biologist with a to.ol which may be
used to evaluate changes in a food web structure.
A change in the
size of any population in the food web may affect the responses of
any or all populations in the system.
Modeling of dynamic changes
involves the adjustment of link values in the stiffnes matrix as
shown in section IV.4 as necessary and a reduction in an appropriate element of the mass matrix.
For the system shown in Figure
V.2a., the set of equations for an undistrubed food chain is shown
below:
The system of equations for the disturbed system shown in Figure
V.2b is shown below:
81
82
Figure V.2.
Undisturbed and Disturbed Food Web Systems
K = 1.0
K = 1.0
Species B
M= 1.0
M
K = 1.0
= 1.0
K = 1.0
M= 1.0
M = .8
Species A
K = 1.0
K = 1.0
(a)
{b)
V.5.
DYNAMIC SOLUTION PROCEDURES FOR NONCONSERVATIVE SYSTEMS
Conservative systems are systems in which peak amplitudes do
not reduce with time.
amplitudes
However, in the case in which the peak
tend to dissipate with time, then they may be labeled
as a non-conservative and correspond to the Rosenweig-MacArthur
damped cycle model (Rosenweig and MacArthur, 1963).
One way of
handling this type of system is to treat the food web as a springmass system with damping.
As in the case of the non-conservative
system, the most basic food web is a single degree of freedom system
with damping.
phenomenon
Damping in mechanical systems a velocity related
which causes a dissipation of each successive peak
amplitude until the system converges on the equilibrium population
size.
Sources of damping in natural systems may be density dependent
factors such as would be associated with predator search times.
The
general equation of oscillation for a single degree of freedom
system with damping is shown below:
mx + ex
+
kx = 0
The general response equation for this system is
2
x(O) + x(O) sw}l/ -swt
w
e
sin(wt+cp)
where tis the percent of critical damping which is the amount of
damping that would prevent oscillatory behavior.
As an example of
a damped system, the same system as was used in the single degree of
freedom system from section V.l will be used.
It will again be
assumed that the equilibrium population size is 100, and the the
displacement is 10.
These numbers are converted to fraction of 1.0
83
84
such that the population size is 1.0 at equilibrium and the initial
displacement is 0.1.
The response equation for the system with a
damping factor (s) of 0.1 is shown below:
x(t)
= 0.1414
-. 1t
e
sin (t + <!>)
where the phase angle (~) is:
tan
-1 "'_ x(O)w
't' .X+.(
. 0-rl)--:-+-x-,("0")s::-W-
Population sizes for a period of 10 years between 0.25 and 10.25
years are shown in Table V.2.
A link by link estimate of damping may not be feasible, but a
system wide estimate of damping may be employed.
An appropriate
method of pro vi ding a system estimate of damping is known as proportional damping (Craig, 1981; Paz, 1980).
The general equation for
a system with damping is shown below:
[M] {x} + [C] {xJ + [K] fxf = 0
Matrix [C], the damping matrix, will be determined using the static
and dynamic matrices.
With some restrictions, the general relation
may be expressed as shown below:
[C] = a 1 [M]
+ a
2 [K]
Constants a 1 and a 2 are calculated as shown later.
The basic procedure follows:
1.
Calculate the natural frequencies and eigenvalues using the
solutiontprocedures from earlier sections.
2.
Estimate a value for the system damping (s).
This value
is generally between 0 and l.O,usually less than 0.15.
85
Table V.2.
Peak Population Sizes at One Year Intervals for a Damped,
Single Degree of Freedom Food Web
YEAR
POPULATION SIZE
0.25
112
1.25
106
2.25
103
3.25
102
4.25
101
5.25
101
6.25
100
7.25
100
8.25
100
9.25
100
10.25
100
86
3.
The following equation is used to calculate values for a1
and a 2.
{~~~
=
1/2 [::
4.
Invert the mass matrix.
5.
Post-multiply the inverse of the mass matrix by the stiffness
matrix.
6.
Pre-multiply the matrix from step 5 by a 1 , a 2 , ... ,an,
and sum the results. The relation is shown below:
. .£1 a . ( [M] - l
1=
7.
[ K]) i
1
The damping matrix is determined by pre-multiplying the
result from step 6 by the mass matrix.
