Optimizing reproduction in a randomly varying environment

J. Theoret. Biol. (1966) 12, 119-129

Optimizing Reproduction in a Randomly Varying Environment DAN COHEN t Electrical

Biological Computer Laboratory, Engineering Research Laboratory, University Urbana, Illinois, U.S. A.

(Received 6 August 1965, and in revisedform

of Illinois,

8 November 1965)

A model is constructed of optimizing long-term growth rate in a randomly varying environment. Specifically, the model is of an annual plant, the seeds of which can either germinate and yield more seeds in numbers which depend on environmental conditions, or remain dormant in the soil and undergo some decay according to their viability. The optimal germination fraction is derived for any combination of probabilities of the yield of seeds per germinating seed, and of various values of the viability of the seeds. The relevance of the model is discussed.

1. rntmduction Most living organisms are faced with a considerable risk of failure when attempting to reproduce. One obvious way to survive and reproduce in a risky environment is to spread the risk so that one failure will not be decisively harmful. 2. The Model Consider an organism such as an annual plant that reproduces only once in its lifetime, each growth and reproduction cycle being completed within a discrete time interval (a year). The average number of seeds per germinated seedling, Y, is a random variable depending on environmental conditions, and is assumed to be independent of the population density. Of the total number of seeds present, a fraction G germinates every year. Of the seeds that do not germinate, a fraction D decays every year. Our aim is to find that value of G which maximizes the long-term expectation of growth, i.e. the optimal strategy. In mathematical terms, the one-step transformation is St+l = St-S,.G-D.(S,-S,.G)+G.Y,.S, (1) t Present address: Research Laboratory Technology, Cambridge, Mass., U.S.A.

of Electronics, 119

Massachusetts Institute

of

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where S is the number of seeds present. For a sequence of N steps, s, = s,.n [(I-~).(i-~)+~.yi]n~ (2) I where ni is the number of times that a particular Yi occurred in the sequence. Taking the logarithms and dividing by N gives log ----=SN N

logs, N ~ ~ ~log[(l-G).(l-D)+G.

When N approaches infinity,

1% 7 so= 0, and 2

= Pi,

lim !%S? = C Pilog[(l-G).(l-D)+G. N I

N+m

Yi]. SO

(3)

that YJ

(4)

1%SN is equal to the associated with Yi. Lim ___ N long-term average, or the mathematical expectation, of the specific growth rate of the seed population. It is clear from equation (4) that decreasing the decay constant D, and increasing the yield of seeds per germinating seed, Yi, will always increase the growth rate expectation. We would like to show how the value of the germination factor G influences the expectation of growth at any particular combination of D, Yi and P, and what values of G give the maximal growth rate.

where Pi is the probability

3. The Case of Two Outcomes

Considering first the case where Y, can assume only two values, 0 and Y. Equation (4) then takes the form lim 1~=(1-P,)log[(1-G).(1-D)]+P,log[(1-G).(1-D)+G.Y](5) N

N-rm

where P, is the probability of lim 1%

of having Yj = Y. We then computed the value

for various combinations

of P,, Y, D and G. Some charac-

teristic results are shown in Figs 1 to 4. Examining equation (5), it can be seen that when Y or D are large enough so that G. Y is large relative to (1 - G)(l - D), equation (5) can be simplified to give lim &g?! N-rm

iv

= P,log[G.

Y]+(l-P,)log[(l-@(l-G)]

(6)

FIG. 1. Long-termexpectationof growth rate, GRLT, plotted againstthe germination fraction G for variouscombinationsof Py, Y and D, with i = 2, Y0 = 0, Y, = Y. All the cmvesgo from log (1 - D) for G = 0 to f Pg log Y, for G = 1, whichgoesto -co in this casesincewehave Y0 = 0.

RG. 2. GRLT plotted againstlog (1 - D), showingthe almostlinear relationships betweenthemexceptwhenPy, Y andG areall low.

