A Stochastic Bayesian Regularization Approach for the Fractional Food Chain Supply System with Allee Effects

Abstract

This motive of current research is to provide a stochastic platform based on the artificial neural networks (ANNs) along with the Bayesian regularization approach for the fractional food chain supply system (FFSCS) with Allee effects. The investigations based on the fractional derivatives are applied to achieve the accurate and precise results of FFSCS. The dynamical FFSCS is divided into special predator category $P(\eta )$, top-predator class $Q(\eta )$, and prey population dynamics $R(\eta )$. The computing numerical performances for three different variations of the dynamical FFSCS are provided by using the ANNs along with the Bayesian regularization approach. The data selection for the dynamical FFSCS is selected for train as 78% and 11% for both test and endorsement. The accuracy of the proposed ANNs along with the Bayesian regularization method is approved using the comparison performances. For the rationality, ability, reliability, and exactness are authenticated by using the ANNs procedure enhanced by the Bayesian regularization method through the regression measures, correlation values, error histograms, and transition of state performances.

1. Introduction

The mathematical systems present the natural phenomena using the predator–prey studies with various types [1,2]. The factor of functional response in the modelling of predator and prey plays a significant part in indicating how the prey impacts based on the predators with time. There are various species based on the functional responses, i.e., Holling 1–3 phase [3,4], ratio dependent [5,6,7] and the Beddington–DeAngelis [8,9,10]. The food chain supply system (FCSS) has been applied to associate the multiple predators or prey. The efficient form of the FCSS with mutual qualitative surveys and abundant communications is shown in [11,12,13]. The mathematical formulations have a significant character to indicate the dynamics based on the nonlinear models, e.g., dengue virus [14], SITR model [15], and nervous system of stomach [16].

The Allee effects (AE), defined a century ago by Allee, have a huge importance in the process of FCSS. The AE assign the evolution to lessen the growing ratio with lessor public amount. The AE have abundant applications in plants, vertebrates, fishery, and invertebrates. These effects infrequently designate the negative inspirations using the indulgence-based population dynamics of the fishery. The division of AE is categorized into the multiplication and addition terms [17,18,19,20]. Singh et al. provided the dual form of the AE with the enhanced system of Leslie-Gower (LG) based on predator–prey dynamics that shows the population of prey with numerous junctions connected to the appropriate limits. Vinoth et al. [21] expressed a mathematical system for the dynamical FCSS using the AE with the process of addition [22].

The current investigations are related to the numerical operator routines based on the fractional food chain supply system with Allee effects (FFSCS-AE) by using the stochastic computing platform of the artificial neural networks (ANNs) along with the Bayesian regularization approach. Recently, the stochastic computing procedures have been explored to solve the number of submissions. Some of them are a singular differential functional system [23], a coronavirus dynamical system [24], a HIV infection system [25], a nonlinear smoking system [26], and a differential form of the delay model [27].

The fractional derivatives (FDs) have been implemented to perform the precise performances. The FDs indicate the minute statics using the superfast/slow evolution, which delivers more features of the system dynamics based on the fractional calculus that is difficult to understand the counterparts of the integer order. The FDs represent better and accurate performances instead of an integer order derivative based on the accessibility of the condition. There are various applications of real world where FDs have been implemented to validate the system performance [28,29]. The FDs have been broadly examined to present the solutions of the control networks, physical/mathematical models, and engineering. The execution performances of the FDs broadly applied over the past three decades by using the extensive operators, e.g., Grunwald–Letnikov [30], Weyl–Riesz [31], Riemann-Liouville [32], Caputo derivative (CD) [33], and Erdelyi-Kober [34]. All these mentioned operators describe their individual importance. However, the CDs have abundant applications in the homogeneous/non-homogeneous forms. The CDs are not difficult to conduct in comparison to the other definitions of the operators. By considering these operator applications, the authors are interested in exploring and solving the FFSCS-AE by using the stochastic computing platform of the ANNs along with the Bayesian regularization scheme.

The other parts of the paper are presented as the following: The construction of the FFSCS-AE is presented in Section 2. The ANNs methodology along with the Bayesian regularization is designed in Section 3. The result simulations are given in Section 4. The conclusion performances are presented in the Section 5.

