Dynamics of a Three-Patch Prey-Predator System with the Impact of Dispersal Speed Incorporating Strong Allee Effect on Double Prey

Department of Mathematics, National Institute of Technology Puducherry, Karaikal 609609, India Indian Institute of Engineering and Science and Technology, Shibpur, West Bengal 711103, India Maulana Abul Kalam Azad University of Technology, Kolkata 700064, West Bengal, India Department ofMathematics Education, Akenten Appiah-Menka University of Skills Training and Entrepreneurial Development, Kumasi, Ghana


Introduction
In mathematical ecology, most of the researchers considered the prey-predator system in a homogeneous environment. But in a real life situation, the environment is heterogeneous, which contains di erent patches connected through migration. Many ecologists and researchers ( [1][2][3][4][5]) have studied the impact of predator species migration on preypredator interactions. In their work, the prey density was signi cant and the predator species was considered to remain in the speci ed patch. Many researchers comprehensively studied the dispersal model in a multipatch environment ( [6][7][8][9][10][11][12][13][14][15][16][17][18][19][20]). Amarasekare [13] studied two-patch models of single species with local density-dependent dispersal and spatial heterogeneity. Padrón and Trevisan [18] considered a single species' logistic growth model composing several habitats connected by linear migration.
Stephen Cantrell et al. [19] examined evolutionary stability strategies for dispersal in heterogeneous patchy environments. In most cases, the randomness of the dispersal rate between di erent patches was assumed to be xed. Kang et al. [20] formulated a Rosenzweig-MacArthur prey-predator model in two patch environments. e rate of dispersal has a great in uence on stable prototypes and the persistence of the species shown in various research works ( [21][22][23][24]). Ruxton [22] investigated the stability behaviour of a population model by adding density-dependent migration between nearestneighbour populations. Rohani and Ruxton [23] presented a two-species interaction to establish the non-stabilization of the system for density-independent movement between populations. Fretwell and Lucas [25] proposed ideal free distribution, i.e. individuals in di erent patches possess similar tness. Several researchers have already discussed this concept after that ( [10,26,27]).
In balance dispersion [10], the dispersion rate remains constant when the population is in the equilibrium state, providing ideal free distribution. But constant rates of dispersal for the passive animal can lead the prey-predator system to a stable situation ( [10,28]). Furthermore, passive dispersal may create negative density dependence staffing rates of the population to stabilize the prey-predator system at an equilibrium state. erefore, due to heterogeneity in patch or dispersal rates, an unstable population may behave like a stable population. Dispersal rates cannot be considered very high because they may harmonize local stability behaviour between patches. erefore, to show the effect of dispersal, researchers have developed their models with the help of fixed dispersal rates. e population density alters very fast because of the dispersal process in most of their models. Krivan and Sirot [27] observed that, the stable population dynamics for two competitive species becomes unstable due to rapid adaptive animal dispersal. Abrams et al. ( [29,30]) also observed the profound impact of the dispersal process on the population dynamics.
Modeling the prey-predator system incorporating the dispersal concept is an active field of research. e dispersal rate of prey species steadily increases with the enlargement of predators in a patch ( [31,32]). Another exciting matter in this field is the Allee effect, a mechanism where the prey population shows a negative growth rate at low density ( [33,34]). ere are several reasons to consider the Allee effect: mate restrictions, stumpy probability of victorious meeting [35], food mistreatment, supportive defence, inbreeding dejection [36], etc. Predator evasion [37] for evolutionary alterations is another important cause to consider the Allee effect. e strong Allee effect ( [38][39][40][41][42][43]) has been considered here, whereas the concept of the weak Allee effect ( [44,45]) also exists in this regard. is classification depends on the per capita enlargement rate of the population at low density. e mathematical formulation of the growth equation due to the Allee effect takes the form Ψ(y) � ry(1 − y/l)(y/l 0 − 1) where, y represents the population size at time t, r is the per capita growth rate, environmental carrying capacity is presented by l and l 0 is the Allee edge. Here, 0 < l 0 ⋘. e population change rate continuously diminishes ( [46,47]) and finally departs to extermination ( [42,43]) if the relation 0 < y < l 0 holds. On the contrary, at low population density, the population intensification rate decreases but stays positive in the weak Allee effect. Researchers ( [48][49][50][51]) observed the impact of the Allee effect on the dynamical behaviour of the prey-predator system. Celik and Duman [52] enormously studied the stability behaviour of a prey-predator model with and devoid of the Allee effect. e impact of the Allee effect on a prey-predator harvesting system was investigated by Javidi and Nyamorady [53]. Wang et al. [54] discussed the prey-predator system that exhibits intricate population dynamics due to the Allee effect. Zhou et al. [55] studied the population dynamics of a prey-predator system with the Allee effect. Kent et al. [56] used the concept of Allee effect as prey outfluxes. erefore, numerous works ( [57][58][59][60]) on prey-predator interactions have already been presented in this field of ecological modeling. A small number of researchers studied the dispersal dynamics of the prey-predator system in a patchy environment with a strong Allee effect.
Recently, Pal and Samanta [61] presented the dispersal dynamics of the Allee effect in one prey of a prey-predator system in two patch environment. Saha and Samanta [62] studied the dispersal dynamics of a prey-predator system in two patch environments with two prey species considered the Allee effect. Consequently, we are introducing a preypredator system with dispersal and strong Allee effect in three-patch environments, namely Patch 1, Patch 2 and Patch 3. Every patch consists of a pair of prey-predator species and a strong Allee effect in the prey population escalation in the first two patches. Our main objective is to study the impact of dispersal speed and the Allee effect on the population dynamics of our desired system. Ecology always demands a balance among its inhabitants, so stability of the population is always expected in this regard. We have studied the effect of dispersal and the Allee effect on the persistence of species. So, we hope that this study may help ecologists in their development of the environment. e next section reflects the mathematical structure of the prey-predator interaction in three-patch environment. Section 3 is equipped with the positivity and boundedness characteristics for our proposed prey-predator system. Equilibrium points and their stability conditions have been discussed in Section 4. Bifurcation behaviour of the proposed model has been analysed in Section 5. Numerical verifications have been done in Section 6 and the concluding remarks have been given in Section 7.