OQ

Oc5

GRLT = f(P,) for varying

x D. and G

GRLT=f(ln

Y) ot P=@5

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which is linear with respect to log (1- 0) and log Y. These relationships can be seen in the linear portions of Figs 2 and 3, respectively. Equation (5) is also linear with respect to P, over all possible values of the other parameters, as can be seen in Fig. 4. Differentiating equation (5) with respect to G gives Y+D-1 l-’ =-l-G+P,. (7) (l-D)(l-G)+G.Y’ Setting equation (7) equal to zero and solving for G,,, gives 03)

Examining equation (8) it can be seen that G,, is a linear function of P,, and becomes approximately equal to P, when Y is large enough or when D or P, are close enough to one so that the second term in the equation can be neglected. The value of G,,, is then independent of D and Y. These relationships are illustrated in Fig. 5, where, except for a combination of low Y and low D, the GmaXpoints all fall on the identity line G,,, = Py. I.c

of )-

uE

C)’

-0.:

,

FIO. 5. G,, plotted against Py, demonstrating the linearity of the relationshipunder all conditionsand the approachto the identity G,, = Pr for large Y and D.

Gmax cannot assume negative values. A negative G,,, according to equation (8), means that G,,,,, = 0, i.e. that in such conditions no positive growth rate is possible.

124 The conditions

D.

for a positive

COHEN

long-term

growth expectation,

i.e. for

lim log SN > 0, as a function of P,, Y and D, after optimization had already N taken place, i.e. when G = G,,,,, are illustrated in Fig. 6. In this figure the

N-m

FIG. 6. A plot of the relationship between D and Pr for various values of Y, after optimization had already taken place, i.e. G = G,,,, when GRLT = 0. The dotted lines are approximate extrapolations to the two boundary conditions of Pu = 0, D = 0 and D = 1, Py = 1.

values of Pr and D for zero long-term growth rate have been plotted for several values of Y. Each one of the curves divides the unit square into an area of positive growth towards higher Py values to the right and lower D values at the bottom, and an area of negative growth in the opposite directions. The two limiting conditions of P, = 0, D = 0 and D = 1, P, = 1 have been calculated by setting equation (5) equal to zero, with G = G*ax = PY. 4. The Case of Maoy Outcomes In the case of a finite set of Yi values it is possible to take the derivative of equation (4) Y,fD-1 slim, 1% SN aG=CPi (9) i (1~G).(l-D)+G. x I to set it equal to zero, and to compute G,,, for any values of the other parameters. For the sum to equal zero it is necessary for at least one of the

iN

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terms to be negative, i.e. Yr+ D < 1, and for at least one to be positive, Yi+ D > 1. That means that at least one Yi > 1 and one Y, 1 < are necessary for 0 < G,, < 1. Since the expansion of equation (9) results in a polynomial of G to the (i- 1)th degree, it is impossible to obtain analytically the value of G,, for i > 4. However, since each single term in (9) is a decreasing function of G for 0 I G 5 1, 0 I; D 5 1, and 0 I Yi, when Yr- (1-D) is either positive or negative, their sum must also be a decreasing function of G under these conditions. From this is follows that the derivative can be equal to zero at most at only one value of G between zero and one, which means that the long-term growth expectation as a function of G has at most a single maximum in this range. For 0 < G,, < 1, the derivative must be greater than zero for G = 0, and smaller than zero for G = 1. For G = 0, equation (9) gives a Iirn

log 7

& ac=pi.+$I

1

i

(10)

and for G = 1

Equation (10) is greater than zero for c P, Yi > 1, which is also a necessary condition

for having a positive 1ong:term growth rate. Equation (11) is 1-D negative when C Pi 7 > 1, from which it follows that the conditions i 1 for Max c 1 are given by the inequality (12)

The expression on the right-hand side of (12) is the harmonic mean of Y1, symbolized by Hr. We thus have that for G,,, < 1 it is a sufficient condition that 1-D > Hr. The harmonic mean bears the following relationship to the arithmetic mean Pand the variance oi, of Yi,

(13)

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provided that (Y- y)3/H3 is small, compared with (Y- P)/y Using (12) and (13) we get the necessary condition for Gmax < 1 as P l-7 4 Thus, it is only when the variance of the yield becomes large enough in relation to the mean yield and to the viability of the seeds that it becomes advantageous to postpone the germination of some fraction of the seeds. Even if we do not use the approximate relationship between the harmonic mean and the variance, as in equation (13), it is possible to make use of the fact that the harmonic mean is always less than the geometric mean. A suficient condition that G,,,,, < 1 is thus given by 1 -D Taking logarithms

> geometric mean of Yi.

