THEORETICAL
POPULATION
The Dynamics
BIOLOGY
39, 129-147 (1991)
of a Size-Classified
Benthic
with Reproductive
Subsidy
Population
MERCEDES PASCUAL AND HAL CASWELL
Biology
Department,
Woods Hole Oceanographic
Woods Hole, Massachusetts
02543
Institution,
Received October 2, 1989
This work presents a discrete time model for the dynamics of a size-classified
benthic population with planktonic larvae. Recruitment, decoupled from local
reproduction by larval dispersal, is represented in the model as an external subsidy
to the local population. Analysis of the model reveals the importance of recruitment
and growth plasticity in determining the stability of an equilibrium which always
exists. Growth plasticity promotes stability, while recruitment plays the opposite
role. The stability results provide a scale to which observed levels of recruitment,
mortality, and growth can be compared, in terms of their effects on population
dynamics. 0 1991 Academic Press, Inc.
INTRODUCTION
The life cycles of many marine invertebrates are characterized by long
range dispersal of their planktonic larvae (Mileikovsky,
1971; Harrison et
al., 1984). As an extreme example Scheltema (1971) describes the teleplanic
larvae of some gastropods as capable of trans-oceanic dispersal. In studies
of the population
dynamics of sessile marine organisms, the scale of
dispersal of the larvae may be well beyond the arbitrary scale selected for
observation, and planktonic processes may uncouple recruitment from local
reproduction (Loosanoff, 1966). A model describing the dynamics of such
an open marine population was developed by Roughgarden et al. (1985)
for a population with space-limited recruitment. They obtained three main
results: (1) the population establishes a locally stable equilibrium if the percentage of free space at equilibrium exceeds 50 %, (2) sufficiently high levels
of settlement can destabilize this equilibrium,
and (3) the equilibrium
is
globally stable if the expected area occupied by a cohort declines with age.
Roughgarden et al. (1985) classified organisms by age. However, for
many if not most sessile marine invertebrates, size rather than age is the
relevant factor determining the demographic fate of an individual, and
their highly plastic growth rate produces a lack of correlation between
129
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Copyright
0 1991 by Academic Press, Inc.
All rights of reproduction
in any form reserved.
130
PASCUALANDCASWELL
size and age (Hughes, 1984; Caswell, 1985, 1988; Jackson, 1985). Since
size-classification is known to affect the results of demographic analyses
(Caswell, 1989), in this paper we extend Roughgarden’s model to populations classified by size. The equilibrium of the model and conditions for its
stability are related to settlement, mortality, and growth. In particular,
settlement and growth plasticity are shown to play a central role in the
dynamics of the population. The mathematical analysis of this model (see
also the appendix) closely follows that found in Roughgarden et al. (1985).
The analytical results are then illustrated by a numerical example and
possible extensions of the model are considered.
DERIVATION
OF THE MATHEMATICAL
MODEL
The following model describes the dynamics in discrete time of an open
marine population with space-dependent recruitment. Organisms are sessile
and larval dispersal decouples recruitment from local reproduction. Settlement is proportional to the amount of free space I;, at time t and the total
space available has constant area A. Define the settling parameter s as the
number of larvae per unit of free space settling during one time period and
surviving to the next census. Let n,, f be the number of organisms in size
class x at time t and let a, be the average area of an organism in that class.
An organism in size class x at time t may either survive and remain in this
class with probability
R,, or survive and grow to the next class with
probability P,. The size classes and the projection interval are chosen so
that sa, < 1 and so that individuals cannot grow more than one size class
between t and t + 1. The condition sa, < 1 is necessary to apply the model
to an initially empty substrate: if sa, > 1 then, after one time period, the
occupied space exceeds the total area A. Given the above assumptions the
dynamics of the local population are described by the equations
n O,r+l=~Ft+Rono,,
(la)
x = 0, ...) w - 1.
n,+,,,+,=P,n,,+R,+,n,+,,,
Furthermore,
(lb)
since total space is preserved,
Ft=A--
i u,n,,,.
x=0
(lc)
By substitution of Eq. (lc) in Eq. (la), the previous equations can be
written as the matrix model
n ,+,=Bn,+sAe,,
(2)
RECRUITMENT-SUBSIDIZED
131
POPULATION
where
-ssa, + R,
PO
B=
i:
-sa,
R,
---a2
0
..’
