Professor Subhas Chandra Mukhopadhyay

Exeley Inc. (New York)

**Subject:** Computational Science & Engineering, Engineering, Electrical & Electronic

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Afef Ben Saad
** ^{*}**
/
Ali Hmidet
/
Olfa Boubaker

**Citation Information : **
International Journal on Smart Sensing and Intelligent Systems. Volume 11,
Issue 1,
Pages 1-16,
DOI: https://doi.org/10.21307/ijssis-2018-010

**License : **
(BY-NC-ND-4.0)

**Published Online: ** 03-September-2018

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In recent years, dynamical relationship between species in ecology has been intensively investigated and will continue to be one of the most significant themes. The dynamics of predator–prey’s systems are at the heart of these studies. Such models are generally depicted by nonlinear polynomials and exhibit many complex nonlinear phenomena. In this paper, not only a prey–predator model displaying richer dynamical behaviors is analyzed but also its electronic circuit is also designed via the MultiSIM software proving the very good agreement between biological theory considerations and electronic experiments.

Ecology is the study of interactions among organisms and their environment. There are many useful applications of ecology as natural resource management and city planning (Laktionov et al., 2017; Teay et al., 2017; Umar et al., 2017; Visconti et al., 2017). However, ecology is also concerned by understanding and analyzing the dynamical behavior of populations in ecosystem in order to predict their undesired dynamics (Feng et al., 2017; Zhang et al., 2017b). On the other hand, understanding population dynamics are leading nowadays to many interesting optimization algorithms such as *Particle Swarm* algorithms (Mehdi and Boubaker, 2011, 2016;), Ant Colony algorithms (Pang et al., 2017; Yongwang et al., 2017) and Prey-predator algorithms (Sidhu and Dhillon, 2017; Zhang and Duan, 2017).

It is obvious that it is not at all evident to construct a mathematical model that will fit entirely any natural population interactions. In the literature, several models for predator–prey’s systems have been already proposed and analyzed incorporating some specific effects (Bürger et al., 2017; Elettreby et al., 2017; Li et al., 2017a, 2017b; Liu et al., 2017, 2018; Liu and Dai; Liu and Wiang, 2018; Yuan et al., 2018; Zhang et al., 2017a). The first proposed one was the Lotka Volterra predator–prey model (Volterra, 1928). However, to the best of our knowledge, there is no research work that attempt to reproduce the complex dynamics of such systems using an equivalent electrical circuit. While, certain biological systems such as biological tissue and biological neural network are modeled by electrical circuit (Le Masson et al., 1999; Gómez et al., 2012). As such predator–prey systems exhibit complex dynamics as chaotic behaviors (Luo et al., 2016), such circuits can be adopted in the future in communication for encryption tasks (Lassoued and Boubaker, 2016; Lassoued and Boubaker, 2017; Kengne et al., 2017).

In this paper, the predator–prey system, already analyzed in Ben Saad and Boubaker (2015, 2017), and incorporating nonlinear polynomials in its complex model is considered for the weak Allee Effect case study. The objective is to design a suitable electronic circuit permitting to understand and conceptualize many aspects of ecological experimentation and theory by using only simple analog electronic components. It is clear that proving a good agreement between biological theory and electronic experiments is not evident due to the imperfections and uncertainties related to electronic components. Despite, these undesirable effects, the main nonlinearities of the biological model will be validated by means of experimental data. The circuit related to the prey–predator model will be then designed and simulated using the MultiSIM software.

This work is arranged as follows: Sections “PREDATOR–PREY MATHEMATICAL MODEL” and “WEAK ALLE EFFECT CASE STUDY ANALYSIS” recall the nonlinear model as well as its basic dynamics. Then, numerical simulation of the global dynamic behavior of the system is presented in Section “NUMERICAL SIMULATIONS VIA MATCONT SOFTWARE”. In Section “EXPERIMENTAL VALIDATION OF MAIN NONLINEARITIES”, the circuit design, simulation results via the MultiSIM software as well as the experimental data of the main nonlinearities of the predator–prey model are investigated. In Section “CIRCUIT DESIGN VIA MULTISIM SOFTWARE”, the circuit design of the whole complex model is finally proposed and a good agreement between simulation results via MATCONT and MultiSIM softwares are shown. Finally, an experimental implementation is illustrated in section “EXPERIMENTAL IMPLEMENTATION OF THE PREDATOR–PREY MODEL”.

Consider the predator–prey model with Allee Effect described by Ben Saad and Boubaker, (2015, 2017):

Two case studies for the Allee effect ℓ can be considered (Dhooge et al., 2008):

In the following, only the Weak Allee effect case study will be considered and the parameters *k*, *e*, and *m* are taken as *k* = *e* = 1 and *m* ϵ [0 1].

