# Population Invariants, an Application to Relationships in between the Species of Acarofauna on Plum Untreated Orchards

S. CHIPEVA^{1} and J. T. MEXIA^{2}

^{1}* University of National and World Economy, Department of Statistics and Econometrics, BG-1000 Sofia, Bulgaria*

^{2}* Universidade Nova de Lisboa, Faculdade de Ciencias e Tecnologia, Portugal*

### Abstract

CHIPEVA, S. and J. T. MEXIA, 2002. Population invariants, an application to relationships in between the species of acarofauna on plum untreated orchards. *Bulg. J. Agric. Sci.*, 8: 49–52

Relationships between phytophagous, predator and indifferent acarofauna on plum-untreated orchards were investigated. Since there are no independent variables in the biological systems a multivariate approach to their study must be taken. One of the best-known multivariate techniques is principal component analysis. A position symmetrical to the usual practice to concern the components with larger variances was taken centering our attention on principal components with small variances, the population invariant. It is shown how to use these invariants to express part of the initial variables as linear functions of remaining ones. An application was presented to express the population densities of 7 species of *Acarina* as linear functions of the corresponding densities for the previous observed moment. Population density models, identifying factors and their influence on the density level, were developed for each family. This application leads to short-periods predictive models.

The population density of each of species depends in great extent on abiotic factors, namely temperature and humidity. The analysis presented in the work shows that the population density level of species at the previous observed moment also contributes to determine the real density level in the present moment. Joining the both lines of influencing, by the abiotic factors in one side and by the relationships in between species in another, we can implement more precise and accurate short-period forecasts for expected population density of species. This is a better way to reduce the chemical applications in orchards keeping phytophagous population density up to certain not dangerous level by using the tools of biological pest control.

*Key words:* Principal Component Analysis, Short-period Predictive Models, *Acarina* Species, Population Density