Genetic erosion is the loss of genetic diversity, including the loss of individual genes and combinations of genes. The main cause of genetic erosion in crops is the replacement of local varieties by improved or exotic varieties and species (FAO, 1996). Erosion can occur when a small number of new varieties replaces a larger number of older varieties and/or the newer varieties have a different gene base to the old one.

While some indicators of genetic erosion have been developed (FAO, Annex 1.1; Reid et al., 1993), there have been few studies with quantifiable estimates of the rates of genotypic or allelic extinction of CGRs. This chapter proposes a proxy measure for the rate of erosion of agro-biodiversity: the change over time in the number of potential accessions from in situ sources to ex situ collections. In addition to being a concrete measure, it has the benefit of making the link between in situ and ex situ conservation explicit.

This chapter assumes that genetic erosion takes place in poor countries but generates negative externalities for rich countries, which use agro-biodiversity as an input to breeding new crop varieties. It assumes further that rich countries bear the cost of agro-biodiversity maintenance, and that farms participating in in situ conservation programmes are price takers on the world market.

Let AM0 be the estimate of the number of distinct existing cultivars uncollected at present. Chang (1993), for example, presents estimates of this value for various crops. Let AMt be the number remaining at date t:

where It = investment by rich countries in agro-biodiversity conservation in target areas; St = vector of socio-economic factors in the programme area that are causing AMt to decrease over time (see e.g. Brush et al., 1992); and It and St > 0. The expected signs are 9Amt /9It > 0 and 9Amt/9St < 0. It is assumed tli at 9 AMt/d t < 0.

Let At be the number of these distinct cultivars collected (accessions) and put into ex situ collections in period t. In addition, it is likely that cultivars not existing at period 0 may be developed by breeding activities of the indigenous farmers in the programme target area. Let these new varieties be denoted by AN. The production of ANt would be expected to be part of a conservation programme if the programme goes further than just paying farmers to conserve landraces they already have, but allows conservation to be a dynamic process

and encourages the breeding of new varieties using their existing stock as a base. The accessions are functions of the following variables:

where EA and EANt (both >0) are the collection efforts ($) (i.e. amount spent to collect CGRs during t in the programme area, on existing and new varieties, respectively). Note that EA and EANt are ex post accounting values: the search agent spends Et over the period and then calculates how much was spent on At and ANt in simple proportions to the amounts of each retrieved from the field. The expected signs are 9At/9It > 0, 9At/9Et > 0 and 9A/dSt < 0. There is no reason to expect that ANt would not behave in the same way as At in that investment It and Et will turn up both existing and new cultivars, and hence 9 AN/d It > 0, 9 AN/9) Et > 0 and 9AN/3 St < 0.

Of course, AMt, ANt, and At cannot be known with certainty. Since AMt is censored at 0 and discrete, the probability of these levels occurring is best modelled by a discrete distribution such as the negative binomial, at least in small samples.4

This approach differs from Simpson et al. (1996) in which biodiversity prospecting is modelled as a sequence of independent Bernoulli trials, where each species could either yield a 'hit' in the search for a new product, or prove useless. Here instead, in the sequence of independent Bernoulli trials that make up the negative binomial, a 'hit' is simply turning up a new (i.e. uncatalogued) distinct variety.

The probabilities of failure PA t, PANt, and PAMt are assumed to be a function of It and St. Since the true specification of Pt is unknown, Pt is chosen to be logistic, producing a tractable closed form specification. For a random variable xt , this functional form is:

1 + exp kt xt-mt where mt and kt are the mean and scale parameters, respectively. For simplicity, it is assumed that the transformation of (xt — mt)^t = Gt = f (I, St), where 9Gt /9St > 0, 9Pt/dEt < 0, 9Pt/dIt < 0, and 9Pt /9St > 0. Given d&( /9tPt < 0, then 9tF>dPt/9)Pt9It > 0.5

Given that accessions in any period are negative binomial distributed random variables, mean ANt is:

As shown above, AMt is non-increasing, so that e (AMt) = j> 0[1 - argmax{PAM 1,

PAMt>]/argmax{P

AM.t-1, Pam.}- Given the logistic specification for P, e(AMt) can be rewritten as:

where e(AM_1) = tp0 (i.e. PAM0 is set equal to 50%).

Unlike ANt, At is restricted by the previous levels of At and by AM. Let t be the unrestricted value of At, and have a mean value of q>AtQA /PA t, then At is:

t-1 t-1 t-1 AMt - X Ak, AMt - X Ak > 0 and AMt - X Ak <Yt k=0 k=0 k=0 t-1

where AMt — Ej,=o Ak is the uncollected CGRs from AMt that are remaining on the field at time t. Note that since 9Amt /91< 0, the uncollected CGRs can fall to 0 without Hk=0Ak reaching the level AM0, i.e. some potential accessions known at time 0 may be irreversibly lost.

Given annual accessions At and ANt, the size of the ex situ collection at period t is denoted as TE and is simply:

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