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This paper uses a methodology based in Fuzzy Sets Theory in order to describe the interaction between the prey, Aphis glycines (Hemiptera: Aphididae)—the soybean aphid, and its predator, Orius insidiosus (Hemiptera: Anthocoridae) and to propose a biological control to soybean aphid. Economic thresholds were already developed for this pest. The model includes biotic (predator) and abiotic (temperature) factors, which affect the soybean aphid population dynamics. The dynamic model results in a fuzzy model that preserves the biological meaning and nature of the predator-prey model. The paper also includes a comparison between the fuzzy model and real data reported in the literature. Subsequently, we propose a biological control to soybean aphid by another fuzzy rule-based system. This model has allowed to predict timing and releasing number of predators for soybean aphid biological control. On the one hand, the soybean aphid has still not found in Brazil. Therefore, before any eventual invasion, a predictive model to enhance biological control program is desirable. On the other hand, the soybean aphid has become the most devastating insect pest of soybeans in the United States. Brazil is the second largest exporter of soybean at present, after the USA and before Argentina. According to the Bureau of Agriculture of the USA, it has been estimated that Brazil will be the largest soybean exporter in 2023.

The soybean aphid, Aphis glycines (Hemiptera: Aphididae), is an invasive herbivore new to North America. It was first discovered in North America in Wisconsin in late July 2000 infesting soybean crop. High soybean aphids densities damage soybean plants

by reducing plant height, pod number and total yields, and yield reductions measured from grower strip trials have ranged from 12 to 45% [

Predation is an example of the interaction between two populations that results in negative effects on growth and survival of one population (prey) and in positive or beneficial effects to the other (predator). A predator is an organism that hunts and kills other organisms for food [

・ If the number of predators is small, the number of prey increases;

・ If the number of predators is large, the number of prey decreases;

・ If the number of prey is large, the number of predators increases;

・ If the number of prey is small, the number of predators decreases.

According to these hypotheses [

Next we develop brief reviews of the concept of fuzzy set and fuzzy rule-based system, and we detail the fuzzy model suggested in this paper. Fuzzy Sets and Fuzzy Logic have become one of the emerging areas in contemporary technologies of information processing. Fuzzy Sets Theory was first developed by Zadeh [

The fuzzification is the process in which the input values of the system are converted into appropriate fuzzy sets of their respective universes. It is a mapping of the domi-

nion of the real numbers led to the fuzzy dominion [

Finally, in defuzzification, the value of the output linguistic variable inferred from the fuzzy rule is translated to a real value. The output processor task is to provide real- valued outputs using defuzzification, which is a process that chooses a real number that is representative of the inferred fuzzy set. A typical defuzzification scheme adopted in this paper is the centroid or center of mass method [

This work describes a method based in Mamdani fuzzy control [

The variables of the fuzzy system are number of prey -x-, number of the predators -y- (input variables) and their variations (output variables). The fuzzy sets of the input variables, defined by experts, are {small, small medium, large medium, large} (that is, the triangular functions defined in the interval [0,1000] for the number of prey and in the interval [0,0.3] for the number of predators) and the fuzzy sets of the output variables are {small positive, large positive, small negative, large negative} (that is, the triangular functions defined in the interval [−0.4,0.4] for the variation of the prey -

Considering the hypotheses of the predation and qualitative information from expertise, in particular by an entomologist, allow us to propose rules that relate to the variables of state, with their own variations. We have elaborated 16 rules of the system (S1):

・ If (x is small) and (y is small) then (

・ If (x is small-medium) and (y is small) then (

・ If (x is large-medium) and (y is small) then (

・ If (x is large) and (y is small) then (

・ If (x is small) and (y is small-medium) then (

・ If (x is small-medium) and (y is small-medium) then (

・ If (x is large-medium) and (y is small-medium) then (

・ If (x is large) and (y is small-medium) then (

・ If (x is small) and (y is large-medium) then (

・ If (x is small-medium) and (y is large-medium) then (

・ If (x is large-medium) and (y is large-medium) then (

・ If (x is large) and (y is large-medium) then (

・ If (x is small) and (y is large) then (

・ If (x is small-medium) and (y is large) then (

・ If (x is large-medium) and (y is large) then (

・ If (x is large) and (y is large) then (

Hence, we have obtained the variation rates of the populations from Mamdani Inference method and the fuzzification of the center-of-gravity. Regarding the influence of the temperature abiotic factor in the population of soybean aphids, [

Using an adjustment from least-squares method, we have estimated an intrinsic rate of increase:

Temperature | 20˚C | 25˚C | 30˚C | 35˚C |
---|---|---|---|---|

Intrinsic rate of increase | 0.368 | 0.474 | 0.375 | −0.383 |

where T is the temperature (see

In the numerical simulations performed, we have observed the variation of the number of the prey and the number of the predators considering the temperature. In order to achieve this, we have considered an initial number of aphids,

that is, Euler’s Method, where h is the increment.

Numerical SimulationsWe have obtained the variation of the number of the prey and the number of the predators in the numerical simulation. Let

Simulations of the trajectories produced by the fuzzy model follow the steps below:

1) Given an initial number of the prey population (

2) The fuzzy rule-based system of the predator-prey type gives the output data:

3) Given an initial temperature T (mean temperature in

4) From Equations (2), we find

5)

We used the data available in [

The phase plane by this fuzzy system (dashed line) and the phase plane by the real data (black points) [

Now, we have a set of real data and another of simulated ones (

Deviance measures are applicable when real and simulated data can be paired at the same time. These are based on the differences between the simulated and real

values. According to [

where

The value MRE for 2004 data (

In next section we use this mathematical model to predict the timing and the number of predators released for soybean aphid control.