...
--saw
0
..
-..
R,
..
P,-,
...
0
R,
0 i
e,=
1
0
0 .
1.’
0
0
.I.
P,
and
no,f
n, =
nl,,
n2, ,
Hn w,1
I
6
The size-specific survival probability,
denoted by S,, is given by
of an organism in class x surviving to class x + 1,
independent of the time spent in class x, is given by
P, + R,. The probability
lim (P, + R,P, + RzPx + .. . + Rip,)
f--t co
= P, lim (1 + R, + Rz + . . . + R:)
,+cO
=P,
1
l-R,;
that is, the product of the probability of growing to class x + 1 within one
time step and the mean residence time in class x. As S, increases from zero
to one, P,.( 1 -Rx) also increases from zero to one. Note that if P,
increases, a larger proportion of the organisms in class x grows to the next
class in one time step, and if R, increases, more organisms are available for
growth in subsequent time steps. Now, by keeping S, constant, note that
as R, increases, both P, and P,/(l -Rx) decrease. Thus, the probability
P,/( 1 -Rx) reflects not only the survival rate but also the pattern of
growth given by the expected allocation of survivors to either growing or
remaining in a class.
The age-classified model follows as a special case of model (2) when all
R,=O.
ANALYSIS OF THE MODEL
The equilibrium
The linear model given by Eq. (2) is nonhomogeneous since it involves
the constant forcing term sAeI. The general solution of such a model is
132
PASCUAL
AND
CASWELL
n,=n,+n,,
(3)
where np is independent of time and corresponds to a particular solution
of (2), and nh depends on time and denotes the general solution of the
corresponding homogeneous model n, + , = Bn, (LaSalle 1986). A particular
solution can be obtained by determining the equilibrium for Eq. (2), which
always exists. Let ri, be the equilibrium number of organisms in class x,
and E the equilibrium free space. Substitute fi, for n,, , and n, f + , , and p
for F, in Eqs. ( 1a) and (lb). Then
for
x= 1, .... w.
Three components determine the number of organisms in a particular class
at equilibrium:
recruitment (given by sp), the mean residence time of an
organism in class x (given by l/( 1 -Rx)), and the probability of surviving
from class 0 to class x (given by n;&r Pi/( 1 - Ri)). This product, corresponding to the survivorship function I, for the age-classified population,
plays an important role in the following stability results. To simplify the
notation, let
x ~ I
go=1
and
g,+
pi
i=o 1-R,
for
From Eqs. (lc) and (4), the free space of equilibrium
p=-
A
1 + sA,’
x = 1, .... W.
is given by
(5)
where
A,=i a,g,
x=01-K’
(6)
In this expression, each area a, is multiplied by the mean residence time in
the corresponding class and by the probability of surviving from recruitment to that class. The sum A,, corresponding to A, = C, axlx for the age
classified model (Roughgarden et al., 1985), gives the area controlled by a
larva that has settled into the system, cumulated throughout its life. The
two factors s and A,, determining the value of I’, will also be shown to play
an important role in the following stability results.
RECRUITMENT-SUBSIDIZED
POPULATION
133
Stability
We now consider the roles played by recruitment, mortality, and growth
in determining the stability of fi. An equilibrium
of a nonhomogeneous
model is locally stable if and only if every homogeneous solution n,, tends
to zero as time increases; in other words, if all the eigenvalues of B lie
inside the unit circle in the complex plane (Luenberger 1979). For the
age-classified model, Roughgarden et al. (1985) showed that local stability
holds if the percentage of free space at equilibrium
exceeds 50%. In
Appendix A we show that the 50% free space rule remains valid for the
dynamics of a size-classified population described by model (2).
THE 50 % FREE SPACE RULE. If the percentagefree spacep/A % exceeds
50%, then the equilibrium is locally stable.
From Eq. (5),
Thus, the equilibrium is locally stable when sA, < 1. Stability is promoted
by factors decreasing sA,, including low settlement rates, high mortality
rates (1 - S,), and low growth rates resulting from small areas or from a
large proportion of survivors remaining in a particular class.
