Analysis of failure time in clonally propagated plant populations

ELSEVIER

Analysis of Failure Time in Clonally Propagated Plant Populations JOHN BISHIR Department of Mathematics, Box 8205, North Carolina State University, Raleigh, North Carolina 27695

AND JAMES ROBERDS Department of Genetics and USDA Forest Service, Box 7614, North Carolina State University, Raleigh, North Carolina 27695

Received 30 November 1993; revised 13 May 1994

ABSTRACT A problem originating in forest tree breeding concerns the number of clones needed in clonaily propagated plantings to manage risk of failure due to an unforeseen catastrophic event. In this paper, we present a model for and analysis of time to failure for clonally propagated populations, assuming that in each year there is a chance for attack by an insect or pathogen. We develop the probability distribution of the number of years until population failure, T. A surprising finding is that in some circumstances increasing the number of clones can increase, rather than decrease, the chance of population failure. This suggests that laws, such as those current in the European Community, mandating minimum numbers of clones to be used in reforestation, may not achieve their intended effects, and that further investigation is needed to clarify the situation.

1. I N T R O D U C T I O N In several plant species it has become practicable to establish commercial populations by use of clonal propagules. Using methods of mass propagation, entire plots may be planted in blocks or mixtures containing replicate populations of genetically identical reproductions of a few highly productive clones. In extreme cases, populations may consist of replicates of a single, potentially most productive, clone. The advantage of such practices from a management point of view is that populations are phenotypically uniform and have potential for high yields. A disadvantage is that such populations have reduced genetic diversity compared to those of seed origin. Specifically, there is danger that the

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clones or clone chosen are particularly susceptible to attack by an insect or pathogen which was unforeseen as a problem at the time of clonal selection. Such an attack could result in disastrous consequences for these populations. Examples of catastrophic losses in the presence of genetic uniformity are the southern corn leaf blight epidemics of 1970 and 1971 in the United States [9,20], the severe impact of the appearance of Marssonina brunnea on poplar culture in Italy because of widespread use of the highly susceptible clone 1-214 [4], and the devastating effects of poplar leaf rust on Australian poplar stands, Populus deltoides Marsh [16]. A still unresolved problem in clonal forestry is the number of clones to be employed in reforestation to ensure genetic diversity and protect against risk of extensive loss [2,5]. Several countries have established legal minima for the number of distinct clones that must be used in reforestation [14,15]. However, such mandates are controversial and opinions differ as to what minima, if any, should be imposed. It is our purpose here to address some theoretical issues associated with such processes. Previous research has concentrated on the number of clones necessary to reduce risk of population failure to acceptable levels, assuming that a catastrophic event occurs [6,11,18]. In this paper, we focus on the distribution of the number of years T until a population subject to periodic attacks by insects or pathogens fails, that is, becomes extinct or degraded to the point of having no commercial value. Since all clones, if grown long enough, will eventually succumb to an unforeseen pest or pests, time to failure is an important variable of concern. A failure time that is short is undesirable because then populations will be apt to fail before harvest age. On the other hand, an extremely long time to failure is of little value since populations will tend to reach harvest age long before failure occurs. In Section 2, we present a formal description of our model. Sections 3 and 4 discuss the probability distribution of T, the time to failure, first as the number of clones Q becomes large, and then for moderate values of Q. Numerical examples of possible effects of Q on the distribution of T appear in Section 5. A subsequent paper will present a more detailed description of the biological issues and include extensive computational examples of the theoretical results presented here. That paper also will address the problem of the appropriate number of clones to deploy in mixtures in order to maintain risk of failure below a desired fixed value. 2.