£
. 1
[C] = [M]
a. ([Mr 1 [K])i
1
1=
8.
The response of the system may be determined using the
normal mode method.
The damping matrix is diagonalized
using the following equation:
9.
The general form for the equation of oscillation is shown
below:
t4
10.
1i 1. +
C~. + K11.
1
1
=0
The general form for the equation of oscillation is shown
below:
~j = ~2(0) +
~(0) + ~(O) ~.w.
1 1
Wi
ll/2 e - s.w.t sin
1 1
(w. t + ¢)
1
87
Initial conditions are determined as in the normal mode
solution.
11.
The solutions are then transformed into the general
coordinate system using the relation shown below:
x(t)
= [¢]
fYJ(t) J
88
Example V.4.
Determine the response equations and natural frequencies
for a damped food web shown in example V.l.
Estimate
the natural damping for both modes to be s1 =
1.
~]
-1]
[K] = [2
_,
2
[M]
=[:
w, = 1 cycles/year
w2 = 1.732 cycles/year
[ct>J
= [:
_:]
Estimate system damping.
r.,
= .1
{,2 = . 1
3.
2 = 0.1
Stiffness and mass matrices, natural frequencies and mode
shapes are obtained from example V.l.
2.
~
Determine a 1 and b
1
{;:} =
l/2
al = . 22113
a2 = .02113
[1.:32
89
4.
Invert the dynamic matrix.
5.
Premultip1y the inverse of the mass matrix by the stiffness
matrix.
=[ 1
0
[MT 1 [K]
0] [2 -1]
1
=[ ~1
6.
2
-:]
Multiply the matrix from step 5 by a 1 and a 2 and sum using
the following relation:
= . 2211 3
[
2
-1
-11
- . 02113
2
-.24226]
=[ .35774
.35774
-.24226
7.
-1
Determine the damping matrix
[C]
=[:
0][·35774
1
-.24226
= [.35774
-.24226
-.24226]
.35774
-.24226]
.35774
-1]2
2 -
90
8.
Determine the modal damping matrix
=
9.
[.23096
0
0]
1.2
Determine the uncoupled equations of oscillation.
The
modal matrices from example V.2 are:
[M] =
[~ ~]
The modal equations of oscillation are:
10.
The modal response equations are shown below:
~ (t)
1
= e-. l
t (A cost+ B sin t)
1
1
~ 2 (t) = e-· 1732 t (A 1 cos 1.732 t + B2 sin 1.732 t
These equations reduce to the equations shown below:
~, (t) =
)\1 e-sw1 t
sin (t + )
~2(t) =
x2 e-sw2 t
sin (1. 732 + ¢)
91
where X;
=
!~20
+.
1] 0 + sw; 11 0
Wi
l
thus
lll(t)
11
=
.0866 e-. 1 t sin (t + <!>)
1732 t sin (1.732 t + <!>)
2 (t) = .0866 e-·
where
<1>
is the phase angle for which the formula is shown
below:
tan - 1 <!> =
11.
1'1
'10
w.1
Transform modal responses into the general coordinate
system.
The final solution is shown below:
x,(t) = .0866 (e-· 1 t sin (t + ¢)' +
x2 (t)
=
e-· 1732 t
sin (1.732 t + <!>))
.0866 (e -.l t sin (t + ¢)- e-· 1732 t sin (1.732 t + <!>))
Population sizes will attennate to 1000 from a maximum of 1173
at t = 0 for population A and begin at 100 for population 2.
Both
populations, A and B, will converge upon their equilibrium sizes
of 1000 and 100.
V.6.
DYNAMIC SOLUTION PROCEDURE WITH EXTERNALLY
INDUCED OSCILLATIONS
Population level oscillations may be produced by variations in
the amounts of inorganic materials, available light, and other
environmentally controlled factors.
Such variations may be produced
by seasonal conditions, lunar induced variations such as tides or
even daily variations.
These may be treated as external conditions
which may affect the food web.
The mechanical analogue is the
spring mass system which is subject to an externally forced oscillation (Meirovitch, 1975; Craig, 1981, Paz, 1980).
The general equation
for this type of a system is shown below:
!xl
[M]
+ [K] lxJ
=
IP sin n tJ
For the purposes of the model, P will be assumed to be 1 as its
value may not be identifiable.
