# The Simple Onetrait Oneperiod Model

Plant breeders have two search strategies in their research or inventive efforts. The first of these is the search for 'quantitative' plant traits such as yield. Quantitative traits are controlled by multiple genes (or alleles) and require complex strategies for crossing parental materials and selecting improved cultivars. The second is the search for 'qualitative' traits such as host plant resistance (HPR) to plant diseases or to insect pests. Host plant tolerance (HPT) to abiotic stress (drought, cold, etc.) are also qualitative traits. Qualitative traits are controlled by a single gene (or at least very few).1

Both breeding strategies rely on searching for genetically controlled traits in collections of plant genetic resources (PGRs) which include landraces of the cultivated species (distinct types selected by farmers over centuries from the earliest dates of cultivation and diffused across different ecosystems), 'wild' (related) species and related plants that might be combined.2 PGR collections also include 'combined' landraces including varieties (officially recognized uniform populations of combined landraces often with many generations of combinations). The systematic combining of landraces and evaluation is termed 'pre-breeding'.3 Consider the following representation of the single-trait one-period model. In period 1, the existing breeders' techniques and breeders' PGR collections determine a distribution of potential varieties indexed by their economic value, x. Following Evenson and Kislev (1975), suppose this distribution to be an exponential distribution:

The cumulative distribution is:

The cumulative distribution of the largest value of x (z) from a sample of size (n) is the 'order statistic' (Evenson and Kislev, 1975):

and the probability density function for (z) is:

The expected value and variance of hn(z) are: 1 n 1

Evenson and Kislev discuss the applicability of equations (7) and (8) to plant breeding research. Equation (7) can be derived from a uniform distribution and this is a very general expression for a broad range of functions f (x). Basically (7) can be thought of as the breeding production function with a very simple marginal product:

XS = f( P, Zt, h, tt\TE(Et_„,It_„ ),TEt_s_j (E-s_r,L_-r) (9)

When a measure of the units over which (z) applies is available (e.g. production in a specific ecosystem), V, the value of the marginal product can be computed and set equal to the marginal cost of search to solve for optimal n:

Figure 1.1 depicts f (x) and En(z) for two traits for a single period and shows the optimum.

Fig. 1.1. Single-period search. TD, technological determination point.

Fig. 1.1. Single-period search. TD, technological determination point.