The complex relationships of a crop with the pest, its natural enemies, and the climate factors exist in all the ecosystems, but the mathematic models has studied only some components to know the relation cause-effect. The most studied system has been concerned with the relationship pest-natural enemies such as prey-predator or host-parasitoid. The present paper shows a dynamical system for studying the relationship host-parasitoid (

The study of the biological control is of obligatory inclusion in the knowledge of the relationships between ecosystem factors to maintain the equilibrium of the components. In agriculture, a variation of this relation carries on an inequilibrium favorable to the pest. In these cases, an additional meddling is necessary for the control [

For this, models of the relationship occurring in an agriculture ecosystem have been developed in order to find the moment and densities permitting an inequilibrium. The Lotka-Volterra model shows the dynamical relation [

Particularly, the Nicholson and Bailey model, or modifications of it, is used to describe the relation host-parasitoid [

With the data of the samples, a statistical model can be found that proves the relationship host-parasitoid, and in the same way, the limit density for the control can be calculated.

In particular, the relation between

The present work is aimed at reducing the number of equations in the system guiding the study to the relation host-parasitoid and using the random sequential samples to do regression models to estimate the parasitoid percent knowing the density host in the crop.

Even when the host has more than one natural enemy, the subsystem plant-host-parasitoid can be studied. This system is different from the system prey-predator [

Three-trophic system crop-host-parasitoid. The direction of the arrows indicates a direct negative influence by one component on another. Here,

The parameters in the models are:

Using Derive 6.0, the equilibrium points

For (

As always,

The most important point is (

Conditions to obtain a stable equilibrium can occur but, in this case, all the eigenvalues must be negative It is important to know the conditions in which all the positive coordinates of (

We used Matlab to solve (

Stable solid line cycle for

Figure

The stable limit cycle for the host-parasitoid relation.

A simple change in the parameter

Host dynamics with

To find a model to estimate the parasitoid percent knowing the pest density, weekly samples were taken during two years (from April 2007 to March 2009). The crop sampled was

Linear and nonlinear models were proved to find a logical relation host healthy-parasite host. The models used were: logistic, saturate growth, exponential, Weibull’s model, geometric, logarithmic, potential, sinusoidal, Gauss’ model, MMF model, and all the models of first, second, third, fourth, and fifth degree polynomials. The analyses were done for one year, the best model was selected using the determination coefficient, and the standard error obtained three models finally. The parameters in the polynomials models were estimated using least-square method, and in the nonlinear models, the Gauss-Newton method was used. The host density whose increment does not produce changes in the parasitism percent was calculated; in this moment the liberations of parasitoid is necessary.

For this analysis, the software CurveExpert 4.0, Derive 6.0, and the Statistical System SAS 9.0 were used.

In 2007 and 2008, the best model was the lineal model because the parasitoid percent grows while the

Relation between

The model

It is possible that the host samples are not available, but if the relationship between the populations and the climate factors is known, an algorithm can be done to predict the host level, and using (

In Cuba, the climate has cyclical periods, and its curve can be estimated using time series. Establishment of Fourier series [

Obtaining of the estimated temperature using Fourier series

Obtaining of the estimated relative humidity using Fourier series

Finding of a function that relates the host density with temperature and relative humidity

Estimation of the parasitoid host density using (

The dynamical system with trophic relation is an excellent estimate of the relationships in the ecosystem. Scott and Lawrence [

Particularly, a regression model could be used to predict a parasitoid host level for a number of hosts. An MMF model was the best model and more than 1200

De Dios et al

Cause-effect relation present in the ecosystem by De Dios et al. [

New proposal of nonequidistant cause-effect relation present in the ecosystem.

It is concluded that the dynamical system can be used to predict the behavior of the host, and the nonlinear regression shows that 1200 nymphs by plant indicate that is necessary applied a control method.