Variable n is the frequency of the
induced oscillation.
An appropriate method of determining the response equations
involves the use of the normal mode method.
The normal mode solution
is carried out to the point of forming the modal equations of
oscillation.
The forcing function is applied at the first level
(species A) in Figure V.3, and is transformed into modal coordinates
using the following expression:
fP/ = [4>]T JP sinntj
The modal equations assume the following form:
~i
+
w~ 11 = P sin n t
92
93
Figure V.3.
Two Degree of Freedom Food Chain with Induced
Oscillations
K = 1.0
Species 8
M = 1.0
K = 1.0
Species A
M = 1.0
P sin Ot
K = 1 .o
94
The general solution will be the sum of the steady-state
response and the transient response as shown below:
~.(t) =A. sin w.t +B. cos w.t + P/K sin ~t
1
1
1
1
1
M(w.2 - n )
1
For a system with all initial conditions equal
to zero, the
equation for the response is shown below:
~i(t) =
-f- (~)
1-r
(sin nt -r sin wit)
Variable r, refers to the value of the forced frequency divided by
the natural frequency.
The modal solution is transformed into
general coordinates to obtain the general solution using the following
expression:
x (t)
=
[<PJI n
(t)
l
Three conditions may occur depending upon the ratio of the forced
frequency divided by the natural frequency.
Should r be greater or
less than 1, the system may be assumed to be stable.
If r equals 1,
the system is unstable since the amplitude of the system will increase
with each eye 1e.
95
Example V.5.
Determine the response of the system shown below.
K1 = K2 = K3 = 1, M1 = M2 =1.
and
Q
Assume P = 1.0
= 1.5 cycles/year.
w1 = 1 cycle/year
Kl
w2 = 1.732 cycles/year
1.
From example V.l the following modal equations were
determined:
2.
Since the system is externally perturbed at the level 1,
lt must be transformed into modal coordinates.
{ ::} =
c Jrs:n nt}
l
sin nt J
sin nt
{ ::} = {:
96
Assume that P
{ pp21
3.
l
=
lJ
1 and n = 1.5
sin 1.5 t
The modal equations of oscillation are shown below:
2n 1 + 2n 1
4.
Ill
=
= sin
1 .5 t
The modal response equations are shown below:
l/2 (
1
(1.5 2 - 1)2
) (sin 1.5 t- 1.5 sin t)
l__ (sin 1.5 t -.866 sin 1. 732 t)
112(t) = l/6 _ _ _
1.5 2
- (1.732)
1
~ 1 (t) = ~
-- (sin 1.5 t- 1.5 sin t)
2.5
~ 2 (t) = ~
(sin 1.5 t- .866 sin 1.732 t)
I .5
5.
After transformation into general coordinates, the
responses are:
x1 (t) = 2 ~ 5 (sin 1.5 t- 1.5 sin t +~(sin 1.5 t + .866 sin 1.732 t)
x2 (t) = 2 ~ 5 (sin 1.5 t- 1.5 sin t) -
1~5
(sin 1.5 t -.866 sin 1.732 t)
V.?.
DYNAMIC LARGE MODEL EXAMPLE USING REAL DATA
As an example of a freely oscillating food web system analysis,
the subtropical food web (Paine, 1966) was analyzed (Figure IV.16).
Four cases were analyzed using the conditions shown below:
Model A--All link stiffnesses and population sizes equal 1.0.
Model B--All population sizes equal 1.0 and link stiffness
represent fractional predator consumption patterns with
respect to numbers of prey.
Model C--Same as model B with link 6 reduced by half.
Model D--Same as model B with links 12 to 16 and population
4 reduced by half.
Table IV.3 shows the link stiffness values used for each model.
Table V.l shows the natural frequencies of each model.
Model A,
when compared to Model B has substantially higher natural frequencies
due to higher link stiffness values that were assumed for the model.
Models C and D demonstrate the effect of the changes in link
structure and population size reductions.
The analysis of Model C
shows that the alteration of only one link stiffness can affect the
natural frequencies of the food web system such that they are
slightly lower than in Model B.