2. Mathematical FFSCS-AE

The current section presents the dynamical FFSCS-AE based on two or more predators–prey. A mathematical FFSCS-AE based on the investigations of the common qualitative together with numerous associations is reported in [35,36]. Some of the scholars studied the multi-levels of trophic based on the FCSS using the special predator category $P(\eta )$, top-predator class $Q(\eta )$ and prey population dynamics $R(\eta )$ [37,38,39,40,41,42,43]. The dynamical FCSS is given as [44]:

$$\left\{\begin{array}{l}\frac{dP(\eta )}{d\eta }=u_{0}P(\eta )-\frac{{\rho }_{0}P(\eta )Q(\eta )}{P(\eta )+r_0}-\frac{s_1}{s_2+P(\eta )}-v_0{P}^2(\eta ), P_0=c_1,\\ \frac{dQ(\eta )}{d\eta }=\frac{{\rho }_{1}P(\eta )Q(\eta )}{P(\eta )+r_1}-u_{1}Q(\eta )-\frac{{\rho }_{2}Q(\eta )R(\eta )}{Q(\eta )+r_2}, Q_0=c_2,\\ \frac{dR(\eta )}{d\eta }=w_3{R}^2(\eta )-\frac{{\rho }_3{R}^2(\eta )}{Q(\eta )+r_3}, R_0=c_3.\end{array}\right.$$

(1)

In the above system, the prey $P(\eta )$ and species $Q(\eta )$ represent the Volterra approach, which shows the predator population decreases exponentially in the absence of the prey. The species relationship $R(\eta )$ and $P(\eta )$ is presented by using the LG scheme that shows the population of predator reduces the accessibility per capita [45,46]. $u_0$ and $w_3$ indicate the growth ratios of $P(\eta )$ and; the protection of the environmental factor for $P(\eta )$ are $r_0$ and $r_1$, whereas per capita reduction in $Q(\eta )$ is $\frac{{\alpha }_2}{2}$ presented in $r_2$; the term $u_1$ presents $Q(\eta )$, which performs the reduction in the absence of $P(\eta )$; $v_0$ is the competition form of the $P(\eta )$; the residual for $R(\eta )$ is the shortage of food, i.e., $Q(\eta )$ is presented by $r_3$; $P(\eta )$ is the maximum performances to reduce per capita and is signified by ${\rho }_0$, ${\rho }_1$, ${\rho }_2$, and ${\rho }_3$; the $\frac{s_1}{s_2+P(\eta )}$ is the hyperbolic function that represents the sum form of AE. If $s_1<s_2$ shows the weak AE, then $s_2<s_1$ is the strong AE; the ICs are presented by c₁, c₂ and c₃. The dynamical form of the FFSCS-AE is presented as:

$$\left\{\begin{array}{l}\frac{d^{\alpha }P(\eta )}{d{\eta }^{\alpha }}=u_{0}P(\eta )-\frac{{\rho }_{0}P(\eta )Q(\eta )}{P(\eta )+r_0}-\frac{s_1}{s_2+P(\eta )}-v_0{P}^2(\eta ), P_0=c_1,\\ \frac{d^{\alpha }Q(\eta )}{d{\eta }^{\alpha }}=\frac{{\rho }_{1}P(\eta )Q(\eta )}{P(\eta )+r_1}-u_{1}Q(\eta )-\frac{{\rho }_{2}Q(\eta )R(\eta )}{Q(\eta )+r_2}, Q_0=c_2,\\ \frac{d^{\alpha }R(\eta )}{d{\eta }^{\alpha }}=w_3{R}^2(\eta )-\frac{{\rho }_3{R}^2(\eta )}{Q(\eta )+r_3}, R_0=c_3.\end{array}\right.$$

(2)

In the above system, $\alpha$ is the fractional order CD to solve the dynamical FFSCS-AE. The fractional derivative values are used in the intervals 0 and 1 to designate the behavior of the dynamical FFSCS-AE. The FD in the FSCS-AE shown in system (2) to present the minute particulars, which is difficult to solve through the counterparts of the integer order as illustrated in model (1). Recently, the FDs have been executed in numerous submissions, e.g., pine wilt virus system using the convex rate [47], heat transfer [48], spatiotemporal patterns based on the Belousov–Zhabotinskii reaction systems [49], predator/prey system with herd presentation [50], HBV mathematical system [51], population growth biological system based on carrying volume [52], and soil animal material content based on the infrared spectrometry [53].

