Simple growth laws and selection consequences

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Consequences E&s Szathmiry It is commonly thought that various types of populalion growth can 6e satisfactorily modelled as deviations from an inherently exponential (mafthusian) growth law. Consideration of kinetic resulls from research on the origin of life, laser physics and moreconventional population dynamics makes it clear, however, Ihal in certain cases the simplest and mechanistically most satisfactory assumption is either a basic subexponential or a hyperbolic growth law. Although these simple growth laws cannot be used instead of more-complicated models of density-dependent population growth when exacl quantities are important, the insighf gained 6y thinking them over can 6e substarrtial. Ideas about species packing, for example, await reconsider&ion. Simple malthusian (exponential) growth is used as a stepping-stone in all population-biology texts. The underlying assumption is that the process of reproduction can be satisfactorily characterized by the formal stoichiometric equation: A (+S)

k --+-nA

(I)

where A is the symbol for an individual, S is the consumed substrate(s), k is the malthusian growthrate constant, and n> I. Without loss of generality, we may take n = 2. A good example is the mitotic division of a protist. The associated rate equation is: dxldl

= i = kx

(21

where x is the density of A and x is the time derivative of x. The concentration of S is assumed stationary, and is thus incorporated into k. The solution of Eqn 2 is of course well known: x(t) = x(0)ekr

When two populations with differing malthusian parameters compete with each other, that with the larger growth rate wins the competition, whereas the opponent is competitively excluded (Box 21. Subexponential growth and survival of everybody It has been a continuing desire of students of the origin of life to achieve enzyme-free replication of nucleic-acid-like molecules in the test tube. The attempts have recently met with some success: a hexanucleotide5 and a tetranucleotide analogue6 were shown to replicate by themselves in vitro, incorporating smaller building-blocks. The kinetics of growth were rather peculiar. Instead of following the growth rate of Eqn 2, it was found to follow the approximate dynamics: x = a. + /(y’L

(I II

where CL is the rate constant for spontaneous template formation. The reaction mechanism is displayed in Box 3, which explains how the dynamics of the system can be simplified for practical purposes. The square-root dependence of growth on oligonucleotide concentration is due to the fact that ( I) only the single-stranded oligonucleotide has template activity (new building blocks cannot be aligned by basepairing unless the double strand opens itself); (2) the immediate product of synthesis is the replicationally inactive double-stranded form; and (31 the concentration of

1991

single strands is approximately proportional to the square-root of the total concentration. Szathmtiry and Cladkih” were able to make the prediction that different oligonucleotide sequences should coexist under the described conditions. This statement follows from their proof that if entities grow subexponentially, any variant should be able to invade when rare (Box 4). The same result can be obtained for any number of competing populations and for any exponent, p, between 0 and I.0 replacing ‘/2 in Eqn I I (Ref. 8). From an ‘ecological’ point of view, the figure in Box 3 implies stronger intrathan interpopulation interference for the following reason: although different short sequences will bind to not only their complementary but various ‘mutant’ strands as well, base-pairing rules ensure that the strongest binding (expressed by the highest a:b ratio) will be between fully complementary strands. lntrapopulational consumer interference during consumption of a limiting resource (the concentration of which was thought to be constant in the molecular example) has been modelled by DeAngelis etal.9, using the following equation: i; = (f~+~ll(b

+ x, + dx,)

( I71

where x, and x, are the resource and consumer density, respectively, and f, band dare constants unrelated to those denoted by the same letters previously. Keeping x, constant, we see that if x, < b + dx,, then the growth of the consumer population is approximately exponential, whereas if x, is sufficiently large, then it is approximately linear. Qualitatively, subexponential growth lies between these two extremes. However, the applicability of Eqn I I

(3)

where x(O) is the original density of A and x(t) is the density at time 1. This indicates that without ecological constraints, the population density would grow exponentially. Ginzburg’ argues that exponential growth is the ‘forceless’ (by analogy with physics), or undisturbed, motion of growing populations (Box I). EorsSzathm6ry isat the Laboratory of Mathematical Biology, MRC National Institute for Medical Research. The Ridgeway. Mill Hill. London NW7 IAA, UK.