Formation of Three-Patch Prey-Predator System
We opt for a six species (three each prey and predators) prey-predator system in three-patch (i � 1, 2, 3) environments based on the following notations and assumptions. Notations: x i : prey population density in patch i. y i : predator population density in patch i. r i : growth rate of prey species in patch i. k i : environmental carrying capacity of the prey in i th patch in nonattendance of predation and dispersal. k i : threshold value of the Allee effect in patch i without predation as well as dispersal (0 < k i ≪ k i ). λ i : predation rate in patch i. δ: dispersal speed among patches. d ij : probability of dispersion from the i th patch to the j th patch (j � 1, 2, 3; i ≠ j). e i : conversion rate of prey biomass to predator biomass in patch i. m i : mortality rate of the predator species in patch i. Assumptions: (i) Prey population growth rate is affected by the Allee effect in the first two patches. (ii) All three predator species are free from the Allee effect. (iii) Prey species are movable to higher fitness patches. (iv) Conversion rate of prey biomass to predator biomass less than the predation rate e i ≤ λ i ∀i. e graphical view of our proposed system of threepatch environment system is presented in Figure 1.
Based on the above notations, assumptions, and flow diagram, the population-dispersal dynamics can be presented by the following set of nonlinear differential equations: (1) with primary information x i (0) > 0 and y i (0) > 0(i � 1, 2, 3). Since we considered the balanced dispersal, and hence mathematically, we can write d 12 /d 21 � k 2 /k 1 , d 13 /d 31 � k 3 /k 1 , d 23 /d 32 � k 3 /k 2 . erefore, the carrying capacities, which are identical to the population abundances in all patches, always imply no mesh progress among the patches. Consequently, this situation is precisely the same as without dispersal. erefore, without dispersal speed, the proposed model equation (1) converts to the following system of equations: with initial densities x i (0) > 0 and y i (0) > 0 for all i � 1, 2, 3. When δ � 0, the population in each patch progresses separately. In that case, the non-trivial equilibrium can be found by solving the system of equations f i � 0(i � 1, 2, 3, 4, 5, 6). Our primary target is to find the interior equilibrium point (x 1 , x 2 , x 3 , y 1 , y 2 , y 3 ) of the proposed system equation (1) with respect to k 1 , k 2 , k 3 , m 1 , m 2 , m 3 and we shall also observe the impact of the dispersal rate δ on the said equilibrium point.

