A statistical model of the reproduction of aphids

Summary The logistic function has been generally used to describe the reproductive process of a “population” of animal. However, this model can not give us any information about the reproductive process of “individuals” in the population. In this study a statistical model on the basis of the reproduction of individuals of barley aphid is presented to find the proportion of the mature individuals, the heterogeneity in reproductive ability of the aphids, etc. The model is constructed as follows: The probability that j insects are found on a plant at time t 0 is represented as Q(j) . The probability that h individuals of j have reproductive ability, say, mature individuals, in the period t 0 to t 1 is represented as B(h/j) = j C h w h (1− w ) j−h , where w is the proportion of mature individuals. In a population with a homogeneous reproductive ability, the probability that each parent lays i offspring in the period t 0 to t 1 is represented as P(i/m) = e −m m i / i !, where m is mean. And, in a population, m changes according to the gamma distribution. Hence the probability that a parent lays i offspring between t 0 and t 1 is represented as P ( i ) = q − k( i + k − 1 ) ! i ! ( k − 1 ) !p q i , where p and k are parameters of negative binomial distribution. The probability that h parents on a plant lays s offspring is represented as N e ( s / h ) = q − h k( s + h k − 1 ) ! s ! ( h k − 1 ) !p q s . From the assumptions mentioned above, the probability that s offspring are to be found at time t 1 on a plant with the original j individuals at time t 0 is represented by R ( s ) =Q ( 0 ) + Σ j = 1 ∞ B ( 0 / j ) + Σ j = 1 ∞ Q ( j ) Σ h = 1 ∞ B ( h / j ) N e ( 0 / h ) f o r s = 0Σ j = 1 ∞ Q ( j ) Σ j = 1 ∞ B ( h / j ) N e ( s / h ) f o r s ⩾ 1.The experimental populations were demonstrated to fit well to the model.