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J. theor. Biol. (1973) 40, 429-439
A Methodology for Evaluation of
Parent-Mutant Competition using a Generalized
Non-linear Ecosystem Model
RAYMOND L. CZAPLEWSKI
Department
of Zoology and Physiology, University
Laramie, Wyoming 82070, U.S.A.
of Wyommg,
(Received 11 September 1972)
A generalized, non-linear population dynamics model of an ecosystem is
used to investigate the direction of selective pressures upon a mutant by
studying the competition between parent and mutant populations. The
model has the advantages of considering selection as operating on the
phenotype, of retaining the interaction of the mutant population with the
ecosystem as a whole, and of setting a reasonable balance between theoretical manageability and quantitative testability.
The model is used to make generalizations about several aspects of
evolution in a terminal consumer (e.g. top carnivore). Mutations which
avoid over exploitation of the prey species or introduce intra-specific
population regulators will be selected against.
Application of the method is limited primarily by the assumptions of a
non-fluctuating environment, an asymptotically stable steady state before
the mutation, absence of genetic drift, and uniformity of selective pressure
on the mutant population.
The method described in this paper is weil suitedfor handling the high
degreeof complexity experiencedin most ecosystems.Thus the methodology presentmay becomea powerful tool in the approach to certain
evolutionary questions.
1. rntroduction
It is not very difficult to conceive of an abstract model for an ecosystem which
may have a high potential for realism (at the expense of its specificity) and
the ability to generate generalities (because of its low specificity). The problem
with such a model is the difficulty in analyzing it. I use this sort of model
as a medium to investigate competition between a mutant and its parental
stock. A simplification
is presented which allows an otherwise almost
impossible mathematical analysis.
The model is a generalized population dynamics model which may
incorporate an arbitrarily large number of populations and interactions.
429
430
R.
L.
CZAPLEWSKI
Inputs and losses from outside the model system can be included (allowing
the model to be either open or closed). Population growth and loss functions
are not very explicit; this permits a wide variety of such processes to be
assumed. Thus the model can describe probability
functions (e.g. the
Lotka-Volterra
equations) or saturation kinetics equations (e.g. Gallopin,
1971; Verhoff & Smith, 1971).
The outcome of the parent-mutant
competition is studied by adding a
mutant to the model and determining the stability (using eigenvalues) of
this new system. The stability would be impossible to analyze if it were not
for a simplification arising from the assumptions that the system is near an
equilibrium
and the system was asymptotically stable before introduction
of the mutation.
A further restriction on the parental growth and/or loss functions is also
necessary. The restriction made in this paper (as an example) is that the loss
rate of the parental stock is directly proportional
to its population level
(a reasonable assumption for a terminal consumer such as a top carnivore).
This restriction is only one of a host of possible restrictions on the parental
population.
2. Model
The model considered is a generalized population dynamics model. The
growth rate pi of a consumer population Ni per unit consumer is assumed
to be a function bji of the prey population level Nj,
(1)
The intrinsic loss rate wi due to respiration, egestion, and mortality is
assumed to be a function mi of the population level Ni,
u)i = iNi(
(2)
The extrinsic loss rate si due to predation is assumed to be equal to the
predator level (in this case iVj) times a function bij of the prey level (in this
case N,),
&i =
i
j=l
Njbij(Ni).
The net population change per unit time, dN,/dt, equals the increase rate
equation (1) minus the loss rate equations (2) and (3). Thus, changes in a
system with k populations may be described by k differential equations of
the following form:
dNi/dt= Nijil [bji(Nj)]- mi(NJ- j=l2 [Njbij(NJ].
(4)
PARENT-MUTANT
COMPETITION
IN
AN
ECOSYSTEM
431
Utilization of light may be incorporated into the model by allowing one
population to represent ambient light with all primary producers “preying”
upon this light “population”.
Utilization of space and inorganic nutrients
may be considered in a similar fashion.
No restrictions are placed upon the population change functions bji
and mi. For instance, bji of Nj may equal zero for all values of Nj if the
population i does not prey upon populationj;
bj, of Nj may take the LotkaVoltera form
bji(Nj)
= Cji Nj,
(5)
where Cji is a constant; bji of Nj may be of the Michaelis-Menton
form
bj&Nj) = Uji Nj/(Cji+
Nj),
(6)
where Uji and cji are constants; or bji of Nj may be any growth function.