Model D, however, reflects a
one-half reduction in the barnacle population and shows a substantial
reduction in the natural frequencies of the system when compared
to Model B.
This change would be critical if the system was
assumed to be subjected to an induced oscillation.
Should a natural
frequency be equal to the forced frequency, a resonant, dynamically
unstable condition might exist.
For example, should an inducing
97
98
condition have a frequency of 1.0, the biologist must be concerned
about a system with natural frequencies in the neighborhood of 1.0.
For example, in Model B, frequencies of mode numbers 4 to 7 would be
of concern.
In addition model frequencies 5 to 7, and 6 and 7 would
be critical in Models C and D.
All dynamic models were analyzed using SAPIV, a structural
analysis computer program.
The program essentially uses an iteration
procedure to determine natural frequencies and mode shapes (Bathe
and Wilson, 1973).
99
Table V.3.
MODE NO.
1
2
3
4
5
6
7
8
9
10
11
12
13
*cycles/year
Natural Frequencies of Several Food Web Systems
MODEL A*
MODEL B*
MODEL C*
MODEL D*
.6846
1. 383
1. 414
1. 815
2.032
2.236
2.256
2. 415
3.144
3.443
3.840
4.352
4.454
. 5235
. 7706
.8453
.9952
1. 005
1. 020
1. 023
1.142
1.297
1.377
1 .412
1. 746
2.367
.5210
• 7688
.8447
.9306
1.005
1 . 016
1.023
1.133
1. 297
1. 372
1.410
1 .694
2.367
.4911
.6637
. 7261
.7596
. 9524
1. 005
1 .023
1 . 131
1. 215
1. 298
1. 389
1. 742
2.477
VI.
DISCUSSION
Static and dynamic food web modeling procedures preseneted in
previous sections were developed in order to provide a biologist with
a set of simple methods with relatively simple formulation procedures
through which food web structures can be studied.
Static methods,
based upon discretized structural analysis procedures, are capable
of providing a model of nutrient transmission through the web to the
top order predators.
In addition, they are capable of detecting
shifts in the nutrient flow pattern of a food web due to changes in
the populations within the food web.
Dynamic modeling procedures, based upon discretized dynamic
analysis procedures, provide a model of the dynamic behavior of populations in a food web system.
As the population size of any or several
species varies, the effect on the size of the other populations in the
food web may be determined, and the feedback interactions affecting
population sizes may be followed through time.
Free oscillating systems provide a mathematical representation of
the Rosenweig-MacArthur (1963) models, which demonstrate the periodic
variations oscillating system model is similar to the stable cycle
with refuge model - which allows the prey from being hunted to
extinction- shown in Figure VI.l (Rosenweig and MacArthur, 1963),
shown in Figure VI.2, may be represented by the non-conservative
method presented in section V.5.
Although figures show that peak
amplitudes for both predator and prey populations occur simultaneously,
this may not always be the case.
Food web systems may be affected by the variation in nutrient
100
101
Figure VI.l.
Stable Cycle with Refuge
Predator
Prey
102
Figure VI.2.
Damped Cycle
Predator
Prey
103
which may be produced by seasonal, tidal, or daily patterns.
These
induced oscillation systems may be used to model the stable cycle
with refuge (Figure VI.l) for an induced frequency which is lower
than the natural frequency of the system (Rosenweig and MacArthur,
1963).
This produces a pattern of oscillation in which the amplitude
peaks of predator and prey are in phase.
Should the system be forced
at a frequency which is larger than the natural frequency, the peak
amplitudes will be one-half cycle out of phase (Figure VI.3).
In
addition, this pattern may be generated by isolating the second mode
of a two degree of freedom system.
Instability may be produced in a forced system by making the
inducing frequency equal to any of the natural frequencies of the
food web system.
The result is similar to the unstable cycle model
(Rosenweig and MacArthur, 1963) shown in Figure VI.4.
This is the
major cause of concern in a dynamic system in that such a condition
will cause the system to become unstable.
Tropical and subtropical food webs developed by Paine (1966)
provide an ideal set of data for testing both static and dynamic system
models.
The critical contribution of the Paine food webs is the
assignment of links between predators and prey which reflect prey
selection frequencies.
This characteristic allows the biologist to
create precise static and dynamic models of the food web.