THE MODEL

We begin with a general formulation of the model, one which will accommodate a wide variety of circumstances. To supply plantlets of a

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species for planting a particular area, individuals are selected on the basis of phenotypic traits such as form, rate of growth, or fruit production, from a base population. Each chosen individual serves as an ortet for production of genetically identical individuals by vegetative propagation. Such a collection of individuals derived from an ortet constitutes a clone. A desired number of individuals or ramets of each clone are produced and these are grown in field plantings. Such populations of one or more clones are subject to infestation by insects or attacks by pathogens. For simplicity, we shall simply refer to either as a pest attack. The population from which the individual ortets are selected is assumed to be free of pests that may cause a catastrophic epidemic. We further assume that, relative to susceptibility to a pest, there are r genotypes in this population and that the genes for these genotypes segregate independently of genes controlling the phenotypic traits for which the clones are selected. The simplest model involves a single genetic locus with two alleles, in which case r = 3. Definitions of basic terms are collected in Appendix 1. In any particular year, pests may or may not strike the population. Severity of pest attack will be expressed by a sequence of random variables D l, D 2, D 3 . . . . . where D i is the severity of attack in year i. The D's may or may not be independent, and need not have identical distributions although these would be reasonable assumptions. In any particular year there is positive probability that no pest strikes, that is, that D = 0. In the simplest version of the model, if there is a pest attack, the severity is always the same ( D = 1, say), in which case there is a number A for which

P ( D = 0) = 1 - A

and

P ( D = 1) = A.

(1)

The history of pest attack in the area up to year t >~ 1 is defined to be the "severity vector"

Ht = (D1,D2,...,Dt)

(2)

listing the levels of attack in the first t years. The spatialpattern A in which ramets are planted can be described in various ways. We envision the region in which planting occurs as consisting of a grid of plant locations, so A is a vector which lists for each location the genotype of the clone for the ramet planted there. The vector A could be randomly generated if, for instance, planting of ramets takes place without regard to clone or phenotypic trait, or might be completely determined by planting in a prescribed configuration, for example, replicates of the respective clones might be planted in contigu-

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ous blocks. The most extreme instance of this occurs when all ramets are replicates of a single ortet. Each year individual ramets survive or die, with probabilities that depend on the history of pest attack on the population, on the spatial pattern in which the ramets were planted, on their locations within that pattern, and on their genotypes for susceptibility to pests. Death is possible even in the absence of pest attack. Rather than being classified only as alive or dead, a ramet can be assigned a utility, the probability density of which depends on the above listed quantities. For forest trees, utility is typically expressed as timber value, while for other species, it may be value of fruit, of grass (hay), etc. Response to pest attack is measured by functions vjk, where v~(u, ht,a) represents the probability density of the utility at year t of a ramet of a clone genotype j planted in location k, given observed pest history h t and planting arrangement a. Here, h t = ( dl, d 2..... d t) is a vector of observed severity levels. The corresponding conditional distribution function of the utility U of such a ramet is, for u/> 0,

Fj~( u,ht,a ) = P(U <~u lH t = hi, A = a, genotype = j, location = k) =j~k(y,ht,a)dy.

(3)

A population is considered to have "failed" at time t if its total value falls below a prescribed level. Total value or utility can be measured in terms of monetary value, proportion of surviving ramets, fruit production, etc. To compute the probability distribution of the time T until population failure, and the expection of T, we write

P ( T <~t)= fy~a P ( A = a ) P ( T

<~tlHt=y,A=a)fl4,(y)dy

(4)

and

E(T) = £

A =a)E[TIH, = y, A

=

alfH,(y ) dy,

(5)

where frl, is the multivariate probability density for history of pest attack. Integration is over all t-dimensional vectors of possible pest severities and summation over all possible planting patterns. Equations (4) and (5) provide an overall solution to our problem. Specific details of the distribution of T, however, can be difficult to derive. This is particularly true when ramets are planted in a predeter-

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mined, nonrandom arrangement, for then account must be taken of the individual genotype at each site. Ramets of the same clone tend to be susceptible to the same pests so that, although environmental factors will play a role, ramet fates are not independent if, for instance, clone replicates are planted in contiguous blocks. However, if the spatial pattern does not matter, for example, if pests spread only from sources external to the population and environmental contributions to susceptibility at the different plant locations are independent of each other, then properties of T are more easily analyzed. In this case, once H t is specified it is reasonable to assume that ramets are independent in their response to pest attack. Even then, however, individuals are unconditionally dependent because the same H t values apply to all ramets within the population. In the remainder of this paper, we investigate theoretical aspects of the distribution of T, assuming no dependence on spatial pattern and that, given a particular disease history h t, ramets at time t are independent in their response to pest attack. Appendix 2 summarizes the notation that will be used. A future paper will present computational results corresponding to some particular situations in which this assumption obtains. The case in which spatial pattern matters still needs investigation. 3.