(15)

of both sides, we get lo~w~qp-%Y,

(16)

so that if the mean log seed yield per germinating seed is less than the log of seed viability, G,,, will be less than one. The values of G,,, for i > 4 were most conveniently computed from numerical computations of the long-term growth expectation as a function of G and of the other parameters. The value of G,,, was found by graphical interpolation on the plotted curves, or by numerical curve fitting. A most plausible distribution of Pi(Y) is of an approximately normal distribution of P, as a function of log Yi. The growth rate expectation for several such distributions, differing in their means and variances, have been computed (Fig. 7(a)-(d)). The results are qualitatively similar to the case of two-valued Yi, in that when there is some probability for high Y, values (> lo), Cnax tends to equal the weighted probabilities of these high Y, values. It is clear that G,,,,, must be less than one if there is even a very small probability of Yr = 0. Thus, 1 -G,,, ? C Pi for the very small values of Yi(< 0.1). G,,, always increases with D, the effect decreasing with increased c P, log Yi. 1

For any number of Yi values, lim

logs,

-=log(l-D) N

forG=O

(17)

lim~=FPl.logY, N

forG=l.

(18)

N-a,

N-rm

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001 @IO

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005, O-20 100 001 040 G

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000 'I-

is plotted as a function of G and for several values of of P(Y,). The discrete distributions are shown in block diagrams at the bottom of the corresponding curves. FIG.

7. (a) to (d) GRLT

D for 4 different distributions

It follows from equation (18) that with G = 1, any finite probability of Y, = 0 will lead to a - co value for the long-term growth rate expectation, i.e. population size will be zero. The above relationships are illustrated very well in Fig. 7(a) to (d).

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5. Discussion Our model is formally similar to that of economic decision making under risk. G is equivalent to the fraction of capital invested, D to the depreciation of uninvested capital due to rise in prices, fees, taxation, etc., and Y is the return on the investment. The results are formally exactly the same. For the biological model the results show the necessity of having the combination of high yield when successful, the ability of ungerminated seeds to survive for many generations in the soil and a low yearly germination fraction, in order to survive in an environment in which there is a high probability of total failure. Conversely, with a high probability for successful reproduction, optimal germination fraction is high and the ability to survive for a long time becomes less important. Of a particular interest is the relation between G and D. When G,,, is close to one, due to high probability of high yield, the effect of D on the growth rate is negligible (Fig. 1). Thus, in such an environment there will be almost no selection for seeds with better ability to survive and D would remain high. Under conditions of increasing probability of no yields, selection will act to decrease G, following which the advantage of decreasing D will become greater, so that D will be also decreased. The same relations operate in the opposite direction. Where D is very high because of any combination of external or internal factors there would be a strong selection to keep G close to P,. Where D is low, and specially with a low Y, G,,,,, may be much lower than P,. We can expect, therefore, to find in seeds and spores in their natural environment a high positive correlation between the fraction which germinates and the fraction of those that do not germinate which does not survive each generation. Such a relationship is indeed well known in seeds (Mayer & Poljakoff-Mayber, 1963). We can also expect to find the fraction that germinates every year to be approximately equal to the probability of producing a high yield, and the fraction which does not germinate to be approximately equal to the probability of total or near total failure to produce seeds. The results of our model have nothing to say about one important characteristic of population growth, which is the prediction of the actual or the most probable sequence of sizes as the population goes on growing in time. Even the very important measure of the deviation from the expected longterm size, usually measured by the variance, is not given. It can definitely be said that for each long-term growth rate there is an infinite number of possible sequences and an infinite number of different variances, all leading to the same limit of growth expectation. It is intuitively clear, for example,

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that in the case of two-valued Yi, a combination of low Py and high Y will be expected to result in wider fluctuations and higher variance than a combination of high Pr and low Y having the same long-term growth rate expectation. It is hoped that a model will be constructed from which it would be possible to predict the variance of the growth expectation at any point in the time sequence and the general form of the trajectory. Supported by the United States Public Health Service grant GM-10718 (03). I would like to thank Professor H. Von Foerster, head of the Biological Computer Laboratory, for his encouragement and stimulating discussions, and Dr Klaus Witz of the Department of Mathematics, University of Illinois, for his helpful suggestions and criticism. It is also a pleasure to thank Mr B. Lipnitzky for his assistance.

REFERENCJSS MAYER, A. M. & POUAKOFF-MAYBER, A. (1963).Chapter7 ia “The Germinationof Seeds”.

Oxford, New York: PergamonPress.