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Fitness and survival in logistic models
J. theor. Biol. (1978) 74, 23-32
Fitness and Survival in Logistic Models MATTHEW
WIT-TEN?
Department of Biometry, Medical University of South Carolina, 171 Ashley Avenue, Charleston, South Carolina 29403, U.S.A. (Received 5 July 1977, and in revisedform 21 February 1978) The conceptsof fitnessand survival in logistic modelsare shown to be independentif we follow certain intuitive definitions for theseconcepts. This conclusionfollows from a simpletopological analysisof the function fb (x) = bx (l-x), which is just the standardform of the logistic growth equations.
1. Logistic Models The fact that simple, non-linear, iterative dynamical and semi-dynamical systems can behave in extremely complex ways has been well documented by a wide variety of authors. For a concise summary of the literature, see May (1975, 1976). Much emphasis has been placed upon the study of the periodic dynamics, as well as the more complex “chaotic” dynamics of such models (May, 1975,1976; Li & Yorke, 1975; Guckenheimer,Oster & Ipaktchi, 1977). While studies of the preceding dynamical behaviors are of mathematical and biological interest, it is also interesting to study the dynamics from a more general topological standpoint. That is, we will be interested in saying things about the various dynamical behaviors by examining the form of the particular equation defining our model. In particular we would like to examine the topological dynamics of logistic forms. The logistic function fb(x) = bx(1 -x) arises from studying either the continuous logistic growth equation with carrying capacity K
or its discretized version x
Both equation (1) and equation (2) are classical examples of a population model exhibiting density dependent population growth (Maynard Smith, 1968; Witten, 1977a). t Reprintrequeststo: Departmentof BiochemicalEngineering, University of Southern California,LosAngeles, California90004,U.S.A. 23
0022-5193/78/170023+10 $02.00/O
:Q 1978AcademicPressInc. (London)Ltd.
24
M. WITTEN
When one discusses problems of form in mathematics, the easiest concepts to make use of are the topological conjugacy and the conjugacy class. Two functionsfzd --) A and g :B + Bare said to be topologically conjugate if there exists a mapping/:A -P B such that /is continuous, invertible and, for all s in A, /[f(x)] = g[+(x)]. This is equivalent to saying that there is a change of variables that will make the two functionsfand g lie on top of each other. If two or more functions are topologically conjugate to each other, we say that they lie in the same conjugacy class. The conjugacy classes of the function fb(x) = bx( 1 - x) have been classified exactly. For a more mathematical formalization of the conjugacy theorems and conjugacy classes see Witten (1977u) and Webster & Witten (1977). The resultant study has shown that the conjugacy classes of fb(x) are parameterized by the value of the parameter b. These classes are: b = 0, b E (0, 11, b E (1, 2), b = 2, b E (2, 3), b = 3, at least countably many classes in the interval (3,4), and b = 4. Hence, by virtue of the fact that b E (0, l] represents a conjugacy class, the iterates of any point x behave in the same manner dynamically, as long as b E (0, 11. In particular they all tend to zero monotonically. We define the nth iterate of a point x to bef”(x) = f[f”-‘(x)] where fO(x) = x. It can also be seen that dynamical behavior of the iterates of a point x for b E (0, l] is not the same as the dynamical behavior of iterates of x for b E (1,2) in which the iterates approach the fixed point XT = (b- 1)/b, as illustrated in Fig. 1. Further, the dynamics in these two classes is totally
Fro. 1. The dynamical behavior of the iterates of a point x0 under the mapping f*(x) = bx(1 - x) for values of b E (1, 2).
FITNESS
AND
25
SURVIVAL
different from the behavior exhibited by points x for b E (2,3), where the iterates spiral into XT, as seen in Fig. 2. It has been shown (Witten, 19773) that the set of all initial population
FIG. 2. The dynamical behavior of the iterates of a point x0 under the mapping f&x) = 6x(1 - x) for values of b E (2, 3).
values x, which tend to zero under iteration,
denoted b(c), where
8(fb) = x E [0, l] jlim f{(x) = 0 (3) n-rco > 1 can also be parameterized by the value of b. The parameter dependence of b(fJ is given as follows:
WJ = LO,11 b E(0, 0, Q(h) = (0, 11 b E(1,4), Wi)
= tj f-W
u (0, 1)
b = 4,
n=2
where f-“(3) is the set of all points x that map into + after n iterations such that there is no integer m < n satisfyingfern = 4. Finally, the restriction b E [0,4] arises from the fact that for b > 4 the iterates of a normalized initial population value x would become negative. This restriction also makes fa a self mapping. That is fa: [0, l] + [0, l] for b E [O ,4]. The concept of fitness first appears in Fischer (1958) in his fundamental theorem of natural selection. This theorem relates genotypic fitness to population adaptedness. Since 1930, when the theorem originally appeared, the
26
M.
