Spatiotemporal Dynamics of a Reaction Diffusive Predator-Prey Model: A Weak Nonlinear Analysis

. In the realm of ecology, species naturally strive to enhance their own survival odds. Tis study introduces and investigates a predator-prey model incorporating reaction-difusion through a system of diferential equations. We scrutinize how difusion impacts the model’s stability. By analysing the stability of the model’s uniform equilibrium state, we identify a condition leading to Turing instability. Te study delves into how difusion infuences pattern formation within a predator-prey system. Our fndings reveal that various spatiotemporal patterns, such as patches, spots, and even chaos, emerge based on species difusion rates. We derive the amplitude equation by employing the weak nonlinear multiple scales analysis technique and the Taylor series expansion. A novel sinc interpolation approach is introduced. Numerical simulations elucidate the interplay between difusion and Turing parameters. In a two-dimensional domain, spatial pattern analysis illustrates population density dynamics resulting in isolated groups, spots, stripes, or labyrinthine patterns. Simulation results underscore the method’s efectiveness. Te article concludes by discussing the biological implications of these outcomes.


Introduction
Mathematical modeling is an efective tool for describing and comprehending the complex systems' spatiotemporal dynamics.Te behaviours of various species in the natural ecosystem reveal a wealth of dynamical traits.Te habits of diferent species in the natural ecosystem exhibit a wide variety of dynamic characteristics [1].Many species have threatened extinction in recent decades as a result of inadequate resources, overexploitation, pollution, and predation.Most species extinctions are caused by an imbalance in the environment or an ecological system.External support trends to prevent species extinction including refuge and population limitation to a single place [2][3][4].Considering the issue of extinction, many authors [5][6][7][8][9][10] investigated the dynamical behaviour of the same system.However, because we live in a spatial environment, spatial aspects in predatorprey systems should be considered.As a result, to characterize these systems, reaction-difusion equations [11] should be used.Te interaction between prey and predators receives the most attention in the ecosystem because it is so important and happens all over the world [4,12].In the realm of ecology, the dynamics of predator-prey interactions have long been the focus of interest and research.For forecasting and controlling ecosystems as well as deciphering the complex interspecies balance, it is essential to comprehend the spatiotemporal dynamics of these interactions.Mathematical models may be used as a useful tool for analyzing and forecasting the behaviour of predator and prey populations, which is one way to investigate these dynamics [13].
Te spatiotemporal dynamics of a response difusive predator-prey model is an important study area for a number of reasons [14][15][16].First of all, it enables us to investigate the intricate interplay of species in a vast ecosystem.We may investigate how the movement and dispersal of predators and prey afect their population dynamics, geographical distribution, and overall stability by introducing difusion factors.Furthermore, this study issue is extremely important in the light of global environmental changes and habitat fragmentation [17][18][19].Landscape changes and the loss of habitat connectivity can have a signifcant impact on the dynamics of predator-prey interactions.We may acquire insights into the resilience and susceptibility of ecosystems under changing conditions and improve conservation policies aiming at maintaining biodiversity by examining the spatiotemporal dynamics of predator-prey models.Finally, the study of the spatiotemporal dynamics of a response difusive predator-prey model is an extremely interesting research area.We can acquire a better understanding of how spatial dynamics and mobility impact the stability, coexistence, and persistence of predator and prey populations by including difusion variables into mathematical models.Tis study's fndings have implications for biodiversity conservation, ecosystem management, and our general knowledge of complex spatiotemporal systems in nature and beyond [20,21].
Te creation of spatiotemporal patterns is signifcant in describing the dynamics of interacting populations as part of a larger ecological community [22,23].Such populations are scattered heterogeneously over their habitats, resulting in patterns that might be stationary or nonstationary in terms of time.Turing [24] demonstrated that difusion-driven instability might bring out the spatial patterns in a system of coupled difusion equations.Segel and Jackson [25] proved that Turing's theories can also be applied to species diversity.Gierer and Meinhardt [26] also demonstrated a biologically justifed formulation of the Turing-type model.Levin and Segel [27] proposed spatial pattern formation as one of the possible causes of planktonic fakiness.
Nevertheless, when compared to two-species food chain models, pattern development for three-species systems has received far less attention.In the study in [28], White and Gilligan give adequate conditions for a difusive model to induce difusion-driven instability.Chen and Peng [29] established the existence of nonconstant positive steady conditions to prove the endurance of static patterns for a crossdifusion model.Tey also showed that without cross difusion, the corresponding model fails to deliver any stationary pattern.
Te time-dependent spatial patterns can be studied using the amplitude equation.It ofers us a diplomatic classifcation of weak nonlinear studies' pattern formation.It is feasible to derive the given model's amplitude equation using multiple-scale analysis.
A wide range of computational approaches has been employed for the nonlinear prey-predator model, including the fnite diference method [30], B-spline method [31], fnite element method [32], spectral method [33], perturbation method, variational iteration method (VIM) [34], and so on.Researchers, on the other hand, have always sought to identify the most efective numerical method [35].A sinc function interpolation method for the prey-predator system has been developed.
In this article, our objective is to explore the intricate spatiotemporal dynamics exhibited by models involving three interacting species (specifcally, two prey species and their shared predator).Our primary focus lies in analyzing the consequences of the stochastic dispersal of all three species on their respective population distributions.In addition, we aim to delve into fundamental ecological inquiries, such as the phenomenon of extended transience and the infuence of habitat size on the resulting patterns.To address these questions, we thoroughly examine a temporal model based on reaction-difusion principles.Tese systematic investigations hold values in discerning and contrasting the ensuing dynamics and their underlying causal factors, given that these three models form a sequence of nested systems.Te following is a breakdown of the structure of our work.In Section 2, we introduce a temporal model with three species-two prey species and one predator, and discuss the dynamical behaviours briefy as well as provide stability of the model then expand by including spatial components, where we also present a broad range of spatiotemporal dynamics that the model is capable of.Te necessary condition of Turing instability as well as bifurcation is identifed in Section 3. In Section 4, a sinc interpolation approach has been developed using our model.We obtain the amplitude equation by applying a weak nonlinear analysis close to the parameter threshold value for the patterns in Section 5.In Section 6, we perform numerical simulations to support our analytical results.Ten, we show the Turing patterns for some fxed parameter values in Section 7. We discussed and concluded our work in Section 8.