Positivity and Boundedness of the Proposed System
Let us consider the following theorems to ensure that the anticipated model equation (1) is well-posed.

Theorem 1.
Each solution of the anticipated system equation (1) starting from R 6 + stay positive for all time.
If it is false, then there exist T 1 ∈ [0, θ) so that x 1 (T 1 ) � 0, dx 1 /dt < 0 as well as x 1 (t) > 0 for all t ∈ [0, T 1 ). Now, the first equation of the system equation (1) provides, Discrete Dynamics in Nature and Society 3 which contradicts dx 1 (T 1 )/dt < 0 and hence x 1 (t) > 0 for all t ∈ [0, θ). Next, we claim that x 2 (t) > 0 for all 0 ≤ t < θ. If it is false, then there exist T 2 ∈ [0, θ) so that x 2 (T 2 ) � 0, dx 2 /dt ≤ 0 as well as x 2 (t) > 0 for all 0 ≤ t < T 2 . Now, the second equation of the system equation (1) gives which again contradicts dx 2 /dt ≤ 0 and consequently, We further claim that x 3 (t) > 0 for all 0 ≤ t < θ. If our claim is false, then their exist Now, from the third equation of equation (1), we get this also contradicts dx 3 /dt ≤ 0. Hence, x 3 (t) > 0 for all t ∈ [0, θ). Again, from the last three equations of the system equation (1) we can obtain, therefore, from the above discussion, we can conclude x i (t) > 0 as well as y i (t) > 0(i � 1, 2, 3) for all t ≥ 0. Hence, the theorem is proved. Proof. To prove this theorem, we have the following cases.
For x 1 (0) ≤ k 1 , let us claim that x 1 (t) ≤ k 1 . Assume that the claim is false. erefore, there exist t 1 and t 2 in such a way that x 1 (t 1 ) � k 1 as well as x 1 (t) > k 1 for all t ∈ (t 1 , t 2 ) where x 2 (t) < k 2 and x 3 (t) < k 3 for all t ∈ (t 1 , t 2 ) as k 2 , k 3 are the respective carrying capacities of patch 2 and patch 3. Now, for all t ∈ (t 1 , t 2 ) we have, where, 4 Discrete Dynamics in Nature and Society ϕ x 1 (s), x 2 (s), x 3 (s), y 1 (s), y 2 (s), y 3 (s) � r 1 Now, erefore, But for all t ∈ (t 1 , t 2 ), ϕ(x 1 , x 2 , x 3 , y 1 , y 2 , y 3 ) < 0 as us, x 1 (t) < k 1 which is a contradiction and proves that the claim is valid. Hence Similarly, for x 2 (0) ≤ k 2 we can prove that x 2 (t) ≤ k 2 taking d 12 /d 21 � k 2 /k 1 and d 32 /d 23 Again, if we assume that the claim is not true, then there exist t ′ , t ″ such that and x 2 (t) < k 2 as k 1 , k 2 are the respective carrying capacities of patch 1 and patch 2. Now, for all t ∈ (t ′ , t ″ ) we have, where, Ψ x 1 (u), x 2 (u), x 3 (u), y 1 (u), y 2 (u), y 3 (u) � r 3 Discrete Dynamics in Nature and Society Now, en, But , y 1 (u), y 2 (u), y 3 (u)) < 0 as x 1 < k 1 and x 2 < k 2 . Hence x 3 (t) < k 3 which contradicts our assumption. Consequently, our claim is true and For x 1 (0) ≤ k 1 and x 2 (0) ≤ k 2 we can easily prove that x 1 (t) ≤ k 1 and x 2 (t) ≤ k 2 (as in Case 1). For x 2 (t) < k 2 as k 1 , k 2 are the respective carrying capacities of patch 1 and patch 2. Now, for all t ∈ (t ′ , t ″ ) we have, where, Now, en, 6 Discrete Dynamics in Nature and Society But Hence x 3 (t) < k 3 which contradicts our assumption. Consequently, our claim is true and Assume that the claim is false. erefore, there exist t 1 and t 2 in such a way that x 2 (t 1 ) � k 2 as well as as k 1 , k 3 are the respective carrying capacities of patch 1 and patch 3. Now, for all t ∈ (t 1 , t 2 ) we have, where, Now, erefore, Discrete Dynamics in Nature and Society us, x 2 (t) < k 2 which is a contradiction and proves that the claim is valid. Hence erefore, in this case we also get . In a similar way, considering Case 1, we get x 2 (t) ≤ k 2 for Now, k 2 , k 3 are the respective carrying capacities of patch 2 and patch 3. So, x 2 (t) < k 2 and x 3 (t) < k 3 (carrying capacity means the maximum population that can be achieved by the environment). us Again, the first equation of (1) gives, Also, we have, dx 1 is implies that limsup t⟶∞ x 1 (t) ≤ k 1 which contradicts the assumption and so limsup t⟶∞ x 1 (t) ≤ k 1 .
Similarly, we can prove that, limsup t⟶∞ x 2 (t) ≤ k 2 . Lastly let us assume that limsup t⟶∞ x 3 (t)≰k 3 . en Now, k 1 , k 2 are the respective carrying capacities of patch 1 and patch 2. So, x 1 (t) < k 1 and x 2 (t) < k 2 (carrying capacity means the maximum population that can be achieved by the environment). us Again, the third equation of (1) gives, We also have, dx 3 3 . is again contradicts the assumption and we get limsup t⟶∞ x 3 is implies, Now applying the theory of differential inequality, we get, Taking t ⟶ ∞, en all the solutions of (1) that initiate in R 6 + are confined in the region: Discrete Dynamics in Nature and Society 9 is proves the theorem.