The effects of high settlement rates on stability are not addressed by the
50 % free space rule, since it provides only sullicient conditions for stability.
For a special case of the age-classified model, Roughgarden et al. (1985)
showed that increases in s destabilize the equilibrium.
In Appendix A we
prove that sufficiently large values of s yield a locally unstable equilibrium,
without assuming specific forms for growth or survival. Assume for the
moment
that all RX> 0, and let p0 = 1 and pX = flT:i P,. Let
K=CT=,
((-l)“+’
a, PJIJ;=~ Ri). Then the equilibrium
fi is locally
unstable if
for
K>O
for
K-CO.
or
A similar result holds if some or all of the R, = 0 (Appendix A); this
includes the age-classified model in which R, = 0 for all x.
134
PASCUAL
AND
CASWELL
A large number of simulations suggests that the critical value of settlement s, determining the onset of an unstable equilibrium is unique, since
the absolute magnitude of the dominant eigenvalue of B becomes a
monotonic increasing function of s when s > s,.
When the equilibrium is locally unstable, departure from it occurs by
cycles. Assuming the existence of a basis of eigenvectors for matrix B, the
general solution of the model given by Eq. (3) can be written as
n,=il+ c ci12;vi,
i=O
where li is an eigenvalue of B, vi its corresponding right eigenvector, and
ci a constant determined by initial conditions (Luenberger 1979). The
following result, proved in Appendix A, explains the oscillations in the
departure from 8.
If the equilibrium is locally unstable, then an eigenvalue with absolute
magnitude larger than 1 is either negative or complex.
When raised to a power, as in Eq. (8), these eigenvalues produce oscillations of increasing magnitude. Eventually, as the number of organisms in
the different classes increases, the population proceeds to full cover of the
substrate. Thus, s > s, implies that, given ‘the levels of growth and mortality, settlement is too high for the population to reach an equilibrium
where free space is present and the population has the potential to completely cover the substrate. In this case, the dynamics will eventually be
determined by density-dependent effects on mortality and growth, which
are not included in (2). Thus, model (2) cannot describe the dynamics of
the population far from an unstable equilibrium:
after full cover of the
substrate solutions become negative through Eq. (1~).
Even local stability guarantees a positive trajectory to equilibrium only
for initial conditions sufficiently close to this equilibrium. Roughgarden et
al. (1985) prove a result (the decreasing net area function rule) which
guarantees global stability of the equilibrium and guarantees that trajectories originating from positive initial conditions remain positive for all
time. Let 1, denote survivorship to age x. Then the expected area occupied
by a cohort at age x is a,l,; the decreasing net area function rule states
that the equilibrium is globally stable and trajectories remain positive if
and only if a,l, decreases with x.
This result defines conditions under which space-limited recruitment
alone can regulate the population. If it is violated, some other form of
density dependence is required to prevent the occurrence of negative
populations.
The corresponding result for the size-classified model (2) is
RECRUITMENT-SUBSIDIZED
POPULATION
135
THE DECREASING NET AREA FUNCTION
RULE.
The equilibrium is
globally stable if and only if the product a, g, is a decreasingfunction of x.
This result is derived in Appendix B as a special case of an arbitrary
stage-classified model. Here, a,g;, a function of size, plays the role of the
net area function a,l,, a function of age. ‘In both cases, the net area
function is the product of the area corresponding to a particular stage and
the probability of surviving from recruitment to that stage. This function
gives the average area occupied by an organism at stage x, taken over all
organisms, dead or alive, of the same cohort. As for the age-classified
population, when this average area is a decreasing function of x, mortality
overshadows growth. Consequently, for any value of settlement satisfying
sa,, < 1, the population
converges to an equilibrium
where free space is
present. When a, g, decreases with size, a, + I g, + ,/a, g, < 1 and therefore
Px
ax+,
a, (l-R,)”
for
x=0, .. .. w- 1.
Factors decreasing the left-hand side of the above inequality promote
stability. These factors include an increase in mortality rate and a decrease
in growth rate.