366

8 1991. Elbev~e Sc~cnce Puhllshcrs L’d ILKI 0:69-5347’91’502

120

TREE vol. 6, no. 11, hvember

1991

(with cx = 0, i.e. excluding spontaneous generation) to biological populations may be rather limited. Sexual reproduction, hyperbolic growth and survivalof the common Eqn 1 is not a good scheme for sexual reproduction, since it does not reflect that the interaction of two individuals is essential to produce progeny. The simplest scheme accounting for this is 2A(+S)k-

3A

(18)

where for simplicity’s sake the two sexes are not distinguished. Assuming again that the density of the resource S is stationary, the growth rate, k, may be thought to incorporate its value. The corresponding growth equation is: x 27.zh2 (19)

populations of sexually reproducing species should grow hyperbolica11y4. If this is so, sexual populations should exhibit an Allee effect-a decrease in the malthusian (per unit density) growth rate with decreasing population density”: i/x = kx

(211

The potential importance of hyperbolic growth lies in its consequences for selection. Originally, this was called ‘once-for-ever’ selection3s4, but the term ‘survival of the common’ seems more appropriate12. The logic of these terms is

as follows. When two competitors with differing k values g&w together, the one with the larger x(Olk value will ‘blow up”, first, as is apparent from Eqn 20. The same claim follows when resources are limited7Jo - the outcome of selection is a function of the initial conditions. Note also that due, to the Allee effect (Eqn 21 I, a fitter varia+,t (with larger k) in a competitiy sftuatiotr cannot invade when rare; intrinsic fitness is ‘masked’ by the growth law’*. We obtain the same result with any exponent pr I in Eqn 19 (Ref. 4).

the solution of which reads3 x(t) = i/l I/x(O) - kt]

(20)

approaching infinity as r-+ Illx(O)kl. The startling conclusion is that a population of individuals reproducing according to Eqn 18 would grow to infinite density in finite time, if resources were not limiting. The population growth is a hyperbola (Fig. 2); hence the term ‘hyperbolic growth’. Hyperbolic growth was emphasized by Eigen in connection with his hyeercycle modePI of early molecular organization (see also Refs 7 and IO). A hypercycle is a molecular system comprising information carriers that aid each other’s replication along 8 cyclic path of action, i.e. the autocatalytic growth of each member is ‘catalysed by the preceding one in the cycle. A singlemembered hypercycle may be characterized by Eqn 18 in its simplest form. Eigen and Sch,uster also noted that, at low density, 367

TREE vol. 6, no. 11, November

In I938 (i.e. in the ‘Golden Age’ of theoretical ecology), Volterra13, incorporated the Allee effect, taking sexual reproduction into account, into a modified logistic equation: i = kx= - dx - 6x3

(22)

where d and 6 are positive constants. Note that the first negative term represents conventional death, whereas the cubic competition term was constructed mathematically to prevent explosion and is hard to interpret in ecological terms. This equation has three equilibria (provided k* > 4dS). two of which are locally stable, and one of which is unstable. A more satisfactory approach was taken by Philip” in 1957 using the following model: i = (k - dlx - kxe-px

(23)

where I - e--PX is the probability of a female being fertilized. This assumes no sociality and a population whose members are moving around at random in a homogeneous environment (i.e. f3 is positive). The general conclusions are similar to

ll[x(d)k] Fig.

368

I.

Hyperbolic

growth.

t

those drawn from Eqn 22. Philip was careful to point out that sociality would modify the relations. Purely from the dynamical point of view, outcrossing plant species would be considered asocial, and selfing species as social’?. Apart from sexually reproducing asocial populations (whose inherently nonlinear growth dynamics are usually simply ignored), two examples of hyperbolic growth are worth mentioning. ( I ) The growth curve of the global human population rather closely fits a hyperbola (cf. Fig. I), having its asymptote around the year 2040r5. This phenomenon can be explained by the observation that the death rate of the human population decreases because of the development in health care, which in turn results from the activity of the human population. (2) The rate of increase in the number of basic ideas in physics seems to be proportional to the product of the number of ideas already known and the total population number of humansIb. Because the latter number itself increases hyperbolically, it is understandable that the number of ideas does the same. Mutualism Note that there is a formal resemblance between a two-membered hypercycle (where member one catalyses the replication of member two and vice versa) and models of mutualism’7. It is known that LotkaVolterra models of mutualism may cause an explosive increase by hyperbolic growth, because of the positive x,x2 interaction term for both partners. In a Lotka-Volterra model of obligate mutualism the partners either die out or explode hyperbolicallyr8. I will come back to this problem in the final section. Growth laws and species packing Species packing refers to the distribution of coexisting species along a resource gradient19. The basic models are by MacArthurr9 and May*O (also see Ref. 2 I for a review). Genotypes are assumed to differ in their adaptedness to the various values on the resource axis in Fig. 2. The value on the resource axis to which a genotype is best adapted is labelled y. Each genotype has a util-