WITTEN
concepts of fitness and adaptedness have exerted a strong influence on the field of population biology. Lewontin (1974) points out that one component of the concept of fitness is the probability of survival from conception to the age of reproduction. This statement indicates the underlying assumption that the concept of fitness is somehow intertwined with the concept of survival. It is this “intuitive” relationship that we will now investigate. To begin our discussion of the relationship between fitness and survival, it will be assumed that the parameter b is directly related, in some way, to a measure of the Darwinian fitness of the population. (Dobzhansky, 1970). This is not a totally unwarranted assumption, and it can be found in Wilson (1976), as well as Haldane (1966). Fitness, as pointed out by Herskowitz (1973), is best described in terms of the reproductive success of an individual/ genotype/initial population size relative to the reproductive success of other individuals/genotypes/initial population sizes. Hence, as Herskowitz points out “fitness is usually considered to refer to survival and reproduction of a genotype”/individual/initial population size. The construction of the logistic model, both in discrete and continuous forms, defines the parameter b to be the net births minus deaths. (Witten, 1977~; Maynard Smith, 1968; May, 197.5; Shikata, 1967). That is, b represents a finite net rate of increase or the reproductive success of an initial population size. Hence, it is not unreasonable to assume that, within the context of logistic models, b is a measure of the fitness of an initial population size. Dobzhansky (1970) points out further that Darwinian fitness is a measure of the reproductive success of the carriers of a given genotype in relation to that of the carriers of other genotypes. Hence, it is a comparative measure. The innate capacity for increase, which is related to b (see Birch, 1953a) is an absolute measure of one aspect of adaptedness under a given set of environmental conditions. Hence one is not surprised that Darwinian fitness and adaptedness measure different properties of life. Consequently, one expects b to be correlated to the fitness, and hence to the survival. It we initially assume that b E (0, l), then all populations x E (0, 1) will eventually die out. Further, perturbations 6:b + bf i? so that b+6 E (0, l] do not change this outcome. Thus increasing or decreasing the value of b in the interval (0, I] implies that increasing or decreasing the reproductive success does not influence the fitness of these populations. Now, a large enough perturbation of b, enough to send b into the interval (1,4), will change the dynamics so that only two populations will become extinct. Hence, an increase in the fitness leads to an increase in the number of populations that survive, i.e. do not become extinct. Notice once again, however, that changing b in the interval (1,4) doesn’t change the number of populations that survive.
FITNESS
AND
27
SURVIVAL
If we now perturb b from the interval (1,4) to b = 4, a totally new extinction dynamics appears. Under this perturbation, an increase in fitness leads to an increase in the number of populations that will eventually become extinct. In fact, it can be shown that the set of all extinction points, for the case b = 4, is dense with dense complement in the interval [O, 13. (Witten, 1977c). This implies that ~$0 contains at least countably many points, and thus, an increase in b from b = 4 - 6 to b = 4 leads to a striking increase in the number of populations that go extinct. Hence, an increase in the “fitness parameter” leads to a counterintuitive decrease in the survival of a large number of population values. Figure 3 illustrates the bifurcation of &‘(fb> as a function of the parameter b.
I-
O
I
2
3
4
b
FIG. 3. A bifurcation diagram illustrating the parameter dependent nature of the extinction set off&) = bx(l - x).
With all this in mind, one is lead to either one of two possible conclusions. Either we must be much more careful about the meanings we assign to various parameters in models of physical systems. Or else, within the context of logistic growth equations, it is possible for an increase in fitness to decrease the survival change of a population and a decrease in fitness to increase a population’s survival chances. Hence, one is lead to the conclusion that, within the context of logistic growth equations, fitness and survival are not co-extensive concepts.