T H E DISTRIBUTION OF T W H E N T H E N U M B E R OF CLONES IS L A R G E

The legal mandates mentioned in the Introduction specify minimum numbers of clones to be used in reforestation [14,15]. The number required depends on the size of the region and other factors, but the underlying assumption that increasing the number of clones will decrease the chance of population failure is unquestioned. It is of interest, then, to look at implications of our model as the number of clones is increased. Suppose Q ortets are chosen from a population in which there are r different genotypes relative to susceptibility to pest attack. For j = 1,2 ..... r, let pj = proportion of ortets of genotype j in the base population. (6) Ortets are chosen for observable phenotypic traits. Genotypes associated with susceptibility to pests are not observable, so actual proportions of these are unknown. However, our assumption that genotypes for susceptibility are independent of the traits for which ortets are chosen means that, for each j, the expected frequency of genotype j is

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pj, regardless of criteria used to select ortets, l e t Qj be the number of ortets chosen that are of genotype j. Then the strong law of large numbers (see, e.g., [1]) implies that as Q becomes large, the proportion, Qj/Q, of ortets of genotype j, converges to pj, for each j, with probability 1. An essential feature of clonal culture is that each ortet chosen is replicated a large number of times to create a clone, a collection of individuals of identical genetic makeup. Thus, the number of clones is the same as the number of ortets, and we shall henceforth refer to clones, rather than ortets. Let R be the total number of ramets generated for all the clones, and let R i be the number of ramets that are of genotype j(1 ~
1 ~Nj,= ~

Nt "-R=Rj=I

NitRj j = l Rj R "

(7)

By the strong law of large numbers, each N]t/R j converges, with probability 1, to 6j(ht), where

t~j(h t ) = probability an individual of genotype j dies 1 by time t, conditional on h t .

(8)

If in year t an individual ramet is "dead" when its utility drops below a fixed value Udt, then ~j(ht) = Fjk(udt,ht, a) can be computed from (3), with u = udt. In the present setting, our assumption is that 8j does not depend on spatial pattern a or ramet location k. Also, while this discussion has been phrased in terms of death, a convergence result of similar type pertains to the frequency of occurrence of any event, e.g., loss of commercial value, presence of degradation due to pest attack, etc.

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It follows that for each h t

N t / R converges to ~'~ pjt~j(ht).

(9)

j=l

Thus, the effect of increasing the number of clones without bound is to ensure that, in the presence of a particular pest history ht, the proportion of ramets that die by year t is close to

xt(ht) = ~ pjrj(ht).

(10)

j=l

Thus far in this section we have considered time (year) t to be fixed. By letting t vary, we obtain the distribution of random variable T, as Q ~oo, conditional on observed disease history h r Following Roberds et al. [18], we now define T, somewhat more precisely than before, to be the number of years until a specified proportion [3 ~<1 of the original ramets are "dead," at which time the population is no longer commercially viable and must be replanted. We assume that for each genotype j and pest history ht, the probability of death t~j(ht) is zero when t = O, and strictly increases to 1 as t goes to infinity. Thus, 0 = x 0 < x~ < x 2 < •.. and lim t _,=x t = 1, where x t =- xt(h t) is defined in (10). Let m = smallest integer >I [3R.

(11)

The population fails in t years, that is, T ~
P ( T <~t) = P ( ~ >1m ) = P( N t / R >1 [3 ).

(12)

Since N t / R converges to xt with probability 1, the last probability in (12) is, in the limit as the number of clones goes to infinity, and conditional on a particular ht, equal to zero if x t /[3. We summarize these results in T h e o r e m 1, parts (i) and (ii)(a) of which follow immediately. Part (ii)(b) is proved using the central limit theorem [1, p. 254] rather than the strong law. Details are omitted.

THEOREM 1 Assume ramet responses do not depend on the spatial planting pattern. Conditional on any particular pest history h t (i) I f [3 = 1, then for each fixed t,

P(T<~t)-~0,

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JOHN BISHIR AND JAMES ROBERDS as Q goes to infinity. Moreover, E(T) ~.