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ization function, centred around its corresponding y. This function has the same shape for each genotype. Let the density of the species at y be xY. The change in density is as follows: i,, = Ika(x,)R,

- dlx,

(24)

where a(x,) is a function expressing the density-dependent deviation from malthusian growth and the quantity Ry is the total amount of resources available to species y. The critical element of our interest here is the function a(x 1. It is monotonically decreasing r or subexponential, constant for exponential and monotonically increasing for hyperbolic growth. I will discuss these cases in turn. f I) The effect of subexponential growth on species packing has not yet been analysed in this model. As there is an indication from theory that a number of species can coexist in the same environment y (see above), I think that the tightest possible species packing should result from this case. Its potential impact on problems such as the ‘paradox of the plankton’ (the coexistence of many species in an apparently uniform niche)== await detailed analysis. (2) With pure exponential growth, the number of coexisting types along a gradient can be arbitrarily large, as was clearly demonstrated in Roughgarden’s clone-selection mode12’ (note that members of a clone reproduce asexually). The explanation is that if the intrinsic fitness, k, of the different genotypes is the same, then because of the shape of the utilization function (Fig. 2) a population best adapted to the point y cannot be immune to invasion by any population best adapted to a pointy * E, E being an infinitesimal number. Low densities will of course lead to stochastic extinctio$O, but the resulting pattern is far from robust and regular”. (3) Hopf and Hopf” have realized that the Allee effect can have a very important role in species packing. The root of their idea comes from an analogy with laser physics. The ordinary laser results from a process whereby a photon stimulates the emission of another one from an atom according to Eqn I. Photons of any frequency (within certain bounds) can coexist in such lasers.

TREE vol. 6, no. 7 7, November

1991

In certain other laser types, however, stimulation for each frequency alone follows Eqn 18, resulting in a discrete distribution of realized frequencies2j. Similarly, analytical as well as numerical investigations show that the presence of an Allee effect (a cost of rarity in the form of finding a mate) results in a discrete equilibrium distribution of species along the resource gradient”. This result may have a bearing on Darwin’s dilemma of transitional forms: ‘Why, if species have descended from other species by insensibly fine gradations, do we not everywhere see innumerable transitional forms? Why is not all nature in confusion instead of the species being, as we see them, well defined?’ Based on the model of Hopf and Hopf”, Bernstein et a1.25 argue that the clue lies in the growth and selection dynamics of sexual populations. Note that sexual species are easily distinguishable in most cases, whereas asexual ones are usually not26. Comparing data from rodent, hawk and parthenogenetic lizard (Anolis) populations, Hopf could find some empirical support for the concept that the cost of rarity, i.e. the cost of finding a mate, was in fact a crucial factor in shaping packing of sexual species2’. It seems that the phenomenon of the ‘survival’ of the common is not restricted to lasers or biology. In heavily technology-dependent modem industries, contrary to conventional economy, returns are increasing rather than diminishing. Competition between two such firms is not settled by the unbiased judgement of the market, but by the one establishing itself first being likely to win, even if the other one produces ‘intrinsically’ better products. Nonlinearities of the Aileeeffect type are essential for such economic behaviour28. Complications and connections In the classical theory of competition developed by Gause29, based on a two-species Lotka-Volterra system, three different outcomes are possible, depending on the parameter values: stable coexistence, unconditional survival of one species and survival of one species depending on initial densities. We find the same outcomes for two

competitors with subexponential, exponential and hyperbolic growth, respectively, in the following system (see Refs 4 and 81:

x, = k,x,~- x,lk,x,p

+ k,x,PI (25al

X2 = kZx2” - x,lk,x,“

4 k,x,“I

Sl$!!?

B

126a)

44

126bl

b. B -- ‘

where a, b and care rate constants; usually a, b + c. B is the complex of the unmated pair. Such a system grows practically hyperbolically with the rate ‘constant’ Zca/( b + cl at very low density. and exponentially with the rate ‘constant’ cat very high density”’ I’ (cf. Ref. 321. This saturation property of the dynamics of sexual reproduction was essentially realized by Kostitzin in 1940. who proposed I3 the phenomenological growth equation x = kx’(p

+ x) - dx - 6x2

gradient

Resource

gradient,

125bi

where p< I for subexponential. p = I for exponential and p > I for hyperbolic growth, and the sum invariant due XI + x2 = I remains to the negative outflow term. Whether there is some deep mathematical relationship fan appropriate transformationf behind this apparent similarity between the two-species Lotka-Volterra and the system in Eqn 25 is not known. Eqn 18 assumes that there is no ‘functional response’ in sexual reproduction, although this is unrealisticd,‘(‘3’. Consider, for example, two haploid protists reproducing by syngamy and meiosis. Not every encounter of male and female cells is followed by cell fusion; some pairs invariably fall apart due to the finite strength of the interaction of the cell surfaces. Thus, Eqn 18 may be replaced by the following3’: 2AI+

