Fuzzy parameters in a partial differential equation model for population dispersal of leaf-cutting ants

Abstract

This paper proposes a model for the reoccupation of ants in a region of attraction, using evolutive partial differential diffusion–advection equations, in which the population dispersion and velocity in directions x and y are fuzzy parameters. The domain under study contains an attractive region, representing areas with high concentrations of palatable host plants with better nutritious qualities. The algorithm developed here uses information about the foraging behavior of a leaf-cutting ant colony of the Amazon region in northern Brazil. In the first model, we determined the numerical solution of the partial differential deterministic equation with constant dispersion and velocity, without incorporating uncertainties that may occur in this type of biological phenomenon. In the second model, we calculated the numerical solution of the partial differential equation, determining the dispersion at each iteration and for each triangular finite element. The dispersion was determined from a fuzzy rule-based system that depends on the number of individuals of the population and on the characteristics of the terrain. In this model, we also considered the velocity along the x and y directions as a fuzzy parameter that depends on the ant movement characteristics on the terrain in the y and x directions, respectively. The two models produced solutions consistent with ant behavior in response to the proximity of an attractive region, as stated in the papers by Shepherd (1982) [2] and Vasconcelos (1999) [3], but the fuzzy model incorporates uncertainties that do occur in the biological phenomenon under study.