A NUMERICAL
EXAMPLE
The following simulations are based on the hypothetical matrix shown in
Table I, obtained by simplifying the matrix of transition probabilities given
by Hughes (1984) for the reef coral Agaricia agaricites for the year
1978/1979. Three sizes classes are distinguished (O-10 cm*, l&50 cm*, and
5CL200 cm*); the fourth class of the original matrix is ignored since in the
year 1978/1979 colonies did not grow beyond the third size class. The life
cycle described by Hughes (1984) is substantially more complex than the
one described by matrix B in model (2): organisms may not only grow or
TABLE
I
Matrix of Transition Probabilities B and Averages Areas a,(cm2)
of the Colonies in the Three Size Classes x
B=
Note. Size ranges and mortality for the different classes are borrowed from data on the
coral Agaricia agaricifes (Hughes, 1984)
136
PASCUAL
AND
CASWELL
1
Size class index (x)
FIG. 1.
probabilities
size.
The net area function.
The equilibrium
given in Table I is not globally stable
of model (2) for the matrix of transition
since the net area function
increases with
stay in the same class but may also shrink by colony fragmentation or
partial mortality. We ignore these additional transitions between classes and
consider that all the survivors not growing larger remain in their particular
class. Hence, the simulations do not represent in detail the dynamics of
Agariciu ugaricites, but rather serve to illustrate the roles of settlement and
growth plasticity in the behavior of the model. The average areas a, used
in the simulations of Figs. 1 and 2 are specified in Table I; the total area
of available substrate is 12 m2. Given the values of a,, P,, and R,, the net
area function increases with size (Fig. 1) and therefore the equilibrium is
8
“0
10
TIME
(;EARS)
30
I
40
0
IO
$EARS)
30
J ‘J
TIME
FIG. 2. Locally
stable equilibrium.
The value of the settlement parameter
s = .Ol is below
the critical value s, x .06 larvae/year/cm*.
After a small perturbation,
the population
returns to
equilibrium.
n, = number of colonies in size class x; equilibrium:
[no = 892, nr = 555, n2 = 4441;
percentage
free space at equilibrium:
36 %; initial condition:
[no = 1200, n, = 800, n, = 6001.
RECRUITMENT-SUBSIDIZED
137
POPULATION
h
h
-
1
‘/
cT
0
lo
TIME $EARS)
3b
4’0
FIG. 3. Locally unstable equilibrium. The value of the settlement parameter s = .07 being
above the critical value s, z .06 larvae/year/cm’, the population oscillates away from equilibrium. Equilibrium: [no = 1293, FI, = 804, n2 = 6431; percentage free space at equilibrium:
7 %; initial condition: [no = 1200, n1 = 800, rz2= 6001.
globally unstable and there exists a critical value of s producing the onset
of a locally unstable equilibrium. This value, determined by varying s and
obtaining the absolute magnitude of the eigenvalues of B, is s, x .06 larvae/
year/cm2. An initial condition extremely close to equilibrium being chosen,
the system returns to it for s < s, (Fig. 2). However, for the same initial
condition, the system oscillates away from equilibrium when s > S, (Fig. 3).
Eventually, as the number of organisms in the different classes grows,
the free space F, approaches 0 and becomes negative. Thus, when the
equilibrium is locally unstable, the model can only represent the dynamics
of the population until the total area is completely covered.
0.3
Probability
of remaining
0.6
in the second
0 7
stage
FIG 4. The critical value of settlement determining the onset of an unstable equilibrium
increases as a larger proportion of survivors in class 1 remains in that class.
138
PASCUAL
AND
CASWELL
We have shown in the previous section that an increase in the proportion of survivors remaining in a class promotes stability. In more concrete
terms, as the probabilities R., increase, more larvae need to settle into the
system to produce the onset of an unstable equilibrium. To illustrate this
point with the present example (Table I), let the probability of remaining
in the second stage vary from 0 to 0.8 and, to keep mortality constant, let
the probability of growing to the third stage decrease accordingly. Figure 4
shows the resulting increase in the critical value of settlement s,.
DISCUSSION
This work extends the model presented by Roughgarden et al. (1985) to
size-classified populations of sessile organisms with pelagic larvae. In both
models, recruitment is represented as an external contribution to the local
dynamics. Based on work by Connell (1985), Roughgarden et al. (1985),
and Underwood and Denley (1984), Lewin (1986) emphasizes the influence
of settlement rates on the roles played by local processes such as growth,
predation, competition,
and disturbance in shaping the dynamics of a
benthic marine population or community. This paper illustrates the importance of settlement level in the dynamics of a size-classified population.