28
M. WITTEN
The obvious objection can be raised. Most authors feel that the simplicity of fb may justify its occasional use as a model [see Haberman (1977), Lakshmikantham (1977), Smale & Williams (1976)], but that the extinction aspect is an unintended consequence of this simplicity. Hence, one is drawn into examining extinction in slightly more complex functions. An easy function, which is still density dependent, to analyze is the function given by f(x.) = Ix, emax,
I,a > 0.
(4) This model is discussed in Hassell (1972), Southwood (1975), and Cook (1965). Clearly, equation (4) satisfiesf:[O, co) -+ [0, Alea]. One can show that f,,, = Wea< ha, = l/a for all L E [0, 1). Hence all, initial population values will tend to zero under iteration. Further, using the theorems in Witten (1977a), one can show that [0, 1) represents one conjugacy class of equation (4). As L becomes greater that one, the valuef’(0) becomes greater than one, and the only points that go to zero are points x satisfying f"(x,) = 0 for m an integer greater than zero and m # co. One can show that the only value x, satisfying f “(x,) = 0 is x, = cO,m = 1. Hence,one obtains an extinction bifurcation map which is illustrated in Fig. 4. Since equation (4) is just a simplification of Cook’s (1965) model f(x,)
= x, e’(l -xn’k),
FIG. 4. A bifurcation diagram illustrating tion set off(x) = kexp( - ax).
(5)
the parameter dependent nature of the extinc-
FITNESS
AND
SURVIVAL
29
where k is just the inverse of the carrying capacity, and L = e’, we are justified in saying that the intrinsic rate of growth r tunes the survival of the population. However, notice that for Iz E [0, l] the dynamics of any initial population is invariant to a parameter change in L. [By virtue of the fact that ;1 E [0, l] represents a conjugacy class of equation (5)]. Hence, changing the rate of growth does nothing to alter fitness or survival, under the assumption-as before-that r is somehow related to fitness. Thus, even in a less simple, more stable model, there is an extinction bifurcation that illustrates the fact that the survival is insensitive to changes in the fitness parameter, over certain parameter ranges. May (1975) has discussed the model
f(X”)= ; x;-* as an example of a discrete model of a single age-class population with a finite net rate of increase A, as well as density dependence. This function has been demonstrated to fit population data for Drosophila melanogaster, Tribolium custaneum, and Lucilia cuprina. Assume that b < 0 in equation (6). One can show that the extinction set for this model is independent of A/a and is exactly J(f) = [0, bva7). Hence a A/a bifurcation picture-Fig. 5-shows an extinction behavior that is insensitive to changes in the finite rate of increase parameter. Thus, we have another example of survival not being sensitive to the assumed fitness parameter. Witten (19773) has detailed the parameter sensitivities of the extinction sets of a large number of iterative dynamical models. All of these models show insensitivities in the parameters assumed to be interrelating survival and fitness. It has also been proposed that we generate the xn+ i’s in a stochastic manner by applying a random number generator to either the intrinsic rate parameter or the population x,. Let us consider the perturbation of X, with a random number. If we choose an initial population x,, and generate xt =.f(xd+ x2 =f(xl)+rl
r. (7)
x,+1 =fW+r,
30
cOr
M.WITTEN
s w $I$- ---_-----__--_ ----__ --_-_-----.
FIG. 5. A bifurcation diagram illustrating the parameter dependent nature of the extinction set off(x) = (A/u)x~-~.
then as long as there are no strange behaviors, such as the dense with dense complement in fb for b = 4, there will be no change in the dynamics. This arises from the fact that you have not changed the parameter that tunes the dynamics. For example, in the case of equation (2), no matter what one perturbs the population value x, with, if b s 1 that population will go extinct. For the case b = O-5 in the modelf, = bx( 1 -x) we used a uniform random number generator to perturb a grid of 10,000 initial starting values x0. All points were, accurate to sixteen decimal places, zero after 1000 iterations. Notice that for b = 4 the random perturbation might make a significant difference. For example, let xc = 0*5+0.5 ~6% One can show that .c(xo> = 0.