Temporal Model and Stability
In this section, we framed a model of prey-predator interactions involving three species associated with a system of three diferential equations and discussed the stability of this model.
Te parameters used in the model are described in Table 1.
Stability in ecology is frequently described as the capacity to resume the initial structure or functioning following external disturbances.Now, we fnd the equilibrium points, confrm their existence, and examine their stability in order to better understand the dynamical behaviour of the system (1).Solving the following simultaneous equations yields the equilibrium points as follows: 2 International Journal of Diferential Equations System (1) admits fve equilibrium points by solving the above equations.
Proof.Te Jacobian matrix for E 1 is given by the following equation: International Journal of Diferential Equations Te eigenvalues of the matrix are Hence, E 1 is asymptotically stable when the restrictions g 2 < φK 1 and Kμ 1 ] < ρ hold and unstable when g 2 > φK 1 and Kμ 1 ] > ρ.
Proof.Te Jacobian matrix for E 2 is given by the following equation: Here, Terefore, E 2 is asymptotically stable if the above condition holds, otherwise E 2 is a saddle point.
Proof.At E 3 , the Jacobian matrix is given by the following equation: where Te characteristic equation is as follows: Tis can be written as follows: where Te equilibrium point E 3 is stable by the Routh-Hurwitz criterion if υ 1 > 0, υ 2 > 0, υ 3 > 0, and Proof.Te Jacobian matrix at E * of ( 1) is as follows: where International Journal of Diferential Equations Te characteristic equation is as follows: Tis can be written as follows: where Here, Te stability of (1) in the nonspatial case is described in details above.Ten, the Turing instability of system (1) will be investigated by incorporating difusive terms.

Spatiotemporal Model and Bifurcation Analysis
In this section, we extend (1) by introducing the random dispersal of all three species.Te associated spatiotemporal system is given by the following equation: with the following initial conditions: and homogeneous boundary conditions as follows: Here, Λ is a bounded domain in R 2 with the smooth boundary zΛ * ; z/zn represents the outward drawn normal derivative on zΛ * while n is the outward unit normal vector of zΛ * .Te densities of the three species are denoted by a 1 (x, t), a 2 (x, t), and b(x, t), respectively, at spatial position x and time t.Te positive parameters d i (i � 1, 2, 3) denote the difusion coefcient of the species.
Te Jacobian matrix J 0 for the model in (2) at the equilibrium point By implementing a small amplitude spatiotemporal perturbation around the coexistence steady state and by linearizing, we get the following: Here, k * � (k * x, k * y, and k * z) is the wave number vector and k * � |k * | represents the wave number.Te characteristic equation of ( 23) is as follows: Here, 6 International Journal of Diferential Equations where 2) is stable if tr(J λ ) < 0 and det(J λ ) > 0. If it fails to meet the abovementioned conditions, the system is considered unstable.As a result, R(k 2 * ) can be reframed as follows: where ) is a criterion for Turing bifurcation spatial patterns if all other mentioned conditions have been met.
In the next section, system (2) can be solved by using the sinc interpolation method as follows.