Specifically, the stability of an equilibrium for which free space is present
in the system depends on the settlement level. In general, processes freeing
space in the system, such as low settlement and high mortality, promote
stability, and processes occupying space, such as high growth and high
settlement, have a stabilizing effect. Growth plasticity plays a central role:
the more likely individuals are to remain in their size class, the more likely
stability becomes. (Bence and Nisbet (1989) find that extremely fast growth
can stabilize the equilibrium. However, this result seems to depend on the
structure of their model allowing the time lag between settlement and
recruitment to the larger class to become arbitrarily small.)
The importance of the stability results is twofold. First, they provide a
scale to which observed levels of settlement, mortality, and growth can be
compared, in terms of their effects on population dynamics. In particular,
the decreasing net area function rule compares growth to mortality.
Regarding settlement, the point of reference is given by the critical value s,
for which the equilibrium
becomes unstable. Second, the stability results
indicate the potential of the population to occupy space, the limiting
resource. A globally stable equilibrium
corresponds to the permanent
presence of free space in the system. Low variability of settlement and
infrequent disturbance would result in a fairly constant population size and
size structure. Hughes and Jackson (1985) followed the dynamics of
several species of foliacious corals through a 3-year period and reported
RECRUITMENT-SUBSIDIZED
POPULATION
139
remarkably stable size frequencies and number of colonies in spite of the
active underlying changes at the individual colony level. On the other
hand, a locally unstable equilibrium implies a highly variable size structure,
in which free space oscillates until full cover’ of the substrate is reached.
This potential of the population to reach crowded conditions demonstrates
the importance of density-dependent
effects on mortality and growth.
Further theoretical work is required to incorporate such effects in the
model. Roughgarden et al. (1985) assume in their simulations of barnacle
dynamics that mortality increases as free space approaches zero. Consequently, under local stability conditions the population converges to equilibrium even for initial conditions far from this equilibrium, and when the
equilibrium
is locally unstable the population does not proceed to full
cover of the substrate but converges to a limit cycle. Their assumption of
density-dependent mortality derives from an increase both in predation by
starfish and in susceptibility to removal by waves in areas of high barnacle
cover. Yoshioka (1982) describes another example of higher mortality
under crowded conditions for the bryozoan Membranipora membranacea.
The most abundant fish predator of this bryozoan feeds exclusively on
Membranipora
during periods of high density, choosing alternative preys
otherwise. However, under high cover conditions, the first effects of density
are likely to be on growth. Harper and Bell (1979) describe the potential
effect of neighbors on the growth of modular sessile organisms. Densitydependent growth would be easier to incorporate in a size-classified model
than in an age-classified model.
Another extension of this work relates to the highly complex life cycles
of colonial marine invertebrates. Corals provide an interesting example of
such complexity: an individual colony may suffer partial mortality, produce
by fragmentation several smaller colonies, and even fuse to another colony
of the same genet (Jackson and Hughes, 1980). Our global stability results
apply to such life cycles, in which transitions between all classes are
possible.
Finally, the relevance of models in which recruitment is completely
decoupled from local reproduction depends on the scale of dispersal of the
larvae, the mobility of the organisms under study, and the scale selected for
observation. Thus, models for recruitment-subsidized
populations are not
limited to sessile marine invertebrates, but can be applied to the dynamics
of other benthic marine populations. Most reef fishes, for example, are
sedentary animals producing dispersive larvae, the fate of which remains
largely beyond the control of the adult population (Sale, 1977). Studies of
the planktonic component of the life cycle (Gaines and Roughgarden, 1987;
Lobe1 and Robinson, 1986; Robertson et al., 1988; Shanks, 1983) are particularly important to better describe the input term in such models and to
determine the scales at which the dynamics of a population is to be con-
140
PASCUAL
AND
CASWELL
sidered an open process. Beyond such scales, a model recently proposed by
Roughgarden et al. (1988) and studied by Possingham and Roughgarden
(in press) explicitly describes this planktonic component by the inclusion of
physical-oceanographic information related to the dispersal of the larvae.