Likewise, m, of Ni may represent any intrinsic loss function.
3. Steady State
A steady state is the condition in which all net population changes,
dN/dt, are equal to zero. Such a steady state may be found by setting
equation (4) equal to zero. Thus,
(7)
represents k algebraic equations defining the steady state. mi and Rj are
the population levels at steady state. All population levels are assumed
to be greater than or equal to zero.
4. Steady-state
Stability
The stability of a steady state may be determined by introducing an
inCnitesimally small perturbation into the system which is at a steady state.
If the perturbation grows larger with time, the steady state is unstable. If the
perturbation becomes smaller with time, the steady state is asymptotically
stable. This approach has appeared in the recent biological literature with
use by Koga & Humphrey (1967), Canale (1970), Othmer & Striven (1971),
Verhoff & Smith (1971) and Gallopin (1971).
The perturbation is introduced into the system by setting each population
level Ni equal to the steady state level mi plus a perturbation from that
steady state ni,
Ni = mi+ni.
(8)
432
R.
L.
CZAPLEWSKI
Substitution of equation (8) and Taylor series expansions of the functions
bj, and mi into equation (4) yields a system of k differential equations:
dni/dt = (mi + ni) ~ [bji(Rj)
j=l
+ &5i(~j)nj + b~i(Wj)nj”/2! + . . .]
- [mi(RJ + m~(rJ&Zi f m;(Ri)$/2
-
jil
{(Nj+nj>[bij(RJ+
! + . . .]
bG(WJnF/2! f . . .I>.
bij(Wi)ni+
(9)
The primes above the functions indicate the derivative of the function.
Because the perturbations are infinitesimally
small near the steady state,
the product of two or more such perturbations is considered approximately
zero. By using this zero approximation
and subtracting equation (7) from
equation (9) a linear approximation
of the non-linear system (9) results:
+
jil
l.njCNib~i(Nj)
-
bij<mi>]>*
(lo)
j+i
It is important to note that this linear approximation
is valid only near a
steady state.
The linear system (10) may be written in matrix form
dn,/d t N A, Xkuk
(11)
or
(dn,/dt
al3
. . . alj
. . . alk \ f
al2
dn,/dt
a23
. . . a2j
. . . a2k
a22
dn;/dt
\dn;/dt
.
.
.
.
.
.
aj2
Uj3
.
.
.
. . . a,‘]’
. . . ajk
.
.
.
ak2
ak3
. . .
. . .
akj
akk
1 \
where n, is a vector of length k which contains the perturbations, and A, Xk
is a matrix of the order k by k in which the element Uij is the coefficient of
nj in the ith differential equation of the system described by equation (10).
The elements of A are
- bij(mi),
for j # i,
(13)
aij = RibSi
aii = j$l Ebjdmj>l - mXmi> - j$l [mjb;j<Ni>lj#i
j#i
(14)
PARENT-MUTANT
COMPETITION
IN
AN
ECOSYSTEM
433
Integrating the linear approximation
equation (11) yields a solution of
the following general form &(t) is a polynomial which does not alter
qualitative behavior as t approaches infinity)
nk
where i is a set of constants
4 is a vector of length k, e is
time. Note the similarity of
function. The equation for the
(15)
of order k which contains the eigenvalues,
the base of the natural logarithms, and t is
equation (15) with the exponential growth
eigenvalues is
0 =
=
ck enkt
[pk<t>l,
(16)
IAkxk-AIkxkl,
where Akxk is the coefficient matrix which appears in equation (1 l), and
I kxk is the k by k identity matrix. The vertical bars in equation (16) indicate
the determinant of the enclosed matrix.
If the real parts of all eigenvalues (A) are less than zero, the perturbations
ni “decay away” with time and the steady state is asymptotically stable.
If one or more of the real parts of the eigenvalues are positive, the perturbations “blow up” with time, and the steady state is unstable. Thus, an
essential property of asymptotically stable steady states is that the real
parts of all eigenvalues are less than zero.