Resource

(271

where I-L, d and i, are positive constants. Note that this modified logistic equation does not need the unrealistic cubic competition term of Volterra’s equation lEqn 22). The combination of a saturating sexual and a simultaneous simple reproduction leads to interesting dynamic behaviour - two such competitors can stably coexist, and invade when rare (I, hence the result of Hopf and Hopf” is not, contrary to some suggestions!‘, robust for facultative parthenogens.

(b)

.

Fig. 2. An illustration of species packing. la) Utilization functions, f, and equilibrium population densities. y, for two species. Ibl The multi-species equilibrium. This may be the closest packing of species with an Allee effect.

The importance of saturation in a model of mutualism was realized by Vandermeer and Boucher’-’ in the following way x, = x,lx,b,rx,,x,l

- d,I

128at

X2 = x,)x,b,lx,,x,l

.-- d-i

f28bl

where the functions b, and b, are decreasing for both Y, and x:,.-This system has two fixed points in the interior of the positive quadrant: one with lowerdensities is unstable. and the other one with higher densities is locally stable’-‘. The lower instability reflects a generalized Allee effect for the mutualistic pair. The models discussed above do not reflect spatial structure explicitly. Britton’s study’i indicates that this can be very important. He formulated a model of animal populations in which there is an advantage to individuals in grouping together thence there is an Allee effect), and animals move around at random (described by diffusion in space). The surprising conclusion is that aggregation can lead to coexistence of two populations, one of which would otherwise become extinct. The bearing of this finding on the theory of species packingof sexual populations, as modelled by Hopf and Hopf”, awaits scrutiny. Finally, the reader’s attention is drawn to another review covering kindred topics”’ Acknowledgements I rhank the late Fred Hop1 for many excitmg discussions and T Czar~n. Rick Michod. I Scheuring and three anonymous referees for u4uI comments

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References I Ginzburg, L.R. t 19861/. Theor. Biol. 122, 385-399 2 Michod, R.E. t 1981) Br. /. Phi/OS. Sci. 32, l-36 3 Eigen. M. f 1971 I Natorwissenschaften 58. 465523 4 Eigen. M. and Schuster, P. f 19791 The Hypercycle. Springer-Verlag 5 Von Kiedrowski, G. f 19861 Angew. Chem. 98,932-934 6 Zielinski, W.S. and Orgel, L.E. f 19871 Nature 327. 346347 7 Szathmary, E. (1989) in Oxford Surveys in Evolutionary Biology (Vol. 61 (Harvey, P.H. and Partridge, L.. eds). pp. 169-205, Oxford University Press 8 Szathmary, E. and Gladkih, 1. f 19891 1. Theor. Biol. 138, 55-58 9 DeAngelis, D.L.. Goldstein, R.A. and O’Neifl. R.V. f 19751 Ecology 56. 881-982 IO Szathmary. E. II9891 Trends Ecol. Eva/. 4, 200-204 II Hopf. F.A. and Hopf, F.W. 11985) Theor. Popol. Biol. 27, 27-50 12 Michod, R. f 1984) in A New Ecology (Price, P.W.. Slobodchikoff. C.N. and Gaud, W.S.. edsl. pp. 253-278, Wiley 13 Volterra. V. f 19381 Hum. Bio/. IO, l-l I 14 Philip, 1.R. (1957) Ecology 38, 107-I I I

I5 Eigen, M. and Winkler. R. f 1975) Das Spiel: Naturgesetze steuern den Zufa//, R. Piper Verlag I6 Fowler. R.C. f 19871 /. Sci. Explor. I, I l-20 I7 Szathmary, E. f 1986) Bioldgia 34, ‘3-37 I8 Hallam, T.G. ( 19861 in Mathematical Ecology: An introduction (Halfam. T.G. and Levin. S.A., edsl, pp. 241-285, SpringerVerlag 19 MacArthur, R.H. (19721 Geographical Ecology: Patterns in the Distributions of Species, Harper & Row 20 May, R.M. ( I973 1 Stability and Complexity in Mode/ Ecosystems, Princeton University Press 21 Roughgarden. I. II9791 Theory of Population Genetics and Evolutionary Ecology: An Introduction, Macmillan 22 Hutchinson, C.E. ( I9781 An /ntroduction to Population Ecology, Yale University Press 23 Menegozzi, L.N. and Lamb, W.E., fr i 1978) Phys. Rev. A 17, 701-732 24 Darwin, C. ( 1859) The Origin of Species by Means of Natural Selection. fohn IMurray 25 Bernstein, H., Byerly. H.C., Hopf. F.A. and Michod. R.E. i 19851/. Theor. Biol. I I7,665-690 26 Maynard Smith, I. (I9781 The Evo/ution of Sex, Cambridge University Press 27 Hopf. F.A. I I9901 in Organizationa/