APPENDIX
To simplify the notation,
A: LOCAL STABILITY
in the following proofs let
i=x-1
and
PO=1
Px=
n
for
pi
x= 1, .... w,
i=O
where w + 1 denotes the total number of classes.
THE CHARACTERISTIC POLYNOMIAL OF B. The characteristic equation of
matrix B is det(B - 11) = 0. To obtain det(B - AI), let F,(A) be a minor of
ordern+l
ofB-A.
Then,for l<n<w,
R,-1
F”(A) = det
. ..
R,-A
. ..
0
...
0
By expansion of the above determinant on the last column,
Fn(A)=(R,-A)F,-l(ll)+(-l)“C1sa,p,
and Fe(A) = R, - sao- 1. Then
det(B-AI)=F,(A)=
fi
(R,-I)+(-l)“f’sa,p,
X=0
W-l
+s C
(-l)X+‘a,p,
x=0
The eigenvaluesof B satisfy
F,(A)=0
o
JJo (Ri-~)=(-l)“‘sawpw
w-l
+s C (-lYa,p,
.X=0
fj
i=x+l
(Ri-A.)
1
.
RECRUITMENT-SUBSIDIZED
141
POPULATION
Assuming that I- Ri # 0 for all i, divide both sidesof the previous equation
by nrzO (Ri - A) to obtain
l=s
(-l)Xaxpx
i
x=o FI?=o (Ri-2)’
(9)
THE 50% FREE SPACE RULE. Zf the percentage of free space at equilibrium exceeds 50%, then the equilibrium is locally stable.
Proof
By Eq. (7), $/A > l/2 is equivalent to sA, -C 1, and
Assume that Ii1 > 1. Inequality
and, since Iill -R,<
(10) then implies that
(2 - R,I,
On the other hand, A satisfies Eq. (9). By the triangle inequality,
implies that
But, inequality
than 1. 1
(12) contradicts inequality
THE DESTABILIZING
Eq. (9)
(11). Hence, [I( must be smaller
EFFECT OF SETTLEMENT.
For sufficiently large values
of s the equilibrium is locally unstable.
ProoJ: The characteristic
polynomial
F,,,(I)=cO+c,l+c,l*+
of matrix B can be written
... +/I’“+‘,
det B (Horn and Johnson, 1985).
where cO=(-l)“‘+l
criterion (Jury, 1964) gives necessary and sufficient
polynomial to have all its roots inside the unit circle.
these conditions is violated and therefore )I1 > 1 for some
The SchurrCohn
conditions for a
If lc,,l > 1 one of
eigenvalue ,? of B.
142
PASCUAL
AND
CASWELL
We next show that JcOJ> 1 if s is larger than a critical
lc,,l = ldet BI, we seek conditions on s such that
value. Since
ldet BI = ipjo R,+s((-l)“i’U,,l,.
H’-
1
(13)
x=0
-1y+’
a,p,+
c (-l)X+‘axPx
x=0
rI
“i]
i=.x+ 1
M
R,
>1-n
(14)
w-1
-,)w+l
a,p,+
< -1-
i
.x
=0
c
(-l)X+laxPx
= 4.I
W’
i=x+
1
R,.
.X=0
Assume that all R, > 0, divide both sides of the above inequalities
n:=, R,, and let
by
Then (14) becomes
I
1
SK>n,"=o R,/
(
SK<
If K > 0, (15) is equivalent to
(15)
or
-1
n,w=, Rx-'.
RECRUITMENT-SUBSIDIZED
143
POPULATION
The second inequality in (16) cannot be satisfied by s since its right-hand
side is negative.
If K< 0, (15) is equivalent to
or
I
S’
(17)
(
-1
ryvEO Rx-l
>
K-l.
The first inequality in (17) cannot be verified by s since its right-hand side
is negative. The first inequality in (16) and the second inequality in (17)
thus provide sufficient conditions for [co1> 1.
When some R, = 0, let 2 = max{ x 1 R, = 0} and define
W-1
g=(-l)w+’
%Pw+
1
* = i
(-lY+‘axPx
;iL
Ri.
Then, (13) is equivalent to
(note that for the age-classified model 1x1 = a,p,).
1
DEPARTURE PROM EQUILIBRIUM.