The role
of this degree of precision is demonstrated int he static case in
the analyses of Models A and B in section IV.?.
It was shown that
the prey dependence on number of prey of the sea star Heliaster sp.
in Model A was most concentrated on Muricanthus sp ..
However,
104
Figure VI.3.
Stable Cycle
Predator
Prey
105
Figure VI.4.
Unstable Cycle
Predator
Prey
106
barnacles compose more than half of the diet of Heliaster sp. while
Muricanthus sp. is rarely preyed upon.
In contrast, Model B was constructed to reflect prey selection
patterns.
The result was a more precise analysis which showed that
.505 of the diet of Heliaster sp. was composed of barnacles and .001
of its diet was Muricanthus sp ..
Likewise, in the dynamic analysis of both Models A and B, it was
found that the natural frequencies of Model A were substantially higher
than those determined in Model B.
Again, the assumption that all link
stiffnesses were 1.0, as in Model A, reduces the precision of the
dynamic model.
The analysis of Model B shows that the natural frequen-
cies of the food web are generally lower when prey preference is
considered.
The primary application of both static and dynamic modeling
procedures lies in the analysis of changes in the populations in
the food web.
The static analysis of Model C may be compared to
Model B in section IV.?.
This comparison demonstrates the effect
of a reduction in the food dependence of Muricanthus sp. on bivalves
and on the diet composition of Heliaster sp ..
The effect was
undetectable using the static model since Heliaster sp. has only a
minimal dependence on Muricanthus sp. thus an indirect, weak dependence on bivalves fed upon by Muricanthus sp.
As a result, only a
minimal change in the static food web model should be expected.
In contrast, it was shown that in comparing the dynamic analyses
of Models Band C in section V.? the natural frequencies may be
altered.
The result of the reduction of the dependence of Muricanthus
sp. on bivalves was the general lowering of the natural frequencies
107
as shown in Model C.
A comparison of the static analyses of Model D and B reveal
the effect of a one half reduction of the barnacle population on the
rest of the food web.
Since barnacles are a major prey source of
several predators in the food web it would be expected that a substantial differnce would be produced by this change.
The analysis
of Model D shows that Heliaster sp. will reduce its direct dependence
on barnacles from .505 to .338 and increase its dependence on bivalves
from .308 to .41.
More minor increases and decreases occur in the
other links in the food web.
The surprising result is that a one
half reduction in the link stiffness between barnacles and Heliaster
sp. is not matched by an equivalent reduction in its prey dependence.
This causes the sum of the stiffness values of the links connecting Heliaster sp. to the food web to be .745 instead of 1.0.
Since in
the disturbed food web, the stiffness value for the barnacle to
Heliaster sp. link is .255, roughly one-third of the available prey,
therefore barnacles will still compose one-third of the diet of
Heliaster sp ..
The shift to a more substantial dependence on bivalves
and other species compensates for the loss of a portion of the barnacle
population.
As in the case of the comparison of the static analyses of
Models B and D, there was a substantial difference in the system
natural frequencies when Model D was compared to Model B.
The
reduction in the barnacle population size and the corresponding
reduction in stiffnesses of links connected to all predators causes
a general reduction in the natural frequencies of the food web system.
108
Should the food web system be subject to an induced oscillation, a
change in the natural frequencies may produce unstable, resonant
condition.
A change in the oscillation of any one species produces a
response in all other species in the food web.
Thus, the oscillatory
behavior of any one species includes the effects of its own response
plus the effects of the responses of all other species in the web.
The result is a cyclical variation in the oscillatory peaks of each
population.
For example, the system from example V.2 has a repeating
cycle once in every 5.2 units of time.
If the units are defined as
years, the peaks would occur once every 5.2 years or if controlled
by seasons, once every 5 years with an occasional 6 year period.
This model is capable of providing an explanation for the 4 and 10
year cycles found in some simple Arctic food webs in which few predator
and prey species exist.
A further resolution of the role of a dynamic
model with repsect to this situation is a topic for further study.
A similar modeling technique was presented by Odum (1983),
which uses a multiple degree of freedom thermodynamic system analogy.
Since both thermodynamic and mechanical systems are based upon the
same basic differential equations, they are quite similar in construction.
The difference between them lies in the units used.