Equilibria and Stability of the Proposed Model
For the non-trivial equilibrium point of the proposed model (1) in the absence and presence of dispersal, we have δ � 0 and δ ≠ 0 for the respective cases.
is a positive solution of the following system of equations: From equations (37), and (39) we have Now, let us take that, Also, let (41) holds for both values.
In this case, we have to consider the following set of equations to evaluate the interior equilibrium e positive solution of equations (46)-(51) is the in- where x * 1δ , x * 2δ , x * 3δ , y * 1δ , y * 2δ and y * 3δ are all considered positive.

Stability Condition of the Proposed System without
Dispersal. At this point E 1 (x * 10 , x * 20 , x * 30 , y * 10 , y * 20 , y * 30 ), the variation matrix is provided by, Discrete Dynamics in Nature and Society 11 Proof.
e characteristic equation of the variation matrix V(E 1 ) is given by, Evaluating, we get (see Appendix A), 12 Discrete Dynamics in Nature and Society erefore, we have, Now, we observe that r 3 x * 30 /k 3 > 0 and e 3 λ 3 x * 30 y * 30 > 0. erefore, the roots of (56) are either all negative or have negative real parts.

Persistence of the Proposed System in the Presence of Dispersal.
e biological meaning of persistence is the survival of all populations in future time. For the Kolgomorov type equations, persistence was discussed by Freedman and Waltman [64]. Mathematical definition for persistence of a system can be written as follows:  Proof. Consider the matrix: where, D � (d ij ) is a matrix whose elements are given by, Here D ij denotes the nonnegative diffusion coefficient of the species from the j-th patch to the i-th patch for i ≠ j and D ii � 0, (i � 1, 2, 3, 4, 5, 6). For the continuous diffusion case, the parameter α ij ≥ 0 corresponds to the boundary conditions. Now we have, Here we can recall the theorem which states that, where, s(A f ) � max Reλ: λ ∈ the set of eigenvalues of A f . Interested readers can follow reference [65] about the proof of this theorem. e characteristic equation for the matrix A f is, Which implies, (see Appendix B) l λ + r 1 λ + r 2 λ − r 3 λ 3 +(P + Q + R)λ 2 Proof. e characteristic equation for V(E 2 ) can be written as, Taking And expanding the determinant we obtain,

