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 (

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 (

In this paper, the predator–prey system, already analyzed in

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 _{1} is the size of the prey population and _{2} the size of the predator population, ℓ is the Allee Effect threshold, _{2} ≥ 0 and _{2} ≥ 0.

Two case studies for the Allee effect ℓ can be considered (

The Strong Allee effect when

The Weak Allee effect when

In the following, only the Weak Allee effect case study will be considered and the parameters

For _{i}(

The zero equilibrium _{0} = (0, 0),

The two non-isolated equilibriums _{1} = (

The equilibrium _{2} = (

For _{i} and eigenvalues (_{1i}, _{2i}) are given by_{2}) = ^{2}−4[

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

Predator–prey model analysis.

Equilibrium | Singularities | Phase portraits |
---|---|---|

The global dynamics of the system (1), already detailed in _{0} and _{1} exist ∀_{3} will be shown numerically in the following for the initial condition (_{1} = 0.4; _{2} = 0.3). For

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. _{3} (0.4, 0.36) when

For the second case study, the obtained behavior of the two populations is periodic which is described as a circle around the equilibrium _{3} (0.4, 0.36). When

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.

Temporal evolution of the predator–prey system for

Phase portrait of the predator–prey system for

Temporal evolution of the predator–prey system for

Phase portrait of the predator–prey system for

Temporal evolution of the predator–prey system for

Phase portrait of the predator–prey system for

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 (^{2} and ^{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 ^{2} is realized as follows:

As shown in ^{2} is modeled by one Analog multiplier AD633AN, one Amplifier AOP, two capacitors (_{1}, _{2}) and two resistors (_{2}, _{2}). Due to the existence of internal loss in the electrical components, the ideal value of _{2} chosen to obtain the best result is 0.9 kΩ. However, for the nonlinear term ^{3}, the electronic circuit is modeled in

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

As it is shown, we have chosen the same scale for the two nonlinearities. For the first nonlinearity ^{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 ^{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.

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.

Circuit design of the ^{2} function within MultiSIM software.

Circuit design of the ^{3} function within MultiSIM software.

Simulation results of the ^{2} function with MultiSIM Software.

Simulation results of the ^{3} function with MultiSIM Software.

Electronic circuit of the ^{2} function.

Electronic circuit of the ^{3} function.

Experimental results of ^{2} function.

Experimental results of ^{3} function.

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:_{1} and _{2} are used for the integration of the circuit outputs in order to obtain as output the populations’ density _{1} and _{2}.

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 (_{1}, _{2}, …, _{6}) and capacitors values are fixed with respect to the parameters values. The value of the two capacitors _{1} and _{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 _{6} which corresponds to the parameter _{6} in this case is equal to 50 kΩ.

Then, the temporal evolution and the phase portrait when

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

In the second case study, we consider the mortality rate of the predator _{6} = 25 kΩ. Then, the temporal evolution and the phase portrait are illustrated in

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

In the third case study, the mortality rate of the predator _{6} = 17 kΩ.

For the third case study, we change slightly the initial conditions (_{1} = 0.4, _{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 ^{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

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.

Circuit design of the predator–prey system for

Temporal evolution via MultiSIM software (

Phase portrait via MultiSIM software (

Circuit design of the predator–prey system for

Temporal evolution via MultiSIM software (

Phase portrait via MultiSIM software (

Circuit design of the predator–prey system for

Temporal evolution via MultiSIM 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

The experimental results of the three cases studies are obtained by using the oscilloscope and presented in

Comparing experimental results with numerical results obtained via the MATCONT software shown in

Phase portrait via MultiSIM software (

STM3278 Technology.

Experimental temporal evolution (

Experimental phase portrait (

Experimental temporal evolution (

Experimental phase portrait (

Experimental temporal evolution (

Experimental phase portrait (

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.