5. Mutation
A mutation is introduced into a specific population p by forming a mutated
population m which is identical to the population p except that either the
increase rate pP is changed by a dimensionless function fjP of N,,,, the intrinsic
loss rate CD~is changed by a dimensionless function g, of N,, or the extrinsic
loss rate sP is changed by a dimensionless function h, of N,. The mutant
population m is part of a k plus one by k plus one system,
k+l
k+l
dN,/dt= Nijzl CbjdNj)I- mi(NJ- jzl [Nj bij(NdIy
k+l
k+l
- jJIl CNjbp.J~m)~pjCNrrz)I*
(18)
If the mutation functions fjp, gP, h,j of N,,, are less than one and greater
than zero, the associated gross rates of population change pP, wP, sP are
decreased; if fjp, g,, h, of N,,, equal one, pP, wP, Ed are unchanged; and if
fjP, gP, hpj of N,,, are greater than one, pP, wP, .sPare increased.
434
R.
L.
CZAPLEWSKI
A linear approximation of the non-linear differential
describes population changes in population m,
equation (18) which
k+l
is determined near a steady state in the same manner as equation (4) is
approximated by equation (10).
Of particular interest is the case in which the steady-state level of the
mutated population is zero because it is near this steady state that the mutant
will first experience selective pressure. If iVm equals zero, and the loss rate
functions of the mutated population are zero when N,,, is zero, equations
(20) and (21) will simplify to
U,j=O,
forj#m,
a mm=
jzl Cbjp(NjI&jp(o)l
(22)
kfl
kfl
- ~~CMp(O> - jzl Cflj bLj(o>hpj(o>l-
(23)
The step from equation (20) to equation (22) is extremely important
because it is this simplification which makes the application of an eigenvalue
analysis possible in complex systems.
6. Selection
If the steady state in which the mutated population level R,,, equals zero
is unstable, the perturbations initially caused by the mutation wiIl increase
with time, and the system will move away from this steady state. If this
steady state is asymptotically stable the perturbations due to the mutation
will decrease with time and the mutated population will tend towards
extinction. Thus, selection will initially favor the mutant if the real part
of the eigenvalue of the steady state (in which iii, equals zero) associated
with the mutant population is positive, and selection will work against the
mutant if the real parts of all eigenvalues of this steady state are negative.
Because all coefficients of n, which are off the diagonal in the A matrix
are zero, equation (22), when R,,, is zero, the eigenvalue determinant (24)
PARENT-MUTANT
COMPETITION
IN
AN
ECOSYSTEM
435
of the system with a mutation equals a,,,, - I times the eigenvalue determinant
of the system before mutation, equation (16),
0 = IA~k+l)x(k+l)-W~k+l)x(k+1)
I= hmd)IAkxk-;1Ikxk~.
(24)
Thus, one eigenvalue of the system with a mutation equals a,,.
Assuming that the system was stable before mutation, the real parts of
all eigenvalues described by equation (16) are negative. This forces the real
parts of all eigenvalues described by equation (24) to be negative, except the
real part equal to umm, which may be either positive or negative. Thus, the
initial direction of selective force on a mutant may be determined by analyzing
the sign of equation (23).
7. Selection in a Terminal Consumer
Although the behavior of a mutation in a potentially complicated system
may be studied by consideration of a relatively simple equation (23), this
function is still too complicated to analyze unless restrictions are made on
the growth or loss functions pP, op, E*. One convenient restriction is to
assume that the loss rate from the parent populationp is directly proportional
to the population level NP. The loss functions for this restriction appear as
q = zpNp,
(25)
= 0,
(26)
where zP is a constant. This simplification seems reasonable for a terminal
consumer, which has no loss rate due to predation by other populations.
The differential equation for population
changes in this terminal
consumer is
E,j
k+l
dNp/dt= Npj=lC Cbjp<Nj>I-ZpNp
and the steady state condition for such a population
(27)
is
.
t2f9
A mutation occurring in a terminal consumer results in a mutated population
whose rate of change is described by
k+l
The linear approximation
of equation (29) is simply n, times a,,,
where
k+l
a mm= jzl
Cbjptmjlf&Co)l
- zpSp(“)
(30)
436
R.