Hopf Bifurcation
When the parameter of a dynamical system changes due to a sudden qualitative change in its behavior, then a bifurcation of the system occurs. Hopf bifurcation occurs in the case of nonhyperbolic nonlinear equations for two or more dimensions. It is typically happened in a differential equation for switching the eigenvalues to become purely imaginary. e fixed point switches from a stable focus to an unstable one when the real part of the eigenvalue changes from negative to positive; such a bifurcation is known as supercritical. On the contrary, the bifurcation is subcritical when the fixed point changes from an unstable focus to a stable one for switching the real part of the eigenvalue from positive to negative.
We observed a special type of Hopf bifurcation for the system proposed in this study. We find a pair of complex conjugate eigenvalues that pass through the imaginary axis of the Jacobian matrix, and all other eigenvalues consist of negative real parts. Bifurcated limit cycles can be observed in a supercritical Hopf bifurcation both physically and numerically. To distinguish this type of bifurcation with the same for the eigenvalues on the right half-plane, let us call them a simple Hopf bifurcation. In this study, we are interested in exploring the possible occurrence of a simple Hopf bifurcation around E 2 (x * 1δ , x * 2δ , x * 3δ , y * 1δ , y * 2δ , y * 3δ ) with δ as a bifurcating parameter. e following theorem provides the necessary and sufficient condition for the occurrence of a simple Hopf bifurcation of the system (1): e necessary and sufficient conditions for the system (1) to undergo Hopf bifurcation at δ � δ 0 (point of bifurcation) around the interior equilibrium point Proof. e characteristic equation for V(E 2 ) is where, N i (δ)(i � 1, 2, 3, 4, 5, 6) is a smooth function of δ. e Hurwitz matrix , , Using the condition for simple Hopf bifurcation [66] at Discrete Dynamics in Nature and Society and d∇ 5 (δ 0 )/dδ ≠ 0. Hence the theorem is proved.

Numerical Simulation
For the numerical verification of analytical findings of the previous sections, computer simulations have been performed to obtain a different graphical presentation of the proposed model. We used Matlab and Wolfram Mathematica for this purpose, and the figures obtained are extremely imperative from a realistic viewpoint. We have used a set of hypothetical data to examine the effect of dispersal speed on the patches. So, we decided to do the simulations in two different cases. Case 11 attributes to the prey-predator system when the prey population budges independently in its patches, i.e., dispersal does not happen (δ � 0). For Case 12, we consider the shift in the prey population between patches (δ ≠ 0). To simulate the proposed model, we take the value of the parameters as provided in Table 1.
From Figures 3 to 5, we observe that with the increasing values of k i , the i th (i � 1, 2, 3) predator y i gradually increases and remaining species remain constant throughout. Following the above cases, we found that, the changing carrying capacity only affects a certain patch. Now in Figure 6, we consider that, all the environmental carrying capacities are equal, that is, k 1 � k 2 � k 3 � k (say) and observe that, all predator species gradually increase, while all prey species remain constant throughout as k increases. Now, the impact of the mortality rate of the predator species m i (i � 1, 2, 3) on the equilibrium components of the prey-predator populations is represented through Figures 7 to 10 when there is no dispersal.
From Figures 7 and 8 we observed that, as m i increases in the i th (i � 1, 2) patch, then the x i (i � 1, 2) species gradually increases and the y i (i � 1, 2) species follow the parabolic path and all other species remain constant throughout. It is clear from Figure 9 that, when m 3 increases, x 3 species gradually increases and the corresponding predator species y 3 gradually decreases and all other species of the system remain constant throughout. In Figure 10 we consider the death rate of all predator species as equal i.e., m 1 � m 2 � m 3 � m (say) and observe that the three prey species x i gradually increase, predator species y i (i � 1, 2) follow the parabolic path and the remaining predator species y 3 gradually decrease as m progresses.
We have studied the influence of environmental carrying capacity and death rate of predators on species without dispersal. We found that parameters such as k 1 , k 2 , k 3 , m 1 , m 2 and m 3 play a vital role in the population dynamics of the system (1).
Here we assume the positive dispersal of the system (1) and taking δ � 7.7 and other parameters of the system are the same as given in Table 1. is dispersal speed gives the interior equilibrium point E 2 (2.34375, 2.09974, 1.93939, 7.16543, 2.28463, 0.677693). Now, the main objective is to study the effect of dispersal on the equilibrium point E 2 as well as its stable nature. e time series graph of the system (1) for δ � 7.7 with the initial condition (x 1 (0), x 2 (0), x 3 (0), y 1 (0), y 2 (0), y 3 (0)) � (3.1, 3.5, 3.7, 0.3, 0.4, 0.5) is presented in Figure 11.     Figure 11 reflects the stable behavior of the proposed prey-predator system in the presence of dispersal. Here we found that each of the species of system (1) gradually converges to the inner equilibrium point E 2 as time progresses. is observation concludes that the proposed system (1) persists in the presence of dispersal ( eorem 4).  erefore, the stability of the proposed model is completely dependent on the dispersal speed δ. e effect of k 1, k 2 , k 3 , and k(k 1 � k 2 � k 3 � k) on the system (1) is presented in Figures 12 to 15.
From Figure 12, we observe that the predator species of the first patch gradually increases and all other species remain constant with the progression of k 1 . In Figures 13 and  14, the predator species of the i th patch gradually increase as k i (i � 2, 3) progress. In Figure 15, we consider k 1 � k 2 � k 3 � k (say) and observe that, all predator species gradually increase as k progresses and all prey species maintain a constant level throughout. erefore, we conclude that carrying capacity can affect only the predator levels with the dispersal of prey species.
Again, to observe the effect of predator mortality rate on species, we consider Figures 16 to 19.
From Figure 16, we observe that, the predator species in patch one gradually decreases. Consequently, prey species in patch one gradually increases. Also, predator species in patches two and three increase progressively, and the remaining two prey species maintain constant levels. e same types of observation are seen at in Figures 17 and 18, respectively. In Figure 19, we consider m 1 � m 2 � m 3 � m (say) and observe that the predator species y 1 follow a parabolic type path, the species y 2 and y 3 increases first, then decreases and the three prey species gradually increase as m increases.
erefore, we conclude that the death rate of predators affects all the population levels.
Finally, we observe the effect of dispersal speed δ on the equilibrium components of the system (1). For different values of dispersal, the equilibrium points are provided in Table 2. Table 2 reflects that the population level of the prey species x i (i � 1, 2, 3) at the equilibrium level is independent of δ as we obtained x i � m i /e i for i � 1, 2, 3 in the theoretical section. erefore, equilibrium levels of the prey population are not affected by the dispersal speed in any patch. On the contrary, the population level of predator species is changing gradually. e pictorial presentation of the trophic cascade of the system (1) with increasing dispersal speed is described in Figure 20.
Here we found that, the predator population is very sensitive to changing values of δ and therefore the dispersal speed is important for the persistence of the three patchbased systems. Now, if we gradually increase the value of δ, then the solutions of the proposed model system near E 2 shift from its unstable periodic limit cycle oscillations to the stable steadystate through Hopf bifurcation. is dynamical property via Hopf bifurcation has been shown in Figure 21. Here, we have taken all the parameter values as previously and δ as the bifurcation parameter.
All figures show a subcritical Hopf bifurcation localized at δ � 0.35. e proposed system undergoes a limit cycle for δ < 0. 35. e equilibrium point of co-existence is stable locally and asymptotically for δ > 0.35 as shown in Figure 21.