(ii) / f / 3 < 1, there are two cases: (a) I f no x t equals/3 ( x t is defined in (10)), then as Q ~oo, P(T=~)~I and

E(T) ~ , where tt3 is the smallest integer value o f t for which x t ~ /3; (b) I f x t =/3 when t = tt~, then as Q ~ 0% P ( T = t t 3 ) - - * 1,

1

P ( T = t I 3 + I ) ~ - ~,

and 1

E ( T ) ~ tt3 + 7. In this theorem, all expectations and probabilities are conditional on h r Since x t depends on ht, so does tt3. In the special conditions that h t = 0 (i.e., there is no pest attack during the first t years), and there are no deaths except from pests, then tt3 = ~ in part (ii) (a). Theorem 1 asserts that if Q is large and pest history h t is known, then time of population failure is predictable. However, pest history is itself a random process, so the actual time to failure of a clonally propagated population will still be variable. 4.

DISTRIBUTION OF T W H E N THE N U M B E R OF CLONES IS NOT L A R G E

We continue to assume independence of spatial pattern and that, conditional on a particular pest history ht, ramet deaths are independent. Suppose for the moment that a single clone, consisting of C, ramets, is generated and is of genotype j, the probability of which is pj. Then the number of ramets that die by year t, Nit, is binomial (C1, 6j(ht)), where 6j is defined in (8). Since C 1 is typically large, Nit is approximately normal with mean /xj = C 1~j and variance O32 = CI ~j(1 61.). Dependent only on h t, then, the distribution of Nit is closely approximated by a weighted sum of N(/zj, o3.2) distributions, the respective weights being the pj, j = 1, 2 ..... r.

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If Q > 1 clones are used, each experiences a number of ramet deaths having a distribution similar to that just described. While the respective numbers of ramets generated C 1, C 2 ..... C O may differ, legal mandates require near equality [14]. Thus, we would expect the total number of deaths, being the sum of numbers associated with the respective clones, to be approximately normal. Theorem 2 indicates that under reasonable assumptions this is the case. The proof consists of noting that the numbers of ramet deaths associated with the individual clones are uniformly bounded, and that the variance vt given in (14) below goes to infinity as Q ~ oo. Hence, Lindeberg's conditions for the central limit theorem [1, pp. 254-255] are met. THEOREM 2 A s s u m e ramet responses do not depend on the spatial planting pattern. Further, assume there exist positive constants y, 6, and M , with y < 1, such that for i = 1, 2 ..... Q,

(1-y)M~
(13)

where C i = n u m b e r o f ramets planted from clone i. Let R = ~ = 1Ci be the total number o f ramets planted, and 6j and x t - xt(h t) be as given in (8) and (10), respectively. Conditional on a particular pest history h t

(i) The total number A[t o f ramet deaths by time t is approximately normal, with m e a n Rx t and variance

14, (ii) Alternatively, the proportion o f ramets dead by time t, A[t / R , approximately normal, with mean x t and variance R - a v r

is

If we think of M as denoting the average number of ramets planted per clone, the inequalities in (13) represent a quantification of the legal requirement that in mixtures clones should be present in "equal proportions" [14]. Also, because ]~/a=1 ( C 2 / R 2) ~ Q-l((1 + 6 ) / 1 - y ))Z l / Q = 1 / Q , F_,j=lpj6 r j2 _ x 2 < x t ( 1 - x t ) < ~ l / 4 , and R is large, V a r ( N t / R ) = v t / R 2 typically would not exceed 1 / 4 Q . Thus, N t / R should be close to its expectation, x r 5.

CHANCE OF STAND F A I L U R E AS A FUNCTION OF Q - - T H R E E EXAMPLES

As indicated in Theorems 1 and 2, the effect of increasing Q is to narrow the distribution of actual proportions of dead ramets. In the

JOHN BISHIR AND JAMES ROBERDS

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presence of a particular disease history ht, if x t < / 3 , w h e r e x t is d e f i n e d in (10), we e n s u r e that the p o p u l a t i o n survives t h r o u g h time t by choosing a large n u m b e r of clones. I n this situation the o p t i m a l policy is to choose Q as large as is practically feasible t h o u g h the probability of p o p u l a t i o n survival can be large even for relatively small n u m b e r s of clones as in T a b l e 1. O n the o t h e r hand, if xt > / 3 , choosing a large n u m b e r of clones t e n d s to e n s u r e failure of the p o p u l a t i o n , just the opposite of what we h o p e to accomplish. I n this case, a smaller n u m b e r of clones will provide a larger probability of stand viability at the desired harvest time. Again, these s t a t e m e n t s assume a fixed disease