The
thermodynamic model uses calories as its units of exchange between
populations, while the mechancial analogy uses the numbers of
organisms.
The major advantages of the modeling procedures proposed are
1) The simplicity of model formulation; 2) the sensitivity of the
model in detecting the effects of even the most minor changes in the
109
food web; 3) the demonstration of the effects of changes in any one
population on the static or dynamic behavior of the food web;
4) the adaptability of these methods to computer assisted methods;
5) the ability of the modeling procedure to aid in ecological
prediction.
Some major disadvantages must also be recognized.
Ths most
glaring involves the insensitivity of the models in the detection
of interactions other than predator-prey.
Thus, competition, age
structure and other important factors are ignored by the food web
models.
However, the question must be raised with regard to the amount
of information which is necessary to obtain a reasonable system model.
In some cases, a food web model using linear equations, such as
presented in this paper, may be adequate for ecosystem analysis.
In
other cases it may be necessary to use a more elaborate analysis.
The food web modeling procedures are not sensitive to changes in
prey preference due to changes in prey density.
upon a random prey selection pattern.
These models rely
However, a nonrandom prey
selection pattern may be modeled using a piece-wise analysis in
which the problem is handled in small increments and the matrix
reconstructed to reflect changes in prey selection.
As stated previously, the data necessary for an analysis of a
food web system is merely a food web with predation frequencies
identified.
Should a biologist possess such a set of data, a set of
matrix manipulation subroutines is all that would be necessary to
handle the mathematical procedures presented.
The set of subroutines
should include a matrix manipulation routine, a scalar multiplication
routine, a determinate solving routine, a matrix inversion routine,
110
an eigenvalue solution routine, and a polynomial root finder.
Another
approach would incorporate a matrix assembly routine into a computer
program which carries out the operations mentioned above.
The major goal of this paper was to develop, at least in part,
a means of predicting the response of a food web to environmentallyinduced or man induced changes in a food web structure.
is the development of several analytical methods.
The result
Such methods
may be used in environmental analysis in cases in which an event has
a known effect upon a particular ecosystem.
The results of this
event may then be analyzed prior to its occurrence and remedies might
be suggested in advance.
LITERATURE CITED
Anderson, R. M. and R. M. May. 1978. Regualtion and stability of:
host parasite population interaction. J. Anim, Ecol., 47; 219-234.
Bathe, K. J. and E. L. Wilson. 1976. Numerical methods in finite
element analysis. Prentice-Hall, Inc., New Jersey.
Bathe, K. J., E. L. Wilson, and F. E. Peterson. 1973. SAPIV: A
structural analysis program for static and dynamic response
of linear systems. University of California, Berkeley.
Clark, F. W. 1972. Influence of jackrabbit density on coyote population change. J. Wildlife Manges 36: 343-356.
Clough, R. W. and J. Penzien. 1975.
Hill, Inc., New York.
Craig, R. R. 1981.
New York.
Dynamics of structures.
Structural Dynamics.
McGraw-
John Wiley and Sons, Inc.,
Halfon, E. 1979. Theoretical systems ecology.
Press, Inc., New York.
New York Academic
Lawlor, L. R. 1980. Structural stability of natural and randomly
constructed compatitive communities. Am. Nat. 116(3): 394-408.
Lawlor, L. R. and J. Maynard Smith. 1976. Coevolution and stability
of competing species. Am. Nat. 110(1): 79-99.
Levins, S. A. 1974.
Ecosystem: analysis and prediction.
MacArthur, R. H. 1972.
Geographical ecology.
MacArthur, R. H. and J. H. Connell. 1966.
John Wiley and Sons, New York.
SIAM.
Harper and Row.
The biology of populations.
McGuire W. and R. H. Gallagher. 1979. Matrix structural analysis.
John Wiley and Sons, Inc., New York.
McNaughton, S. J. 1978.
communities.
Stability and diversity in ecological
Maynard Smith, J. 1974.
Press.
Models in ecology.
May, R. M. l973a.
Ecosystem stability.
Cambridge University
Ecology 54(3): 638-641.
1973b. Stability and complexity in model ecosystems.
Princeton University Press.
- - . . - . -.