Discussion and Conclusion
In this study, a three-patch based prey-predator system is presented with a strong Allee effect in the first two patches.
is is the first attempt to study the dynamics of the preypredator system in three-patch environments with the Allee effect in the first two patches. e impact of the dispersal speed on the stability and persistence of the proposed system is elaborately studied in this paper. For a better understanding of the proposed model, we provide the concept of balanced dispersal. We observed that the equilibrium level of the prey species depends on the mortality rate of the predator species as well as the conversion rate of the prey biomass to the predator biomass. ese population levels are not affected by the speed of dispersal. But the speed of dispersal has a definite effect on the equilibrium level of predator species. In addition, the stable nature of the equilibrium point depends on the mortality rate of the predator in each patch. e stability criterion of the system in the absence of dispersal ( eorem 3) can help to maintain the ecological balance. But the system becomes stable in the presence of positive dispersal speed ( eorem 5). erefore, we conclude that the mortality rate of the predator species and the dispersal speed play a vital role in the dynamics of the supposed ecological system. e numerical verification of the analytical findings is vital to the biological relevance of this type of model. Moreover, graphical views always support the findings and contribute to the vast future development of the ecosystem. erefore, the proposed system may contribute to the dynamics of the prey-predator in the assumed situations.
Finally, we conclude that the proposed system in threepatch environments is very interesting and shows complex dynamics. But the model can also be modified by taking the Allee effect in all three patches. Predator species can also be assumed to move between patches, making the model more interesting and realistic.

A. Proof of Theorem 3
e characteristic equation for V(E 1 ) is given by Expanding e authors get, Discrete Dynamics in Nature and Society

Data Availability
No data were used to support this study.

Conflicts of Interest
" e authors declare that they have no conflicts of interest."