TABLE 1 Probabilities for Population Failure at or Before 50 Years, P(T <~50), When Various Numbers of Clones Are Selected from Clonal Collections Having Allele Frequencies z = 0.4, 0.62, or 0.7 for the Recessive Susceptible Allele A Allele Frequency* Number of Clones Q 1 2 3 4 5 10 15 20 25 30 35 40 45 50 60 75 100 oo

z = 0.4 Indept Dept

z = 0.62 Indep Dep

z = 0.7 Indep Dep

0.160 0.294 0.089 0.123 0.117 0.042 0.019 0.011 0.005 0.003 0.002 0.001 0.001 0.001 0.000 0.000 0.000 0.000

0.384 0.621 0.363 0.498 0.505 0.486 0.483 0.485 0.481 0.482 0.480 0.483 0.480 0.480 0.479 0.470 0.475 0.468

0.490 0.740 0.508 0.672 0.660 0.719 0.763 0.803 0.824 0.845 0.863 0.881 0.888 0.901 0.917 0.937 0.955 1.000

0.313 0.526 0.234 0.371 0.184 0.180 0.073 0.075 0.035 0.037 0.019 0.021 0.011 0.012 0.008 0.003 0.002 0.000

0.496 0.745 0.494 0.681 0.493 0.612 0.488 0.572 0.485 0.553 0.483 0.541 0.481 0.532 0.525 0.476 0.501 0.468

0.583 0.825 0.622 0.799 0.651 0.801 0.740 0.830 0.793 0.857 0.830 0.878 0.857 0.895 0.909 0.909 0.944 1.000

*Values are given for the case: UAA 0.1, VAa = Caa = 0.98, /3 = 0.5 and A= 0.2. t,,independent,, (Indep) means all ramets survive or die independently. "Dependent" (Dep) means all ramets of a particular clone die if one does, and all survive if one does; relative to our computations, this is equivalent to using only one ramet of each clone. True probabilities for population failure should lie between the values for these extreme cases. =

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history. The stochastic uncertainty associated with real pest sequences will tend to soften the all or nothing nature of this description. Nonetheless, it appears that an assumption that more clones are always better than fewer is invalid and that a legal requirement of a minimum number of clones can, in particular instances, have unintended consequences. To illustrate the possibilities, values for P ( T <~50) were computed for a single locus, two allele (A and a) model in which allele A is recessive for susceptibility to pest attack. In order to avoid unnecessary computational difficulties, we chose the simplest setting, i.e., (i) Individual ramets are classified as alive (have utility) or dead (no utility); (ii) The virulence of pest attack is always either 0 or 1, with probabilities given by (1); (iii) If no attack occurs, chance of ramet death is small--for simplicity, we assume it to be zero (effectively, there is no death, except from pests); (iv) If an attack occurs, a ramet of type j is either unharmed (probability vj) or dies (probability 1 - vj); (v) The probability vj depends only on the present and not on the past history of pest attack. Table 1 contains probabilities of population failure at or before 50 years for varying numbers of clones selected from three clonal populations having respective allele frequencies z = 0.4, 0.62, and 0.7 for A. Failure was taken to mean that at least half the planted ramets died within 50 years, that is,/3 = 0.5 in (12). The probability of pest attack in a particular year was A = 0.2, while probabilities of surviving a pest attack were VAA = 0.1 and UAa = U a a = 0.98 for the three genotypes. In the "independent" case, ramets in the population at a particular time survive or die independently, as assumed in Sections 3 and 4. For comparison, failure probabilities are also given in the "dependent" case in which all ramets of a particular clone die if one does. In actual plant populations, failure probabilities should lie somewhere between the values for these extreme cases. Formulas used in the computations can be found in Appendix 3. The results illustrate three different behaviors that can occur as the number of clones is increased. At z = 0.4, probability of failure is greatest when the number of clones Q is small, exhibits a general decline as Q increases, and approaches a value near zero as Q approaches infinity. A contrasting pattern occurs when z = 0.7; the probability of failure shows an overall increase as Q increases and approaches a limit close to 1 as Q becomes large. A third pattern is