For *k* = *e* = 1, *ℓ* ∈ [−1 0] and m ϵ [0 1], the system (1) admits three equilibriums *E*_{i}(*i* = 1,…,3):

The zero equilibrium

*E*_{0}= (0, 0),The two non-isolated equilibriums

*E*_{1}= (*ℓ*, 0),The equilibrium

*E*_{2}= (*m*, (*m*−*ℓ*) (1−*m*)).

For *i* = 1, …, 3, the corresponding Jacobian matrix *J*_{i} and eigenvalues (*λ*_{1i}, *λ*_{2i}) are given by

Based on the last results, the singularities and the phase portraits of the neighborhood of all equilibriums are summarized in Table 1.

The global dynamics of the system (1), already detailed in Table 1, will be proved numerically in this section using the Matcont software (Dhooge et al., 2008). Since the two first equilibriums *E*_{0} and *E*_{1} exist ∀*m* ∈ [0 1], only the singularity of the equilibrium *E*_{3} will be shown numerically in the following for the initial condition (*x*_{1} = 0.4; *x*_{2} = 0.3). For *m* = −*ℓ* =0.2, Figure 1 shows the temporal evolution of the dynamics of system (1) whereas the phase portrait of the system is shown in Figure 2.

As it is shown, a heteroclinic orbit appears indicating the existence of a global bifurcation described by an attractor which bifurcates from an unstable focus equilibrium point to a heteroclinic cycle. The two populations’ dynamics oscillate between 0 and 1 density values. Figures 3 and 4 show the dynamic behavior of the system at the non-isolated point *E*_{3} (0.4, 0.36) when *m* = 0.4.

For the second case study, the obtained behavior of the two populations is periodic which is described as a circle around the equilibrium *E*_{3} (0.4, 0.36). When *m* = 0.6, the dynamic behavior of the system is shown in Figures 5 and 6.

The obtained phase portrait shows a stable focus. This latter closes to a fixed point with the following coordinates (0.6, 0.32) proven with the temporal evolution.

The aim of this section is to design an analog circuit that can build the nonlinear terms according to system (1). Therefore, equivalent electronic circuits of these nonlinear terms are designed and simulated via MultiSIM software (Yujun et al., 2010). Due to the complexity of the equivalent circuit design, we designed first the equivalent circuit of each nonlinear term *x*^{2} and *x*^{3}. The first nonlinear term circuit needs one Multiplier AD6333. The multiplying core of this multiplier comprises a buried Zener reference providing an overall scale factor of 10 V. For that, an amplification of the output with a gain of 10 is required. Thus, the equivalent electrical model of *x*^{2} is realized as follows:

As shown in Figure 7, by considering the components datasheet, the electronic circuit of the nonlinear term *x*^{2} is modeled by one Analog multiplier AD633AN, one Amplifier AOP, two capacitors (*C*_{1}, *C*_{2}) and two resistors (*R*_{2}, *R*_{2}). Due to the existence of internal loss in the electrical components, the ideal value of *R*_{2} chosen to obtain the best result is 0.9 kΩ. However, for the nonlinear term *x*^{3}, the electronic circuit is modeled in Figure 8 by two multipliers AD633AN, 2 Amplifiers AOP, four capacitors and four resistors, since its electrical model is realized as follows:

$${\mathrm{S}}_{2}={x}^{3}={\mathrm{R}}_{4}10{\mathrm{R}}_{3}\mathrm{S}\ast x={\mathrm{R}}_{4}{\mathrm{R}}_{2}100{\mathrm{R}}_{1}{\mathrm{R}}_{3}{x}^{3}.$$

The input which will be used for the equivalent circuit of system (1) is continuous. However, in this part, in order to verify the efficiency of the two nonlinear terms equivalent circuits, an alternative signal is used as an input with a weak frequency equal to 1 Hz and amplitude of 2 V. The square and the cube of the signal are presented with a pink curve in Figure 9 and Figure 10, respectively. For high frequencies (1 kHz, 10 kHz, 100 kHz), we obtained the same following results.

As it is shown, we have chosen the same scale for the two nonlinearities. For the first nonlinearity *x*^{2}, the obtained signal amplitude is equal to 4,108 V which obviously the square of the input signal amplitude. In addition, for the second nonlinearity *x*^{3}, the result proves that the obtained signal is the cubes of the input signal since the obtained amplitude is equal to 7,980 V which is almost equal to 8 V the cube of the input amplitude.

Figures 11 and 12 show an experimental implementation for the last simulation results. The oscilloscope traces of the proposed circuit are illustrated in Figures 13 and 14.

