Libro de Resúmenes / Book of Abstracts (Español/English)

Resumenes 39
Figure 1: Plano de fase
A predator-prey model to the citrus sudden death
Citrus Sudden Death (CSD) is a disease that has affected sweet
orange trees grafted on Rangpur lime. Researches have shown that the
ducts which lead nutrients generated by the photosynthesis to the roots,
become clogged and degenerated. Without foods, the roots putrefy, the tree
decays and die.
Preliminary studies have suggested that a virus transmitted by
insects known as aphids causes this disease. Among the most known
predators in citrus belonging to the Coleoptera Order and the Coccinelidae
Family, the Cycloneda Sanguinea (ladybug) is an important agent for
biological control because it eats aphids. Research data indicate that each
larva of these predators can consume up to 200 aphids a day and the adults
prey, in average, 20 aphids a day. This means that the degree of predation
of adults is, in average, 10% of the degree larvae’s of predation and that,
therefore, a mathematical model to represent the interaction between
aphids and ladybugs should take into account such difference in the class of
predators.
In this work, we establish a model of type predator-prey, based on
fuzzy rules, to represent the interaction between the aphids (prey) and
(ladybugs (predators) in the citriculture, replacing the standard models,
from the differential equation system.
The variables (inputs) are the rate of quantity of preys (x) and the
rate of potential of predation ( P y ), given by Py = p1
+ 0. 1*
p2
, where p 1 is the
quantity of larvae of this population and p 2 is the population of adults. The
outputs are the variation of prey, (x’), and the variation of the potential of
predation, ( P y'
). The rules of the rule-based fuzzy system can be
exemplified as follows: "If the quantity of preys is high and the potential of
predation is very low, then the variation of preys has a little increase and
the variation of the potential of predation increases a lot."
From the Mamdani Inference Method and the defuzzification of the
Center-of-Gravity, we have obtained the rates of variation of preys (x’) and
potential of predation, ( P y'
). Using a process of numerical integration, the
Trapezoidal Rule, for example, we finally get the population of preys
(aphids) and potential of predation of the predators (ladybugs), whose
phase-plane is illustrates in Figure 1.