Biological control is the use of natural enemies to control insect pests. It is defined as the regulation by natural enemies of the population density of another organism at a lower average than would otherwise occur ( [

The control of pests below the level at which they cause economic thresholds by the deliberate introduction of exotic natural enemies is referred to as classical biological control [

In this way, we have proposed biological control via fuzzy system, since there is no model available for biological control of soybean aphids.

A method to determine if an insecticide treatment is warranted is a conventional method using the 250 aphids per plant economic as the threshold with 80% of the plants infested and aphids population increasing [

Considering the above-mentioned hypotheses, we have observed the variation of the number of prey and the number of predators in the numerical simulation. Let

In this way, we propose the following rule base of the fuzzy biological control system (S2):

1) If (x is small) and (y is small) then (

2) If (x is small-medium) and (y is small) then (

3) If (x is large-medium) and (y is small) then (

4) If (x is large) and (y is small) then (

5) If (x is small) and (y is small-medium) then (

6) If (x is small-medium) and (y is small-medium) then (

7) If (x is large-medium) and (y is small-medium) then (

8) If (x is large) and (y is small-medium) then (

9) If (x is small) and (y is large-medium) then (

10) If (x is small-medium) and (y is large-medium) then (

11) If (x is large-medium) and (y is large-medium) then (

12) If (x is large) and (y is large-medium) then (

13) If (x is small) and (y is large) then (

14) If (x is small-medium) and (y is large) then (

15) If (x is large-medium) and (y is large) then (

16) If (x is large) and (y is large) then (

We obtain the quantity released of the predators from Mamdani inference method and the fuzzification of the center-of-gravity represented in

From

in

The evolution of the population is contingent on the prey and predators over time given by the fuzzy model are depicted in

Now, it is necessary to determine the day that needs to be done the deliberate introduction of predators, according to the number of iteration in the computer simulations.

Linear regression analysis using the least-squares method was used to estimate the days aforementioned are given by the equation:

where t is the number of iterations in the computer simulations.

In the case of

In the case of

We propose the fuzzy model to simulate soybean aphid population dynamics that in-

cludes biotic (predator) and abiotic (temperature) factors. This model is based on fuzzy system (S1) that relates the input variables (number of prey, number of predators) with the output variables (variation of prey and variation of predators) and Equation (1). The use of a fuzzy rule-based system instead of usual differential equations characterizes the classic deterministic models, because many parameters of the differential equations are not available. In addition, the qualitative information and data reported in the literature [

Results demonstrate that the fuzzy mathematical model provides the phase plane that preserves the characteristics of the phase plane of a predator-prey model, that is, dynamic model results in a fuzzy model that preserves the biological meaning and nature of the predator-prey model.

As demonstrated by the above simulations, we can see that the temperature influences the growth of the aphids population [

Next, we use this mathematical model to predict the timing and the number of predators released for soybean aphid biological control.

The concern about the environment has been increasingly important. Currently, actions aimed at sustainable management of natural resources are goals.

In general, pesticides are toxic, harmful to human health and the environment. One of the most common problems is the contamination of soil, groundwater, rivers and lakes. When the pesticide is used, it intoxicates all life present. Studies show the decrease in the number of pollinating bees and the destruction of bird habitat in environments where pesticides are used [

The abusive use of insecticides may lead to an increase number of pests because pests become more resistant, requiring stronger pesticides that will damage the environment even more and will kill the pests’ natural predators [

There is some evidence that biological control can be favorable where a pest is causing damage in a plantation [

Considering that the time between the appearance of aphids on the plant and the maturation stage of grains is 60 days on average, the Figures 10 and 11 demonstrated the importance of the biological control via predation and a constant monitoring of the plantation from the initial arrival of soybean aphids. In addition, the most likely alternative, chemical control is undoubtedly damaging to the environment ( [

Thus, in this paper we have proposed a biological control based on the introduction of predators in the plantation when the prey population exceeds the economic damage threshold.

This study has suggested that the use of the Mamdani Control Fuzzy in Ecology may represent the interaction among species in the environment where the available data are few or qualitative. We have used intuitive hypotheses of the dynamics of aphids- insidious flower bug and data reported in the literature to elaborate the model without explicit differential equations. It was clear that temperature was an important factor on the growth of aphids population. We would like to highlight the advantages of using fuzzy rule-based models compared to the deterministic models (differential equations, for example):

・ The input and output sets of fuzzy rule-based systems may be easily defined by experts, that is, specialists who may know when the population of a particular species is small, large and so forth of the predators population over time.

・ We have used a rule base instead of systems given by equations and this avoids the difficulty of obtaining the parameters.

・ If it is necessary to know the parameters, they may be obtained through a curve fitting of the solution generated by the fuzzy model. That is, the parameters may be obtained through a curve fitting procedure from the solution obtained from the fuzzy rule-based model.

・ We suggest using this as a policy to biological control of soybean aphids because the fuzzy biological control model provides how often and how much to add the predators in the plantation, instead using insecticides, in a simple, intuitive and a direct way.

・ We will develop further studies on simple and specific method using a fuzzy rule- based system to help the implementation of an integrated pest management system of this one.

The first author acknowledges the Coordination for the Improvement of Higher Education Personnel (CAPES) and the second author acknowledges the National Council for Scientific and Technological Development (CNPq), project numbers 305862/2013- 8, for the financial support.

Peixoto, M.S., Barros, L.C., Bassanezi, R.C. and Fernandes, O.A. (2016) On Fuzzy Control of Soybean Aphid. Applied Mathematics, 7, 2149-2164. http://dx.doi.org/10.4236/am.2016.717171