(8)
However, if we let Xl = .mcJ + ro* Then x1 is not necessarily going to satisfy
(9)
(10) f42bI) = 0. Hence, in the case of&(x) the unintended extinction aspects can create perturbation problems. However, one can show that in a computer simulation of the extinction process forf,(x) the computer will not pick up any extinction points other than zero and one. (See Duncan & Witten, 1978). Hence random
FITNESS
AND
SURVIVAL
31
perturbations of this simple model will be lost in problems of machine simulation. If, as we said, the model is too simple and we wish to ignore the sensitivities of this model for the more accurate density dependent forms of equation (4), then one is bound by the nature of the conjugacy classes of the model. And it is the parameter dependence of this conjugacy class that determines the dynamics of the iterates, not the initial starting point. Hence, the more complex discrete dynamical models do not see statistical fluctuations in the populations as long as the parameters are fixed. If we consider equation (2) and choose to drive b with a random number we arrive at a different situation. In this casewe are randomlyvaryinga system parameter which tunes the dynamics of the model. In this case a number of behaviors can occur. If the sequence of randomly generated b’s remains within one conjugacy class the asymptotic behavior of the initial x0 will remain unchanged no matter how b varies. If, however, b fluctuates through a variety of conjugacy classes, then the asymptotic behavior of x,, may be totally scrambled. We simulated this particular situation by driving b in equation (2) by a random number r, in the interval (-0.5,O.S) where b = 3.157 + r,. Without the r, perturbation all x,, should tend to a stable two point cycle. However, with the perturbation there is no evidence of cyclic behavior after 10,000 iterations. In summary, it appears that the concepts of survival and fitness are not as tightly intertwined as our intuition would have us believe. Further, attempts to determine the fitness parameter in simple logistic models are frustrated not only by our intuitive understanding of how fitness and survival are linked, but also by the fact that the parameters which appear to be likely candidates for involvement in fitness seem to yield extinction dynamics which is insensitive to change in those very parameters. I would like to thank Robert Rosenfor providing his insight, Robert May for his comments,Robert Duncan for his statistical expertise, and the two referees for their comments. REFERENCES Ecology 34,713. BIRCH,L. C. (1953).Ecology 34, 698. COOK, L. M. (1965).Nature 207, 316. D-US, L. (1977).Am. Nut. 111, 1163. DORZHANSKY, T. (1970).Genetics ofthe Etlofutionary Process. NewYork: ColumbiaUniverBIRCH,
L. C. (1953).
sity Press. R. & WIT-TEN, M. (1978).(Submitted). FISCHER, R. A. (1958). The Genetical Theory of Natural Selection. New York: Dover Inc. GUCKENHEIMER, J., OSTER, G. & IPACKTCXI, A. (1977). J. math. Biol. 4, 101. HABERMAN, R. (1977). Mathematical Models. (Prentice-Hall,N. J.) DUNCAN,
32
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HALDANE, J. B. S. (1966). The Causes of Evolution. (Cornell Univ. Press, New York). HASSELL, M. P. & MAY, R. M. (1972). J. unim. Ecol. 41, 693. HERSKOWITZ, I. H. (1973). Principles of Genetics. New York: Macmillan & Co. KIMURA, M. & OHTA, T. (1971). Theoretical Aspects of Population Biology. New Jersey: Princeton Univ. Press. LAKSHMIKANTHAM, V. (1917). Nonlinear Systems and Applications. New York: Academic Press. LEWONTIN, R. C. (1968). Population Biology andEvolution. New York: Syracuse Univ. Press. LEWONTIN, R. C. (1974). The Genetic Basis of Evolutionary Change. New York: Columbia Univ. Press. MAY, R. M. (1975). J. theor. Biol. 51, 511. MAY, R. M. (1976). Nature, 261, 459. MAY, R. M. & OSTER, R. G. (1976). Am. Naf. 110, 573. MAYNARD SMITH, J. (1968). Mathematical Ideasin Biology. New York: Cambridge Univ. Press. LI, T.-Y. & YORKE, J. A. (1975). Am. Math. 82, 985. SHIKATA, M. (1967).J. theor. Biol. 14, 59. SMALE, S. & WILLIAMS, R. F. (1976). .I. Math. Biol. 3, 1. SOUTHWOOD, T. R. E. (1975). Am. Nut. 110, 791. WEBSTER, D. & WITTEN, M. (1977). Generalized Conjugacy Theorems (to appear). WILSON, E. 0. (1976). Sociobiology. Mass.: Harvard Univ. Press. WIITEN, M. (1977). (Submitted). WITTEN, M. (1977). (Submitted). WIZEN, M. (1977). (Submitted). WRIGHT, S. (1969). Evolution and the Genetics of Populations. Vol. 2. Ill. : Chicago Univ. Press.