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observed at z = 0.62. For low Q, probability of failure takes on moderate values but does not change substantially as Q increases; instead, it approaches an intermediate value, 0.468, as Q approaches infinity. What Table 1 shows is that under certain circumstances, use of fewer clones can result in less risk. These circumstances occur when the expected result is failure, more precisely, when the probability of ramet death before time of harvest (here, 50 years) exceeds the maximum acceptable proportion of deaths, /3. Theorem 1 and the discussion leading to (9) and (10) apply when the number of clones is large, and in this situation it pays to increase the variance, or uncertainty, so there is greater chance of success. This is done by decreasing the number of clones. The result is similar to the "risk-prone" versus "risk-averse" strategies in ecological theory [13, p. 75; 17, p. 385; 19, Chap. 6]. That is, when a "safe" strategy is almost certain to result in failure, it is better to use a risky strategy that has at least some chance of success. Probabilities of failure for the independent and dependent ramets models follow similar patterns with increasing Q although values differ for the two models. For all cases, convergence is nonmonotonic, a pattern also noted in [11]. The exact pattern of convergence depends on the value o f / 3 ; details will appear in a later paper. Solutions to the problem of number of clones needed to hold chances of population failure below a prescribed level a differ for z = 0.4 and the other two allele frequencies studied. For example, if a = 0.05, a population of 25 clones will have an acceptable probability of failure for z = 0.4, but when z = 0.62 or z = 0.7, a risk of o~ = 0.05 is unattainable regardless of the number of clones included. At z = 0.7, the smallest probability of failure occurs in populations consisting of a single clone, in contrast with the suggestion of Kleinschmit [7]. For z = 0.62, however, probability of failure retains moderate values throughout the entire range for Q. It is of interest to note that while Q = 25 results in an acceptable probability of failure when z = 0.4 and a = 0.05, this Q-value is substantially below the minimum number of clones required by law for multiclonal mixtures in the Federal Republic of Germany (for principal species, 100 clones are requires for special sites, but only 20 are required for less important species that are grown on small areas [15]) and for conifer clonal mixtures in Sweden that are tested under the least stringent requirements (67 clones). It also falls under the minimum number of 29 clones required in Sweden for clonal mixtures that are tested under the most rigorous restrictions employed. Clearly, the merits of using a large number of clones depend heavily on the allele frequency z. Since z would typically be unknown in a given clone pool,

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these results raise some concern about the possible consequences of laws mandating minimum numbers of clones. 6.

DISCUSSION

Previously studied models have addressed the question of numbers of clones; however, none has incorporated the effect of time. The model described in this paper includes time to failure as a variable and provides a new framework for analysis directed toward determination of the appropriate numbers of clones to be used in plantings. Thus, it opens new avenues of attack on this and related questions. For instance, it has been suggested that numbers of clones necessary to assure acceptable risks in forest populations are lower for species with low harvest ages than for those with harvest ages of greater magnitude [12]. This conjecture can now be subjected to analysis using the time to failure model set out here. In fact, comparisons of numbers necessary to meet a prescribed risk are possible for any array of time profiles. A similar situation exists with respect to spatial pattern for deployment of clones. Opinions differ regarding the advantages and disadvantages in clonal forestry of configurations involving random mixtures of clones as opposed to mosaics of monoclonal patches [3, 8,10,12]. These views are primarily based upon perceived operational efficiencies, factors influencing growth, and opportunity for production of alternative products, with some attention also devoted to effects of spatial configuration on risk of failure [11]. Because of the way in which spatial patterns have been incorporated into the model described here, rigorous analysis of the impact of these effects on probability of population failure is now feasible. For many pest problems, pest populations may increase in size and virulence over time as the pest becomes adapted to the host population. Pests may tend to move from affected individuals to adjacent neighbors and in this manner attacks may spread from initial entry points throughout host populations. Individuals of the host species may be subjected to repeated attacks before their utility is reduced below a critical value that renders them to be considered nonusable. Analyses of population failure for events of this type require that the probability density for utility at time _t for a ramet of a given genotype occupying a particular location be conditional on pest history and spatial configuration. In a pest problem of different type, ramet response to attack does not depend on spatial arrangement. Such is the situation when pests spread from locations outside the host population and when environment effects associated with different plant locations, and which influ-