The novel features of the dynamical FFSCS-AE based on two or more predator–prey are presented as:

• The design of the FDs based on the dynamical FFSCS-AE is provided to scrutinize the accurate and real performances.
• The stochastic computing performances have not been executed to solve the dynamical FFSCS-AE.
• The ANNs computing procedures along with the Bayesian regularization scheme have been used to present the dynamical FFSCS-AE using the FDs in 0 and 1.
• The precision of the ANNs computing procedures along with the Bayesian regularization scheme is provided based on the comparative measures of the achieved and reference results.
• The absolute error measures in good performances are reported, which show the exactness and capability of the ANNs computing procedures along with the Bayesian regularization scheme.
• The regression measures, correlation values, error histograms (EHs), and state transition (STs) performances have been provided using the dynamical FFSCS-AE.

3. Proposed ANNs along with the Bayesian Regularization Approach

In the current section, the computing ANNs procedures along with the Bayesian regularization method have been provided for the dynamical FFSCS-AE. The workflow illustrations are shown in Figure 1 for the dynamical FFSCS using the ANNs to solve the dynamical FFSCS-AE. Figure 1 shows the three blocks, the mathematical FFSCS-AE, proposed scheme, and performance of the outcomes. The construction of the stochastic methodology is presented in two steps.

(i)

The noteworthy ANNs procedures along with the Bayesian regularization method are presented.

(ii)

The execution performances using the computing ANNs procedures along with the Bayesian regularization method for the dynamical FFSCS.

The substantial procedures concerning the interpretation have been illustrated through the Adam method. Nine numbers of hidden neurons have been proposed together with the static selection using the dynamical FFSCS based on the train as 78% and 11% for both test and endorsement. The supervised artificial intelligence aptitudes based on ANNs procedures along with the Bayesian regularization method have been presented with best collaboration in the matrices with intricacy, overfitting, hasty convergence, and underfitting steps. Furthermore, these networks have been adjusted after comprehensive reproduction investigations, knowledge, care, experience, and small disparities of the system network.

The second phase based on the ANNs procedures along with the Bayesian regularization method is stated through the general perception of the solo neuron shown in Figure 2. The single layered structure of the neural network is presented in Figure 2a, while the construction of the design layer, an input single layer vector with nine neurons with three outputs in the construction of the outer layers as presented in Figure 2b to solve the mathematical dynamical FFSCS. The ANNs procedures along with the Bayesian regularization method are implemented by applying the nftool command in Matlab software using the suitable hidden neuron selection, kernel/activation function based on the log-sigmoid, n-fold cross substantiation with n = 0, 1000 generations, confirmation statistics, testing statics, and learning. The stoppage criteria are taken as default, and tolerances and step size are implemented to label the data, training for targets along with the inputs, which are accessed using the numerical standard solutions. While the operation presentations of the ANNs procedures along with the Bayesian regularization approach for the dynamical mathematical FFSCS with setting of the parameters are tabulated in

Table 1. Values of the parameter to perform the stochastic scheme.

Parameter	Settings
Maximum mu performances	1010
Fitness measures (MSE)	0
Hidden neurons	9
Decreeing mu performances	0.2
Increasing mu measures	12
Adaptive parameter (mu)	6 × 10−0.4
Substantiation fail amount	6
Maximum Epochs	720
Minimum gradient	10−0.6
Training data	78%
Validation data	11%
Testing data	11%
Sample selection	Random
Hidden/output/input	Single
Dataset generation	Adam
Implementation and stoppage standards	Default

. The system training is applied by the ANNs procedures along with the Bayesian regularization method.

The ANNs procedures along with the Bayesian regularization technique is presented in Table 1 based on the slight disproportion/alteration/adjustment, which can occur in poor performance (premature convergence). Hence, the setting of the parameter settings will be combined with widespread consideration, after directing the numerical examination and consideration.

