using partial derivative notation (∂NS and ∂EC). This concept of environment includes interspecies and ecological effects. Price describes this as follows: Proof Assume there is an n-person population in which the amount of a certain feature changes. Those n people may be divided into groups based on how much of each feature they have. There can be just one group of all n people (consisting of a single common characteristic value) or as many as n groups of one individual each (consisting of n distinct values of the characteristic). Index each group with so that the number of members in the group is and the value of the characteristic shared among all members of the group is . Now assume that having of the characteristic is associated with having a fitness where the product represents the number of offspring in the next generation. Denote this number of offspring from group by so that . Let be the average amount of the characteristic displayed by the offspring from group . Denote the amount of change in characteristic in group by defined by Now take to be the average characteristic value in this population and to be the average characteristic value in the next generation. Define the change in average characteristic by . That is, Note that this is not the average value of (as it is possible that ). Also take to be the average fitness of this population. The Price equation states: where the functions and are respectively defined in Equations (1) and (2) below and are equivalent to the traditional definitions of sample mean and covariance; however, they are not meant to be statistical estimates of characteristics of a population. In particular, the Price equation is a deterministic difference equation that models the trajectory of the actual mean value of a characteristic along the flow of an actual population of individuals. Assuming that the mean fitness is not zero, it is often useful to write it as In the specific case that characteristic (i.e., fitness itself is the characteristic of interest), then Price's equation reformulates Fisher's fundamental theorem of natural selection. To prove the Price equation, the following definitions are needed. If is the number of occurrences of a pair of real numbers and , then: The mean of the values is: {| border="0" cellpadding="0" cellspacing="0" width="100%" |- | align="left" | | align="right" | |} The covariance between the