Fitness and Survival in Logistic Models MATTHEW
WIT-TEN?
Department of Biometry, Medical University of South Carolina, 171 Ashley Avenue, Charleston, South Carolina 29403, U.S.A. (Received 5 July 1977, and in revisedform 21 February 1978) The conceptsof fitnessand survival in logistic modelsare shown to be independentif we follow certain intuitive definitions for theseconcepts. This conclusionfollows from a simpletopological analysisof the function fb (x) = bx (l-x), which is just the standardform of the logistic growth equations.
1. Logistic Models The fact that simple, non-linear, iterative dynamical and semi-dynamical systems can behave in extremely complex ways has been well documented by a wide variety of authors. For a concise summary of the literature, see May (1975, 1976). Much emphasis has been placed upon the study of the periodic dynamics, as well as the more complex “chaotic” dynamics of such models (May, 1975,1976; Li & Yorke, 1975; Guckenheimer,Oster & Ipaktchi, 1977). While studies of the preceding dynamical behaviors are of mathematical and biological interest, it is also interesting to study the dynamics from a more general topological standpoint. That is, we will be interested in saying things about the various dynamical behaviors by examining the form of the particular equation defining our model. In particular we would like to examine the topological dynamics of logistic forms. The logistic function fb(x) = bx(1 -x) arises from studying either the continuous logistic growth equation with carrying capacity K
or its discretized version x
Both equation (1) and equation (2) are classical examples of a population model exhibiting density dependent population growth (Maynard Smith, 1968; Witten, 1977a). t Reprintrequeststo: Departmentof BiochemicalEngineering, University of Southern California,LosAngeles, California90004,U.S.A. 23
0022-5193/78/170023+10 $02.00/O
:Q 1978AcademicPressInc. (London)Ltd.
24
M. WITTEN
When one discusses problems of form in mathematics, the easiest concepts to make use of are the topological conjugacy and the conjugacy class. Two functionsfzd --) A and g :B + Bare said to be topologically conjugate if there exists a mapping/:A -P B such that /is continuous, invertible and, for all s in A, /[f(x)] = g[+(x)]. This is equivalent to saying that there is a change of variables that will make the two functionsfand g lie on top of each other. If two or more functions are topologically conjugate to each other, we say that they lie in the same conjugacy class. The conjugacy classes of the function fb(x) = bx( 1 - x) have been classified exactly. For a more mathematical formalization of the conjugacy theorems and conjugacy classes see Witten (1977u) and Webster & Witten (1977). The resultant study has shown that the conjugacy classes of fb(x) are parameterized by the value of the parameter b. These classes are: b = 0, b E (0, 11, b E (1, 2), b = 2, b E (2, 3), b = 3, at least countably many classes in the interval (3,4), and b = 4. Hence, by virtue of the fact that b E (0, l] represents a conjugacy class, the iterates of any point x behave in the same manner dynamically, as long as b E (0, 11. In particular they all tend to zero monotonically. We define the nth iterate of a point x to bef”(x) = f[f”-‘(x)] where fO(x) = x. It can also be seen that dynamical behavior of the iterates of a point x for b E (0, l] is not the same as the dynamical behavior of iterates of x for b E (1,2) in which the iterates approach the fixed point XT = (b- 1)/b, as illustrated in Fig. 1. Further, the dynamics in these two classes is totally
Fro. 1. The dynamical behavior of the iterates of a point x0 under the mapping f*(x) = bx(1 - x) for values of b E (1, 2).