Constraints on the Dynamics of Evolution (Maynard Smith, I. and Vida. G., edsf, pp. 357-372, Manchester University Press 28 Arthur, B.W. t 19891 Econ. 1. 99 (3941, I l&I31 29 Cause, C.F. ( 19341 The Struggle for Existence, Hafner 30 Szathmary. E.. Kotsis. M. and Scheuring, I. (19881 in Mathematical Ecology IHallam, T.C.. Gross, L.I. and Levin. S.A., edsl, pp. 46-68. World Scientific 31 Szathmary. E.. Scheuring, I., Kotsis, M. and Gladkih. I. II9901 in Organizational Constraints on the Dynamics of Evolution (Maynard Smith, f. and Vida, G., eds), pp. 279-289, Manchester University Press 32 King, G.A.M. f 1981 I BioSystems 13, 225-234 33 Kostitzin, V.H. (19401 Acta Biotheor. 5, 155-159 34 Vandermeer, f.H. and Boucher, D.H. ( I9781 /. Theor. Biol. 74, 549-558 35 Britton, N.F. (I9891 1. Theor. Biol. 136. 57-66 36 Michod, R. in 1990 Lectures in Complex Systems (Santa Fe Institute Studies in the Sciences of Complexity. Vol. Ill) f Nadel, L. and Stein, D.. edsf, Addison-Wesley iin press)

cesses1,2. However, the various fields have evolved somewhat in isolation of one another, although there are good examples of fruitful interactions, as with the study of aspects of spatial memory that has attracted researchers with backgrounds from psychology as well as those from ecology and behaviour3p4. On the functional side, the accumulated evidence shows that animals are quite efficient in the way they locate and utilize resources, and, as a corollary, that it is therefore possible to study such problems by using economic principles that take into account costs and benefits of behavioural actions in terms of fitnessrelevant criteria. Such analyses can become quite sophisticated and are no longer restricted to simple tasks, but can address stage-dependent behavioural policies over time5 or decisions in stochastic environment@. Furthermore, many of the problems that one would typically associate with searching behaviour, such as allocation of search effort in space, are notoriously difficult to solve. A classic is the travelling-salesman problem, where unconventional methods for numerical solutions7 could pave the way for a new look at the spatial behaviour of animals. On the other

hand, in the instances where the mechanics of problem solving by animals was analysed, despite all of the computational and conceptual complexity of the problem, often the actual decision rules or estimation procedures may actually be quite simple8. What is a characteristic of the topic of ‘searching behaviour’, therefore, is its relevance to a number of different disciplines and thevariety of methodological approaches and concepts used. The publication of Bell’s new treatise on the subject, specifying in its subtitle ‘the behavioural ecology of finding resources’, should thus be a timely addition. The subjects covered by this book explicitly not only include searching per se, but also the utilization of resources after they have been discovered. The book contains an ample collection of examples that in some sense or another relate to these issues. One should therefore applaud the intention of the author to bring together the various aspects in a single volume and make it understandable for a broad audience. In part, this aim has been achieved. It is, for example, useful that each section of the book is summarized, allowing specialists in particular

Book Reviews Behavioural Ecology Searching Behaviour: The Behavioural Ecology of Finding Resources by W.J. Bell, Chapman & Hall, f35.00 hbk (xii + 358 paged 0472292706

7997. lSBN

Almost by definition, one of the most important tasksfor mobileorganisms is to search for the resources they need to survive and reproduce. That includes things such as looking for food, mates or a safe place to evade predators. Animals show a bewildering variety of behaviours associated with this task, and it is likely that the strategies of plants, albeit expressed in different ways and on different time scales, will demonstrate an equally intricate fabric towards the same ends. Traditionally, the more functional questions about searching behaviour have been the domain of behavioural ecology (mostly), while the more proximate questions have been of interest to physiology, ethology or psychology. Moreover, since exploitation of resources, notably of food, has repercussions for populations and species assemblages, this question has always been of concern for ecologists trying to understand the mechanics of ecological pro370