111
112
LITERATURE CITED
May, R. M. 1976. Theoretical ecology:
W. B. Saunders, Inc.
1981.
Theoretical ecology.
principles and applications.
Sinauer and Associates, Inc.
Meirovitch, L. 1975. Elements of vibration analysis.
Book Company, Inc.
Murdoch, W. W. and A. Oaten. 1975.
Adv. Ecol. Res. 9: 2-131.
Odum, H. T. 1983.
Systems Ecology.
McGraw-Hill
Predation and population stability.
John Wiley and Sons, Inc.
Paine, R. T. 1966. Food web complexity and species diversity.
Am. Nat. 100; 65-75.
. 1974 .
--~9~3~-1~2~0.
Intertidal community structure.
1980. Food webs:
community infrastructure.
Oecologia 15:
linkage, interaction strength, and
J. Anim, Ecol.: 667-685.
Paz, M. 1980. Structural dynamics.
New York.
Van Nerstrand Reinhold Company,
Roberts, A. P. 1974. The stability of a feasible random ecosystem.
Nature 251: 607-608.
Rosenweig, M. L. and MacArthur, R. H. 1963. Graphical representation
and stability conditions of predator-prey interactions. Am. Nat.
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Siljak, D. D. 1974. Connective stability of a complex ecosystem.
Nature 249; 280 .
. 1975.
25: 25-50.
----~~~
When is a complex ecosystem stable? Math Biosc.
p •
APPENDIX A.
MATRIX ALGEBRA
Definitions
A matrix can be defined as a rectangular array of the form
shown below:
ai .
J
Numbers a 11 to aij are elements of the matrix with i and j
denoting the row and column that defines the position of the element.
The matrix is defined as an ( i x j) matrix.
Additions
Given the matrices shown below, element a 11 in matrix A is
added to element b , a 12 to b12 and so on.
11
Example A. 1
113
114
Multiplication
Given the matrices shown, matrix multiplication is carried out
as follows:
=
X bll) + (al2 X b21) (all X bl2) +
X bll) + (a22
X
b21) (a21 X bl2) +
Example A.2
[
=
19
22
43
50
(1 X 6) + (2 X 8
(3 X 5) + (4 X 7)
(3 X 6) + (4 X 8-)J
A matrix can be multiplied by a scalar quantity as shown:
(b)
rll
a21
al2J
=
a22
r·ll
ba21
Example A.3
(3)
c :l r:
=
)1
(1 X 5) + {2 X 7)
1:]
bal2]
ba22
115
Transpose
The transpose of a matrix is merely a rearrangment of a matrix
such that aij replaces aji and vice versa.
is denoted by a superscript T.
A transposed matrix
For example [A]T is the transpose
The operation is shown below in general form:
of matrix [A].
Inversion (Cramer's Rule)
Under some restriction a matrix can be inverted. This operation
can be reduced to a three step process as shown below:
1.
Replace each element a .. by its cofactor A..
lJ
lJ
Given the matrix shown, the matrix of cofactors can be
determined as follows:
A' denotes the matrix of cofactors
which consists of all elements not in the same row or
column of the element.
A. . = (- 1) ( i +j) det [A l]
lJ
=
(-1)(1+1)
116
(-1)(2+1)
Matrix .of cofactors [a22
-a21]
. a 11
-a 12
2.
Divide each cofactor by the determinant of the original
matrix [A], where det [A] = a 11 a 22
3.
Transpose the result
[Ar
1
Example A.4
=
1
["22
all a22- a21 al2
-a21
=
[:
because det [B] = 0.
Determine matrix of cofactors.
11
-a12J
a22
:]
Note that the matrix [B]
a
a 21 a 12 f 0
Invert the matrix using Cramer's Rule.
[A] = [:
1.
~
= (-l)(l+l) [4] = 4
al2 = (-1)(1+2) [3]
= 3
:]
will have no inverse
117
a21
=
(-1)(2+1) [2]
=
(-1)( 2+2 ) [1]
=1
Matrix of cofactors
2.
=
2
f_: -~1
Divide each cofactor by the determinant of the original
matrix [A].
=
det [A]
L :]
=
(l X 4) - (2 X 3)
1.
5]
-.5
3.
Transpose the result.
[AT 1
-2
=
[ 1.5
= -2