FITNESS
AND
25
SURVIVAL
different from the behavior exhibited by points x for b E (2,3), where the iterates spiral into XT, as seen in Fig. 2. It has been shown (Witten, 19773) that the set of all initial population
FIG. 2. The dynamical behavior of the iterates of a point x0 under the mapping f&x) = 6x(1 - x) for values of b E (2, 3).
values x, which tend to zero under iteration,
denoted b(c), where
8(fb) = x E [0, l] jlim f{(x) = 0 (3) n-rco > 1 can also be parameterized by the value of b. The parameter dependence of b(fJ is given as follows:
WJ = LO,11 b E(0, 0, Q(h) = (0, 11 b E(1,4), Wi)
= tj f-W
u (0, 1)
b = 4,
n=2
where f-“(3) is the set of all points x that map into + after n iterations such that there is no integer m < n satisfyingfern = 4. Finally, the restriction b E [0,4] arises from the fact that for b > 4 the iterates of a normalized initial population value x would become negative. This restriction also makes fa a self mapping. That is fa: [0, l] + [0, l] for b E [O ,4]. The concept of fitness first appears in Fischer (1958) in his fundamental theorem of natural selection. This theorem relates genotypic fitness to population adaptedness. Since 1930, when the theorem originally appeared, the
26
M.
WITTEN
concepts of fitness and adaptedness have exerted a strong influence on the field of population biology. Lewontin (1974) points out that one component of the concept of fitness is the probability of survival from conception to the age of reproduction. This statement indicates the underlying assumption that the concept of fitness is somehow intertwined with the concept of survival. It is this “intuitive” relationship that we will now investigate. To begin our discussion of the relationship between fitness and survival, it will be assumed that the parameter b is directly related, in some way, to a measure of the Darwinian fitness of the population. (Dobzhansky, 1970). This is not a totally unwarranted assumption, and it can be found in Wilson (1976), as well as Haldane (1966). Fitness, as pointed out by Herskowitz (1973), is best described in terms of the reproductive success of an individual/ genotype/initial population size relative to the reproductive success of other individuals/genotypes/initial population sizes. Hence, as Herskowitz points out “fitness is usually considered to refer to survival and reproduction of a genotype”/individual/initial population size. The construction of the logistic model, both in discrete and continuous forms, defines the parameter b to be the net births minus deaths. (Witten, 1977~; Maynard Smith, 1968; May, 197.5; Shikata, 1967). That is, b represents a finite net rate of increase or the reproductive success of an initial population size. Hence, it is not unreasonable to assume that, within the context of logistic models, b is a measure of the fitness of an initial population size. Dobzhansky (1970) points out further that Darwinian fitness is a measure of the reproductive success of the carriers of a given genotype in relation to that of the carriers of other genotypes. Hence, it is a comparative measure. The innate capacity for increase, which is related to b (see Birch, 1953a) is an absolute measure of one aspect of adaptedness under a given set of environmental conditions. Hence one is not surprised that Darwinian fitness and adaptedness measure different properties of life. Consequently, one expects b to be correlated to the fitness, and hence to the survival. It we initially assume that b E (0, l), then all populations x E (0, 1) will eventually die out. Further, perturbations 6:b + bf i? so that b+6 E (0, l] do not change this outcome. Thus increasing or decreasing the value of b in the interval (0, I] implies that increasing or decreasing the reproductive success does not influence the fitness of these populations. Now, a large enough perturbation of b, enough to send b into the interval (1,4), will change the dynamics so that only two populations will become extinct. Hence, an increase in the fitness leads to an increase in the number of populations that survive, i.e. do not become extinct. Notice once again, however, that changing b in the interval (1,4) doesn’t change the number of populations that survive.
FITNESS
AND
27
SURVIVAL
If we now perturb b from the interval (1,4) to b = 4, a totally new extinction dynamics appears. Under this perturbation, an increase in fitness leads to an increase in the number of populations that will eventually become extinct. In fact, it can be shown that the set of all extinction points, for the case b = 4, is dense with dense complement in the interval [O, 13. (Witten, 1977c). This implies that ~$0 contains at least countably many points, and thus, an increase in b from b = 4 - 6 to b = 4 leads to a striking increase in the number of populations that go extinct. Hence, an increase in the “fitness parameter” leads to a counterintuitive decrease in the survival of a large number of population values. Figure 3 illustrates the bifurcation of &‘(fb> as a function of the parameter b.
I-
O
I
2
3
4
b
FIG. 3. A bifurcation diagram illustrating the parameter dependent nature of the extinction set off&) = bx(l - x).
With all this in mind, one is lead to either one of two possible conclusions. Either we must be much more careful about the meanings we assign to various parameters in models of physical systems. Or else, within the context of logistic growth equations, it is possible for an increase in fitness to decrease the survival change of a population and a decrease in fitness to increase a population’s survival chances. Hence, one is lead to the conclusion that, within the context of logistic growth equations, fitness and survival are not co-extensive concepts.
