30
Dynamic Modeling with Difference Equations
replace P + r P(1 P) by these approximations in Pt+1 = Pt +
r Pt (1 Pt ). Use this to determine the stability of the equilibria. Your
answer should agree with the preceding two problems.
1.3.11. Many biologica
64
Linear Models of Structured Populations
in this matrix is saying about the population. Be careful in trying to
explain the meaning of the .11 in the upper left corner.
2.2.9. a. Show that Ax = Ay does!not "
necessarily
y by
! "mean x = !
" calculat21
5
60
Linear Models of Structured Populations
Although we will not prove it here, it is possible to show that, for square
matrices, if Q P = I , then P Q = I . So, if Q is the inverse of P, then P is
the inverse of Q.
Before we try to calculate the inverse o
58
Linear Models of Structured Populations
x0 = (0,50,50,0)
2000
Population size xt
1800
1600
1400
1200
1000
800
600
400
200
0
0
2
4
6
8
10
12
Time t
Figure 2.2. Simulation of plant model; on the right side of graph, classes are in order 1,
2, 3, and 4 fr
4
Dynamic Modeling with Difference Equations
It may seem odd to call Pt+1 = (1 + f d)Pt a difference equation, when
the difference !P does not appear. However, the equations
Pt+1 = (1 + f d)Pt
and
!P = ( f d)P
are mathematically equivalent, so either one
viii
Preface
Our writing style is intentionally informal. We have not tried to offer definitive coverage of any topic, but rather draw students into an interesting field.
In particular, we often only introduce certain models and leave their analysis
to ex
66
Linear Models of Structured Populations
Table 2.2. Linear
Model Simulation
with Eigenvector as
Initial Values
t
xt
0
1
2
3
.
.
v
Av = v
Av = 2 v
A2 v = 3 v
.
.
The practical consequence of this is that although people might speak of
(5, 3) as the eigen
28
Dynamic Modeling with Difference Equations
the flour beetle tribolium, that exhibits chaotic dynamics (see (Cushing et al.,
2001).
Problems
1.3.1. The equilibrium points of a model are located where the graph of Pt+1
as a function of Pt crosses the lin
74
Linear Models of Structured Populations
the eigenvalues i are complex, if 1 is strictly dominant
so |1 | > |i | for
! !
! !
i = 2, 3, . . . , n, then by part (c) of the theorem, ! 1i ! < 1 as before, and so
! !t
! i !
! 1 ! approaches 0 as t increases.
12
Dynamic Modeling with Difference Equations
P/P
r
P
K
Figure 1.1. Per-capita growth rate as a function of population size.
Of course we cannot say exactly what a graph of !P/P should look
like without collecting some data. Perhaps the graph should be c
46
Linear Models of Structured Populations
Matrices (the plural of matrix) are usually denoted by capital letters,
such as A, M, or P. For instance, we might say
!
"
.9925 .0125
P=
.0075 .9875
is the projection, or transition, matrix for our forest model
80
Linear Models of Structured Populations
While our argument above also applies to larger matrices (provided you
learn to compute determinants of larger matrices in some other course), solving the characteristic equation will be much harder because for a
70
Linear Models of Structured Populations
Definition. An eigenvalue of A that is largest in absolute value is called a
dominant eigenvalue of A. An eigenvector corresponding to it is called a
dominant eigenvector.
Notice that we did not say the dominant
68
Linear Models of Structured Populations
Technical remark: Not every matrix has eigenvectors that can be used as
columns to form an invertible matrix. However, it is possible to prove that if a
matrix does not have this property, then by changing the en
54
Linear Models of Structured Populations
To capture the effects on population growth, we might begin modeling a
human population by creating five age classes with:
x1 (t) = no. of individuals age 0 through 14 at time t,
x2 (t) = no. of individuals age 1
16
Dynamic Modeling with Difference Equations
next_p = p+.7*p*(1-p/10)
20
Population at time t+1
18
16
14
12
10
8
6
4
2
0
0
2
4
6
8
10
12
14
16
18
20
Population at time t
Figure 1.4. Cobweb plot of a nonlinear model.
to the parabola to find the point (P1
6
Dynamic Modeling with Difference Equations
1.1.2. In the early stages of the development of a frog embryo, cell division
occurs at a fairly regular rate. Suppose you observe that all cells
divide, and hence the number of cells doubles, roughly every hal
24
Dynamic Modeling with Difference Equations
When linearizing to determine stability, it is vital that you are focusing
on an equilibrium. Do not attempt to decide if a point is a stable or unstable
equilibrium until after you have made sure it is an equ
48
Linear Models of Structured Populations
W (Dx0 ), which we could compute relatively easily by matrix multiplication:
!
"!
" !
"
.9925 .0125
106.45
116.82
x2 =
.
.0075 .9875
893.55
883.18
A more interesting question is can we find a single matrix that w
14
Dynamic Modeling with Difference Equations
next_p = p+.7*p*(1p/10)
Population P
12
10
8
6
4
2
0
0
5
10
15
Time
Figure 1.2. Population values from a nonlinear model.
If we measure population size in units such as thousands, or millions of
individuals, t
20
Dynamic Modeling with Difference Equations
intuitively? (Note that r will be very small, because we are using a
small time interval.) The logistic growth model is sometimes also
referred to as the autocatalytic model.
1.3. Analyzing Nonlinear Models
Un
86
Nonlinear Models of Interactions
3.1. A Simple PredatorPrey Model
Imagine two species, one of which, the predator, preys on the other, the prey.
To keep things simple, we imagine that the predatorprey interaction between
these species is the most impor
40
Dynamic Modeling with Difference Equations
Problems
1.5.1. (Calculus) The logistic differential equation is
dN
= r N (1 N /K ).
dt
a. Show that
N (t) =
K
1 + Cer t
where C =
K N0
N0
is a solution with initial condition N (0) = N0 .
b. Graph N (t) for K
52
Linear Models of Structured Populations
find the following without a computer. Then check your answers with
MATLAB.
a. A + B
b. AB
c. B A
d. A2 = A A
e. 2A
f. Show C(A + B) = C A + C B.
! "
! "
r s
x
2.1.5. For A =
and x =
and c a scalar, show A(cx) =
To J., R., and K.,
may reality live up to the model
MATHEMATICAL MODELS IN BIOLOGY
AN INTRODUCTION
ELIZABETH S. ALLMAN
Department of Mathematics and Statistics,
University of Southern Maine
JOHN A. RHODES
Department of Mathematics,
Bates College