L.
CZAPLEWSKI
and fl,,, is near zero. The notation and mathematical steps in equations (27)
to (30) exactly parallel those in the previous sections.
The sign of equation (30) can be determined by comparison with equation (28). Iff&(O) is greater than g,(O), ammwill be positive. Iffj,(O) is less
than g,(o)~ a,, will be negative.
The qualitative behavior of the system when the mutated terminal consumer level is not near zero may be predicted by analyzing the remaining
steady states which involve different levels of the mutant and parent
populations.
If only one, positive, non-zero steady state level of population p exists,
and the equilibrium points associated with positive parent levels are asymptotically stable, the steady state with a zero parent population level is unstable.
At least one of the eigenvalues of this system has a positive real part. If a
mutant population at a zero level is added, the eigenvalues will be the same
as those of the system before mutation with the addition of one eigenvalue
equal to a,,. Regardless of the sign of a,,,,,,, this steady state will have at
least one eigenvalue with a positive part. Thus, the steady state in which
both the mutant and parent population levels equal zero is unstable, and
the system will move away from this steady state.
The eigenvalues of the mutant system at steady state in the case of a
zero parent population level with a positive mutant level equals the eigenvalues of the k by k system (with the mutant but not the parent population)
plus one eigenvalue equal to app,
If app is positive or the k by k system has at least one eigenvalue with a
positive real part, the steady state of the system with both parent and mutant
populations is unstable. If aPP is negative and the k by k system is asymptotically stable, the k+ 1 steady state is asymptotically stable.
A steady state in which both the parent and mutant population levels are
greater than zero leads to a contradiction, which can be seen by comparing
the steady state conditions of the parent population, equation (29, with
the steady state condition of the mutant population,
k+l
0 = jC1[bj,(~jlf,,(w,)l-z,S,(wd’
(32)
The trivial exception to this impossibility is if the mutation functions have
no effect (fjP = 1, gP = 1) or the beneficial effect exactly offsets the detrimental effect of the mutation. The reason such a steady state cannot exist
is much akin to the reasoning behind the principle of competitive exclusion.
Three general types of population trends are possible in the k+ 1 system
PARENT-MUTANT
COMPETITION
IN
AN
ECOSYSTEM
437
which includes the mutant population. If the first steady state (Rm = 0,
EP > 0) is asymptotically stable and the second steady state (flm > 0,
N, = 0) is unstable, the system will eventually tend towards the first steady
state which is the only stable steady state. This indicates selection against
the mutant. If the first steady state is unstable and the second is asymptotically
stable, the system will tend towards the second steady state. If both steady
states are unstable, the system will oscillate and neither population will
become extinct.
8. Density Independent Mutation
The simplest type of mutation to consider in a terminal consumer is one
which changes the growth or loss rates by a constant value, independent
of the population level N,,,. In this case, the mutation functions would be
described by
fjp(Nm) = qjp
s&%J
= rpy
(33)
(34)
where ~j~ and rp are constants greater than or equal to zero.
Mutations decreasing the population growth rate (4j~ < 1) or increasing
the loss rate (rp > 1) will be selected against (a,, < 0). Mutations increasing
growth rate (4j~ > 1) or decreasing loss rate (r, < 1) will be selected for
(a,, > 0). This suggests that a possible selective force which favors inefficiency in a top carnivore does not function in a steady-state regime.
9. Density Dependent Mutation
A more complicated mutation is one which leads to density dependent
manipulation of growth or loss functions. Such a mutation function in which
fjP(0) is less than one or gP(0) is greater than one will be selected against
(a,, < 0). A mutation function in which&(O) is greater than zero or gJ0)
is less than zero will be selected for (a,, > 0).
However, a density dependent function may well have no effect when the
population level is very near zero (fj,(O) = gP(0) = 1). This would cause
a to equal zero. In order to determine the direction of the selective force
oy such a mutation, it is necessary to analyze the stability of the steady
state in which the parent population level is zero and the mutated population
level is greater than zero. If this steady state is asymptotically stable, the
mutant will be selected for, and if it is unstable, the mutant will be selected
against.