28
M. WITTEN
The obvious objection can be raised. Most authors feel that the simplicity of fb may justify its occasional use as a model [see Haberman (1977), Lakshmikantham (1977), Smale & Williams (1976)], but that the extinction aspect is an unintended consequence of this simplicity. Hence, one is drawn into examining extinction in slightly more complex functions. An easy function, which is still density dependent, to analyze is the function given by f(x.) = Ix, emax,
I,a > 0.
(4) This model is discussed in Hassell (1972), Southwood (1975), and Cook (1965). Clearly, equation (4) satisfiesf:[O, co) -+ [0, Alea]. One can show that f,,, = Wea< ha, = l/a for all L E [0, 1). Hence all, initial population values will tend to zero under iteration. Further, using the theorems in Witten (1977a), one can show that [0, 1) represents one conjugacy class of equation (4). As L becomes greater that one, the valuef’(0) becomes greater than one, and the only points that go to zero are points x satisfying f"(x,) = 0 for m an integer greater than zero and m # co. One can show that the only value x, satisfying f “(x,) = 0 is x, = cO,m = 1. Hence,one obtains an extinction bifurcation map which is illustrated in Fig. 4. Since equation (4) is just a simplification of Cook’s (1965) model f(x,)
= x, e’(l -xn’k),
FIG. 4. A bifurcation diagram illustrating tion set off(x) = kexp( - ax).
(5)
the parameter dependent nature of the extinc-
FITNESS
AND
SURVIVAL
29
where k is just the inverse of the carrying capacity, and L = e’, we are justified in saying that the intrinsic rate of growth r tunes the survival of the population. However, notice that for Iz E [0, l] the dynamics of any initial population is invariant to a parameter change in L. [By virtue of the fact that ;1 E [0, l] represents a conjugacy class of equation (5)]. Hence, changing the rate of growth does nothing to alter fitness or survival, under the assumption-as before-that r is somehow related to fitness. Thus, even in a less simple, more stable model, there is an extinction bifurcation that illustrates the fact that the survival is insensitive to changes in the fitness parameter, over certain parameter ranges. May (1975) has discussed the model
f(X”)= ; x;-* as an example of a discrete model of a single age-class population with a finite net rate of increase A, as well as density dependence. This function has been demonstrated to fit population data for Drosophila melanogaster, Tribolium custaneum, and Lucilia cuprina. Assume that b < 0 in equation (6). One can show that the extinction set for this model is independent of A/a and is exactly J(f) = [0, bva7). Hence a A/a bifurcation picture-Fig. 5-shows an extinction behavior that is insensitive to changes in the finite rate of increase parameter. Thus, we have another example of survival not being sensitive to the assumed fitness parameter. Witten (19773) has detailed the parameter sensitivities of the extinction sets of a large number of iterative dynamical models. All of these models show insensitivities in the parameters assumed to be interrelating survival and fitness. It has also been proposed that we generate the xn+ i’s in a stochastic manner by applying a random number generator to either the intrinsic rate parameter or the population x,. Let us consider the perturbation of X, with a random number. If we choose an initial population x,, and generate xt =.f(xd+ x2 =f(xl)+rl
r. (7)
x,+1 =fW+r,
30
cOr
M.WITTEN
s w $I$- ---_-----__--_ ----__ --_-_-----.
FIG. 5. A bifurcation diagram illustrating the parameter dependent nature of the extinction set off(x) = (A/u)x~-~.
then as long as there are no strange behaviors, such as the dense with dense complement in fb for b = 4, there will be no change in the dynamics. This arises from the fact that you have not changed the parameter that tunes the dynamics. For example, in the case of equation (2), no matter what one perturbs the population value x, with, if b s 1 that population will go extinct. For the case b = O-5 in the modelf, = bx( 1 -x) we used a uniform random number generator to perturb a grid of 10,000 initial starting values x0. All points were, accurate to sixteen decimal places, zero after 1000 iterations. Notice that for b = 4 the random perturbation might make a significant difference. For example, let xc = 0*5+0.5 ~6% One can show that .c(xo> = 0.