When the equilibrium is locally unstable,
an eigenvalue A with absolute magnitude larger than 1 is either negative or
complex.
Proof Assume that 2 is real and L > 1. Then, the characteristic equation (9) is not satisfied by 1 because each term in the sum of (9) is
negative. 1
APPENDIX
B: GLOBAL STABILITY
Consider a life cycle where transitions
and let, in model (2),
L . .
b,--saO
B=
between all classes are possible
b!o
b,, -sa,
b .”
b,, - sa,
b~2
b,o
b,,
bi2
where b, denotes the transition
probability
...
:I’
..I
bo,-sa,
b 1W
.
b,,.
,
!
from class j to class i.
144
PASCUAL
AND
CASWELL
Then, the model has a globally stable equilibrium
M’
( z,
if and only if
for ,j=O, .... w.
a, &)-a,<0
(18)
//
.r # ,
The sum in the left-hand side of inequality (18) gives the area that an
organism in class j is expected to occupy by direct transition to other
classes independently
of the number of time steps required for this
transition. Denote this expectation by E(aj). Then, by (18), global stability
holds if the change in area given by E(a,) - a, represents a loss for all
classes j.
Proof.
Let V,, I be the area occupied by the organisms in size class x at
time t. Then V,X,f = a,n, , and the dynamics of the population is described
by the equations
V o,f+l
=aOnO,r+l
w
= a,
sF~ + C bojnj, t
j=O
=aosFt+ao
f
(19a)
b Vj,,l
aj
1=0
and, for x = 1, .... w,
V .X,t+1 =axnx,t+l
=a, Jto $f
(lgb)
vj,t.
J
Also,
F 1+1 =A-
fj
Vx,t+l
X=0
=F,+
i
Vx,, - a,sF, - jto : boj Vi,, - i
,X=0
=F,(l-a,)+
x-1
J
i
>
i=O
Vi,l.
i
j=O
2 bxj 6, *
J
(19c)
From Eqs. (19a), (19b) and (19c), the dynamics of F, and V,, , can be
represented by the matrix model
V r+~=Mv,,
RECRUITMENT-SUBSIDIZED
145
POPULATION
6
I
IIV,,l
where
V 0, I
V
Lf
V 2,
v,=
and
l-a,.9
l-
f a”b,
x=0 a0
1-i
2 boo
UOS
a”bxl
x=0
...
a1
0
5 blo
a0
f
“b””
x=0
w
z bol
-..
3 b,,
49
a,&,
. ..
2 b,,
a0
M=
l-
a1
%
. .
0
5 bwo
a0
a”bw,
. ..
a1
f&s b,,
%
The sum of the elements of M along any column
Furthermore, all the elements of M are non-negative if
is equal to one.
for j= 0, .... w.
(20)
xgo 2 b, < 1
J
Under these conditions, matrix M has a dominant eigenvalue A= 1. Given
any positive initial condition, the vector V, has non-negative components
for all time and converges to a stable distribution given by the right eigenvector of 1.
Since fragmentation
and reproduction
are only represented by offdiagonal elements of B, the diagonal elements bjj correspond to the
probability of remaining in a class and bjj < 1. Thus, inequality (20) can be
written
C
x=0
a,
&<Uj
1 - bjj
for j=O,...,w.
(21)
Conditions (21) are also necessary for global stability: if one of them is not
satisfied, there exist solutions that become negative via matrix M. l
653:39/:-3
PASCUAL
146
CASWELL
AND
THE DECREASING NET AREA FUNCTION RULE. For the size-classified
population only 6, = R, and b, + 1., = Pi are different from 0 in matrix B.
Thus, (21) becomes
aj+I PI
(l-R,)‘aj
a,+1 gj+l
-
Cl
,for j = 0, .... w - 1,
aigf
and a, g, decreasing with size is a necessary and sufficient condition for
global stability.
ACKNOWLEDGMENTS
The authors are grateful to V. Andreasen, C. Castillo-Chavez, A. P. Dobson, and
S. A. Levin for helpful comments on an early version of this manuscript. This research was
supported by NSF Grants BSR-874936 and OCE-89231, and by DOE Grant DE-FGOZ89ER60882. WHOI contribution number 7300.
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