Assuming that the k by k system which includes the mutant population m
438
R.
L.
CZAPLEWSKI
but excludes the parent population p is stable (the real parts of all eigenvalues
less than zero), the stability of the system which includes both parent and
mutant populations may be determined by finding the sign of the coefficient
app, equation (31). One eigenvalue equals app because all off diagonal coefficients in the pth row of A~k+l~x~k+l~ matrix equal zero. The real parts of
other eigenvalues are negative.
The sign of equation (31) may be determined by comparison with the
steady-state condition of equation (32). If the mutation function represents
some form of negative feedback wherefjP(N,J is less than one or g,(n,J is
greater than one, the mutant will be selected against (app > 0). Thus, evolution of intra-specific negative feedback mechanisms does not appear to be
a steady-state phenomenon in top carnivores.
10. Discussion
The model and analysis procedure may be considered as a new tool in
studying evolution. Rather than considering the influence of selection on
allele frequencies, the model considers the influence of selection on the
population levels of the parent and mutant populations. Fitness is considered
in terms of net productivity rather than probability
of survival (see
McNaughton,
1970).
One of the principal strengths of this approach is that it treats selection
as operating on the phenotype rather than on the genotype. This is an advantage because it is the phenotype that actually determines if an organism is
to survive. The genotype is only vaguely considered in that it is assumed
the phenotype has the potential to be preserved by the recording ability
of the genotype.
Another strength of this approach is that the parent and mutant populations experience selection in the context of an ecosystem rather than in
isolation. Thus, the influence of the entire ecosystem upon the fitness of the
parent and mutant populations can be evaluated. This may be an especially
important consideration in resource limited populations.
Because of the relative ease with which the method presented can handle
complexity in the ecosystem, a high degree of realism is attainable. For
instance, the problems associated with the common assumption in population
dynamics models of homogeneous populations may not be very serious in
this model. The heterogeneity often experienced in populations can be
handled by dividing a population into homogeneous sub-populations. Also
the influence of environmental grain on selection can be considered either
by subdividing a population into several populations which experience
different patches (for a coarse grained species), or by allowing a population
PARENT-MUTANT
COMPETITION
IN
AN
ECOSYSTEM
439
to experience different patches of environment at various frequencies (for
a fine grained species). In this case, it may be reasonable to allow a single
mutation to occur in several related populations. Finally, the model is not
specific about the type of growth and loss functions, and it can incorporate
a wide variety of such processes.
Besides the fact that the method presented can be handled analytically
with relative ease, the model can also be tested quantitatively
because
growth and loss processes are measurable in an ecosystem. In fact, it may
be easier in some cases to measure these processes rather than estimate
the fitnesses of various genotypes, especially when differences in genotype
are difficult to detect.
Although the analysis of aspects of evolution in terminal consumers is
included as an example of the methodology, several additional conclusions
can be drawn from this exercise. The hypothesis that selection may tend
to avoid over exploitation of the food species does not find support in the
mathematical study of steady-state selection in terminal consumers. Mutations which increase exploitation rates are selected for; if a mutation increases
the net growth rate of a top consumer to the point of causing an unstable
predator-prey relationship, the parent and mutant populations will merely
oscillate without exclusion of either population. The analysis also predicts
that intra-specific negative feedback mechanisms (e.g. cannibalism, territoriality, density stress) are selected against in the terminal consumer. This
suggests that evolution of such mechanisms may have occurred in a nonterminal consumer or in a non-steady state regime.
I wish to express my gratitude for critical readings of this manuscript to
Professors L. Weinberg (who reviewed the mathematics), M. Parker, and G. T.
Baxter. N. Stanton, A. Be&s, and G. Pike were kind enough to offer a number of
helpful suggestions. The manuscript is based on a M.S. thesis submitted to the
Graduate School of the University of Wyoming by the author.
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A. E. (1967). Biotechnol. Bioengng 4, 375.
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S. J. (1970). Am. Nat. 104, 337.
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H. G. & SCFUVEN, L. E. (1971). .J. theor. Biol. 32, 507.
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