(8)
However, if we let Xl = .mcJ + ro* Then x1 is not necessarily going to satisfy
(9)
(10) f42bI) = 0. Hence, in the case of&(x) the unintended extinction aspects can create perturbation problems. However, one can show that in a computer simulation of the extinction process forf,(x) the computer will not pick up any extinction points other than zero and one. (See Duncan & Witten, 1978). Hence random
FITNESS
AND
SURVIVAL
31
perturbations of this simple model will be lost in problems of machine simulation. If, as we said, the model is too simple and we wish to ignore the sensitivities of this model for the more accurate density dependent forms of equation (4), then one is bound by the nature of the conjugacy classes of the model. And it is the parameter dependence of this conjugacy class that determines the dynamics of the iterates, not the initial starting point. Hence, the more complex discrete dynamical models do not see statistical fluctuations in the populations as long as the parameters are fixed. If we consider equation (2) and choose to drive b with a random number we arrive at a different situation. In this casewe are randomlyvaryinga system parameter which tunes the dynamics of the model. In this case a number of behaviors can occur. If the sequence of randomly generated b’s remains within one conjugacy class the asymptotic behavior of the initial x0 will remain unchanged no matter how b varies. If, however, b fluctuates through a variety of conjugacy classes, then the asymptotic behavior of x,, may be totally scrambled. We simulated this particular situation by driving b in equation (2) by a random number r, in the interval (-0.5,O.S) where b = 3.157 + r,. Without the r, perturbation all x,, should tend to a stable two point cycle. However, with the perturbation there is no evidence of cyclic behavior after 10,000 iterations. In summary, it appears that the concepts of survival and fitness are not as tightly intertwined as our intuition would have us believe. Further, attempts to determine the fitness parameter in simple logistic models are frustrated not only by our intuitive understanding of how fitness and survival are linked, but also by the fact that the parameters which appear to be likely candidates for involvement in fitness seem to yield extinction dynamics which is insensitive to change in those very parameters. I would like to thank Robert Rosenfor providing his insight, Robert May for his comments,Robert Duncan for his statistical expertise, and the two referees for their comments. REFERENCES Ecology 34,713. BIRCH,L. C. (1953).Ecology 34, 698. COOK, L. M. (1965).Nature 207, 316. D-US, L. (1977).Am. Nut. 111, 1163. DORZHANSKY, T. (1970).Genetics ofthe Etlofutionary Process. NewYork: ColumbiaUniverBIRCH,
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HALDANE, J. B. S. (1966). The Causes of Evolution. (Cornell Univ. Press, New York). HASSELL, M. P. & MAY, R. M. (1972). J. unim. Ecol. 41, 693. HERSKOWITZ, I. H. (1973). Principles of Genetics. New York: Macmillan & Co. KIMURA, M. & OHTA, T. (1971). Theoretical Aspects of Population Biology. New Jersey: Princeton Univ. Press. LAKSHMIKANTHAM, V. (1917). Nonlinear Systems and Applications. New York: Academic Press. LEWONTIN, R. C. (1968). Population Biology andEvolution. New York: Syracuse Univ. Press. LEWONTIN, R. C. (1974). The Genetic Basis of Evolutionary Change. New York: Columbia Univ. Press. MAY, R. M. (1975). J. theor. Biol. 51, 511. MAY, R. M. (1976). Nature, 261, 459. MAY, R. M. & OSTER, R. G. (1976). Am. Naf. 110, 573. MAYNARD SMITH, J. (1968). Mathematical Ideasin Biology. New York: Cambridge Univ. Press. LI, T.-Y. & YORKE, J. A. (1975). Am. Math. 82, 985. SHIKATA, M. (1967).J. theor. Biol. 14, 59. SMALE, S. & WILLIAMS, R. F. (1976). .I. Math. Biol. 3, 1. SOUTHWOOD, T. R. E. (1975). Am. Nut. 110, 791. WEBSTER, D. & WITTEN, M. (1977). Generalized Conjugacy Theorems (to appear). WILSON, E. 0. (1976). Sociobiology. Mass.: Harvard Univ. Press. WIITEN, M. (1977). (Submitted). WITTEN, M. (1977). (Submitted). WIZEN, M. (1977). (Submitted). WRIGHT, S. (1969). Evolution and the Genetics of Populations. Vol. 2. Ill. : Chicago Univ. Press.