Comparing the different results, it is proven that there is a good qualitative agreement between the numerical simulations with Matlab, the electrical simulations with MultiSIM and the experimental results for the two nonlinearities.

In this section, the agreement between biological theory and electronic experiments of the predator–prey model will be analyzed by considering the three cases studies presented numerically by MATCONT. Therefore, a transformation of the biological predator–prey model (1) to an equivalent electrical model is realized as follows:

The electronic circuits corresponding to these cases are designed by the software MultiSIM and presented in the following. We have used three multipliers, five AOP, two capacitors and 12 resistors. The resistors (*R*_{1}, *R*_{2}, …, *R*_{6}) and capacitors values are fixed with respect to the parameters values. The value of the two capacitors *C*_{1} and *C*_{2} is fixed at 100 nF. In addition, two interrupters S1 and S2 are used to introduce the initial conditions of the prey and predator density. In order to analyze the three case studies, we vary the value of the resistor *R*_{6} which corresponds to the parameter *m* since
${\mathrm{R}}_{6}={\mathrm{R}}_{5}\mathrm{m}$
. In the first case study, we consider the mortality rate of the predator *m* = 0.2. Therefore, the fixed value of *R*_{6} in this case is equal to 50 kΩ. Figure 15 describes the electrical circuit of the predator–prey model (5).

Then, the temporal evolution and the phase portrait when *m* = 0.2 are presented in Figure 16 and Figure 17, respectively.

We obtained two phase-shifted alternative signals. The pink curve corresponds to the dynamic evolution of the prey while the blue one corresponds to the dynamic evolution of the predator. Based on the scale chosen in the temporal evolution, the amplitude of the blue and pink curves are almost equal to 0.956 V and 0.836 V, respectively. Comparing with the numerical simulation presented in Figures 1 and 2, we conclude that the permanent regime of the system dynamic obtained via MultiSIM is similar to that obtained using the Matcont software.

In the second case study, we consider the mortality rate of the predator *m* = 0.4. Thus, Figure 18 describes the electrical circuit of the model with *R*_{6} = 25 kΩ. Then, the temporal evolution and the phase portrait are illustrated in Figure 19 and Figure 20, respectively.

As it is illustrated in the temporal evolution, the maximum amplitude of the prey population is equal to 0.488 V and that of the predator population is equal to 0.441 V. The obtained values are obviously very close to the numerical values shown in Figure 3. Furthermore, the phase portrait proves the existence of the center singularity which is demonstrated numerically via the Matcont software in Figure 4.

In the third case study, the mortality rate of the predator *m* is fixed at 0.6. Therefore, Figure 21 presents the electrical circuit with *R*_{6} = 17 kΩ. Figures 22 and 23 illustrate the corresponding temporal evolution and phase portrait.

For the third case study, we change slightly the initial conditions (*x*_{1} = 0.4, *x*_{2} = 0.1) to obtain the clearest results. The obtained temporal evolution and phase portrait prove that the model dynamic tends toward a fixed point. In Figure 22, it is noted that the dynamic behavior oscillates during the transitional regime, whereas in the permanent regime, it becomes continuous reaching a constant value equal to 0.586 V for the prey population and to 0.326 V for the predator one. The pink curve corresponding to the prey population dynamic reaches a pick of 0.978 V. However, the blue curve corresponding to the predator population reaches a pick equal to *x*^{2} = 0.399 V. This temporal behavior is described by a stable focus in the obtained phase portrait. These electrical results are almost similar to the numerical ones presented in Figures 5 and 6.

For the three cases study, we conclude that the electrical results obtained by the software MultiSIM obviously prove the numerical results obtained by the Matcont software.

In this section, an experimental implementation of the equivalent electrical circuit will be realized by placing the different electrical components on a bread board within the laboratory.

As it is mentioned previously, mathematical predator–prey model has several nonlinearities which are modeled by multipliers AD633 in the equivalent electrical circuit. The cascade of components AD633 and AOP with gain of 10 amplifies most likely the imperfection and the uncertainties of the electronic components and consequently distorts the experimental results. Therefore, in order to resolve this problem, the experimental validation is realized by using the technology STM3278 shown in Figure 24.

The experimental results of the three cases studies are obtained by using the oscilloscope and presented in Figures 25–30.

Comparing experimental results with numerical results obtained via the MATCONT software shown in Figures 1–6, we can conclude that good agreement is obtained between simulation results and experiments.

In this paper, an electronic circuit is designed for the model of a prey–predator model and its complex behavior is proved via numerical and electrical results. Furthermore, some experimental investigations are also shown and demonstrate that they can be used to characterize the ecological dynamics faster. In the future, we will try to prove, via experimental results, the chaotic dynamics of such nonlinear systems in the presence of seasonally effects in order to use such circuits for encryption/decryption fields.