4. Results Based on the Dynamical FFSCS

This section shows the three cases based on the FDs that have been provided by using the ANNs procedures along with the Bayesian regularization method. The mathematical formulation of each case is presented as:

Consider the dynamical FFSCS with $\alpha =0.5,$ $u_0=1.5$, $u_1=1$, $v_0=0.06$, ${\rho }_0=1$, ${\rho }_1=2$, ${\rho }_2=0.405$, ${\rho }_3=1$, $w_3=1.5$, $s_1=s_2=0.1$, $r_0=10$, $r_1=10$, $r_2=10$, $r_3=20$ and $c_1=c_2=c_3=1.2$ is given as:

$$\left\{\begin{array}{l}\frac{d^{0.5}P(\eta )}{d{\eta }^{0.5}}=1.5P(\eta )-\frac{P(\eta )Q(\eta )}{P(\eta )+10}-\frac{0.1}{0.1+P(\eta )}-0.06P^2(\eta ), P_0=1.2,\\ \frac{d^{0.5}Q(\eta )}{d{\eta }^{0.5}}=\frac{2P(\eta )Q(\eta )}{P(\eta )+10}-Q(\eta )-\frac{0.405Q(\eta )R(\eta )}{Q(\eta )+10}, Q_0=1.2,\\ \frac{d^{0.5}R(\eta )}{d{\eta }^{0.5}}=1.5R^2(\eta )-\frac{R^2(\eta )}{Q(\eta )+20}, R_0=1.2.\end{array}\right.$$

(3)

Consider the dynamical FFSCS with $\alpha =0.7,$ $u_0=1.5$, $u_1=1$, $v_0=0.06$, ${\rho }_0=1$, ${\rho }_1=2$, ${\rho }_2=0.405$, ${\rho }_3=1$, $w_3=1.5$, $s_1=s_2=0.1$, $r_0=10$, $r_1=10$, $r_2=10$, $r_3=20$ and $c_1=c_2=c_3=1.2$ presented as:

$$\left\{\begin{array}{l}\frac{d^{0.7}P(\eta )}{d{\eta }^{0.7}}=1.5P(\eta )-\frac{P(\eta )Q(\eta )}{P(\eta )+10}-\frac{0.1}{0.1+P(\eta )}-0.06P^2(\eta ), P_0=1.2,\\ \frac{d^{0.7}Q(\eta )}{d{\eta }^{0.7}}=\frac{2P(\eta )Q(\eta )}{P(\eta )+10}-Q(\eta )-\frac{0.405Q(\eta )R(\eta )}{Q(\eta )+10}, Q_0=1.2,\\ \frac{d^{0.7}R(\eta )}{d{\eta }^{0.7}}=1.5R^2(\eta )-\frac{R^2(\eta )}{Q(\eta )+20}, R_0=1.2.\end{array}\right.$$

(4)

Consider the dynamical FFSCS with $\alpha =0.9,$ $u_0=1.5$, $u_1=1$, $v_0=0.06$, ${\rho }_0=1$, ${\rho }_1=2$, ${\rho }_2=0.405$, ${\rho }_3=1$, $w_3=1.5$, $s_1=s_2=0.1$, $r_0=10$, $r_1=10$, $r_2=10$, $r_3=20$ and $c_1=c_2=c_3=1.2$ presented as:

$$\left\{\begin{array}{l}\frac{d^{0.9}P(\eta )}{d{\eta }^{0.9}}=1.5P(\eta )-\frac{P(\eta )Q(\eta )}{P(\eta )+10}-\frac{0.1}{0.1+P(\eta )}-0.06P^2(\eta ), P_0=1.2,\\ \frac{d^{0.9}Q(\eta )}{d{\eta }^{0.9}}=\frac{2P(\eta )Q(\eta )}{P(\eta )+10}-Q(\eta )-\frac{0.405Q(\eta )R(\eta )}{Q(\eta )+10}, Q_0=1.2,\\ \frac{d^{0.9}R(\eta )}{d{\eta }^{0.9}}=1.5R^2(\eta )-\frac{R^2(\eta )}{Q(\eta )+20}, R_0=1.2.\end{array}\right.$$

(5)