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ence susceptibility, are independent of each other. Pests that spread by wind and have life cycles that require successful colonization of alternate hosts, plus others that contribute to sudden epiphytotic episodes, often are responsible for attacks of this form. For some pests of this type, individual ramet response to attack may still depend on prior disease history. Our Theorems 1 and 2 are valid for these types of behavior. Finally, for a third type of pest problem, ramet response to attack is not conditional on spatial arrangement or on history of pest attack. This is the form of behavior assumed for the cases illustrated in our three examples. APPENDIX 1. DEFINITIONS FOR BIOLOGICAL TERMS AS USED IN THE DESCRIPTION OF THE MODEL Clone:

The collection of individuals derived by vegetative propagation from a single ancestral individual. Ortet: The ancestral individual that is vegetatively propagated to produce a clone. Ramet: A single vegetatively-produced individual derived from an ortet. All ramets of a clone have the same genetic constitution as the originating ortet. Pest: An insect or pathogen that has potential to attack an individual and cause damage. Genotype: The genetic constitution of an individual with respect to susceptibility to a pest. APPENDIX 2. NOTATION T Q

Time until population failure Number of clones used (same as number of ortets) aj Number of clones of genotype j used Proportion of ortets of genotype j in the base population pj R Total number of ramets planted Rj Number of ramets of genotype j planted Total number of ramets that die by time t Nt Nj, Number of ramets of genotype j that die by time t h, Observed pest history up to year t; a vector listing severities of pest attack in years 1,2 ..... t aj(h,) Probability a ramet of genotype j dies by time t, conditioned on ht x,(ht) Expected proportion of all planted ramets that die by time t, conditional on h t

FAILURE TIME IN CLONAL POPULATIONS /3 A vj

123

Proportion of ramet deaths at or above which the population is deemed to have failed In the simple model used in the examples, the constant probability of pest attack in a particular year In the examples, the probability a ramet of genotype j is unharmed by a pest attack

APPENDIX 3. T H E COMPUTATION OF TABLE 1 Since genotypes Aa and aa have the same survival probabilities, it suffices to know the number, X, of the Q chosen ortets that are of genotype AA. For c = 0,1 ..... Q,

P( X=c) = ( Q )( z2)C(1- zZ) Q-c,

(15)

where z is the gene frequency for allele A. The number Y of years in which pest attack occurs, is also binomial: for j = 0,1,2 ..... 50,

P( Y= j) = ( 50 ) Aj(1- A)s°- J.

(16)

If Y = j, the probability a ramet of genotype AA survives is (0.1) j, while that for a ramet of type Aa or aa is (0.98) y. Suppose X = c and Y = j. If each clone is replicated a large number of times, and ramets survive independently, the strong law of large numbers (SLLN) implies that, with high probability, the proportion of ramets that survive is close to

sj-oc (0.1)j

+

(17)

The population fails if this proportion is less than 0.5. Hence, in the independent case, the probability of population failure is close to 50 Q P(failure) = E • j=0c=0

P(X=c)P(Y=J)lt,j,o.51,

(18)

where P(X = c), P(Y = j), and sj are given by (15), (16), (17), respectively, and lIB 1 denotes the indicator for event B. (In the actual computations for Table 1, we assumed each clone was replicated 1000 times and used the central limit theorem (CLT) rather than the strong

JOHN BISHIR AND JAMES ROBERDS

124

law. However, the results are virtually identical to those produced by (18).) When ramets are not independent, neither the SLLN nor the CLT is applicable, and the formula is more complex. This time P(failure)

= E

E

P(X=c)P(Y=j)

of()

j=0c=0

× ~

i=0

e-c i

k=0

i

(('98)J) (1-('98)i)

k (.1/) (1-(.1y) Q-c-i

]

l[(k+i)/Q<~°'5] "

(19)

Details of derivation are omitted.

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