Figure 3 presents the STs values using the optimal performances of the dynamical FFSCS. The STs along with the MSE output performances using the substantiation, train, and best curves plotted in Figure 3 based on ANNs procedures along with the Bayesian regularization process for the dynamical form of the FFSCS. The achieved best performances of the dynamical FFSCS have been demonstrated at epochs 132, 56, and 6, which are given as 1.03 × 10⁻¹¹, 6.74 × 10⁻¹⁰, and 4.37 ×10⁻¹⁰. The second part of Figure 3 authenticates the gradient measures using the ANNs procedures with the Bayesian regularization process to solve the dynamical FFSCS. The gradient performances are presented as 5.15 × 10⁻⁰⁹, 5.70 × 10⁻⁰⁹, and 6.67 × 10⁻⁰⁸. These representations specify the exactness of the ANNs procedures along with the Bayesian regularization process for the dynamical form of the FFSCS. The valuations of the results-based training outputs, authentications targets, test marks, fitness, and error curves are shown in the first part of Figure 4. Though the EHs values of test, train, and authentication, the zero error performances are shown in the second part of Figure 4 for the dynamical FFSCS. The EHs are reported as 4.45 × 10⁻⁰⁷, 1.93 × 10⁻⁰⁶, and 1.2 × 10⁻⁰⁶ for the dynamical form of the FFSCS. Figure 5 authenticates the correlation operator values using the train, authentication, and test measures for the dynamical FFSCS. The correlation is reported as one that represents the perfect model. The convergence of the MSE represent the complexity measures, train presentations, authentication, generation, test, and back propagation are authentic in

Table 2. ANNs procedures with the Bayesian regularization process to solve the dynamical FFSCS.

Case	MSE		Epoch	Gradient	Performance	Mu	Time
	Test	Train
1	4.35× 10−11	1.02 × 10−11	132	5.15 × 10−09	1.03 × 10−11	50	4
2	8.812 × 10−10	6.743 × 10−10	56	5.70 × 10−09	6.74 × 10−10	5	2
3	2.331× 10−11	4.37 × 10−10	6	6.67 × 10−08	4.37 × 10−10	50	2

using the ANNs procedures with the Bayesian regularization process to solve the dynamical FFSCS.

Figure 6 and Figure 7 represent the comparative soundings using the result comparisons and AE measures to solve the ANNs procedures with the Bayesian regularization process to solve the dynamical FFSCS. Figure 6 presents the exactness of the ANNs procedures with the Bayesian regularization scheme using the comparison procedure for each variation of the dynamical FFSCS. Figure 7 authenticates the values of the AE for each variation of the dynamical FFSCS using the ANNs procedures with the Bayesian regularization process. The AE measures for the special predator category $P(\eta )$ are performed as 10⁻⁰⁴ to 10⁻⁰⁷, 10⁻⁰⁴ to 10⁻⁰⁵, and 10⁻⁰⁵ to 10⁻⁰⁷ for case 1 to 3 for the dynamical FFSCS. The AE top-predator class $Q(\eta )$ provided as 10⁻⁰⁴ to 10⁻⁰⁶, 10⁻⁰⁴ to 10⁻⁰⁵, and 10⁻⁰⁴ to 10⁻⁰⁷ for case 1 to 3 case to solve the dynamical FFSCS. The AE for prey population dynamics $R(\eta )$ are measured as 10⁻⁰⁵ to 10⁻⁰⁷, 10⁻⁰⁴ to 10⁻⁰⁶, and 10⁻⁰⁵ to 10⁻⁰⁸ for case 1 to 3 of the dynamical FFSCS. These AE depictions validate the exactness of the ANNs procedures using the Bayesian regularization process to solve the dynamical FFSCS.

5. Concluding Remarks

The purpose of these investigations is to provide a stochastic computing platform using the artificial neural networks along with Bayesian regularization to solve the fractional food chain supply system with Allee effects. The investigations based on the fractional derivatives are applied to achieve precise results of the FFSCS. The dynamical FFSCS is categorized into special predator category $P(\eta )$, top-predator class $Q(\eta )$, and prey population dynamics $R(\eta )$. Some concluding remarks of the current study are presented as:

• The computing numerical performances for three different variations of the dynamical FFSCS have been provided by using the ANNs along with the Bayesian regularization technique.
• The data selection for the dynamical FFSCS is selected for train as 78% and 11% for both test and endorsement.
• The accuracy of the designed ANNs along with the Bayesian regularization approach has been approved by using the comparison of obtained and reference solutions.
• The AE for each variation of the mathematical dynamical FFSCS are performed in good measures, which presents the exactness of the scheme.
• For the rationality, ability, reliability, and exactness are authenticated by using the ANNs procedure enhanced by the Bayesian regularization method through the statistical performances.

In future, the ANNs along with the Bayesian regularization approach will be executed to indicate the fractional and functional kinds of systems.