Interpolation Method
Te sinc method is a very efective numerical technique [36].It is frequently used in many diferent areas of numerical analysis, including quadrature, integral and diferential equation solution, and interpolation [37].When singularities or unbounded domains are present, the method may efectively handle them.As the number of bases rises, the method's error converges exponentially to zero.System (2) can be solved by dividing the interval [0, 2π] into N equally spaced nodes, each of which corresponds to a regular region ξ � [0, 2π] × [0, 2π].A periodic sinc function S N [38] is defned as follows: where h � 2π/n.On the other hand, S N (x α − x β ) is an N order unit matrix.
For functions a 1 (x), a 2 (x) and b(x) defned on [0, 2π], let I N be the interpolation operator.We have [39] the following equations: Te following relation holds at the collocation nodes (x u , y v ): Here, a 1 , a 2 , and b are as follows: respectively.As a result, (33) can be expressed as follows: where N and D d N , and D (0,0) � I N ⊗ I N .I N is an N order unit matrix.Te discrete form of ( 19) can be written as follows using equations ( 32), (34), and (35), 8 International Journal of Diferential Equations Here, Terefore, To get the numerical solution of (19), we use the numerical scheme NDSolve in Mathematica software for solving various initial conditions.
Te Turing patterns near the bifurcative value were described using amplitude equations.For the aforementioned models, the dynamics obtained by varying the bifurcative values were interpreted using weak nonlinear analysis in the following section.

Amplitude Equation and Stability Analysis
In this section, to obtain the amplitude equations that make up the various Turing patterns, we will use the multiple-scale analysis method.We can then determine the necessary conditions for the appearance of various Turing patterns by examining the stability of the amplitude equations, allowing us to construct various Turing patterns through numerical simulation.
We use Taylor's expansion to expand (2) at equilibrium E * , since we know that the amplitude of the equation cannot be determined directly.Te system solution could be expanded as follows [40]: c � c 0 +  3 i�1 Z i e ik i r + Complex conjugate.Furthermore, system (2) can be written as follows: where c � When Ω approaches Ω c , we must examine the dynamical behaviour and then expand as follows: where ϵ is a sufciently small parameter and Ω is the critical value.
As the series form of ϵ, we expand c and N as follows: Here, L can be expressed as such that z/zt � z/zT 0 + ϵz/zT 1 + ϵ 2 z/zT 2 + . . .T i is a dependent variable that we use as a base for our time calculation, whereas amplitude is a slow variable.
By incorporating the prior equations into ( 19) and extending to diferent orders of ε, we get the following: An example of frst order ϵ is considered.Since the initial linear operator of the system is Lc, then ((a 1 ) 1 , (a 2 ) 1 , (b) 1 ) T is the linear combination of the eigenvectors corresponding to the zero eigenvalue.
and have deduced that x i � Hy i .Assuming y i � 1 and 10

International Journal of Diferential Equations
Let us now consider the ϵ 2 case.By Fredholm's solubility condition, the vector function of the preceding equation must be orthogonal to the operator L + c 's zero eigenvectors.L + c 's zero eigenvectors are as follows: 1 Te orthogonality criteria implies that By only investigating at e − ik 1 r in the following, we get another case by modifying the subscripts as follows: Te same procedures can be used to achieve the same results as follows: For ϵ 3 case, After simplifcation, we derive the coefcient expressions as 1 /K), and g � (hBH + hH 2 + hB + hH)(g 2 a 2 2 /K).As a result, the amplitude equation is as follows: International Journal of Diferential Equations Te equation for the amplitude can be written as follows: where the phase is ϖ � 2π/3 and the mode ς j can be defned as ς j � |W j |.From (53), Considering the real part, we get the following: We derive the following mathematical relations by substituting the value of (53) in (52) and taking the real part of the equations into account as follows: When ϖ � 0 and ϖ � π, the value of f is positive and negative, respectively.If we consider f > 0, then the system's pattern will be stable and defned as F 0 , if we consider f < 0, then the system's pattern will be stable and defned as F π .As a result of (56), the following system of equations is obtained as follows: We examine the stability properties of the amplitude equations using (57).Theorem 6. Te prerequisite for stationary state stability is Ω * < Ω * 0 and the condition for the unstable is Ω * > Ω * 0 , where Ω * 0 � 0.
On simplifcation, we get the following equation: Tus, the hexagonal pattern is stable if we get the following: On comparing (67) and (70), λ 1 is positive.Now, calculating λ 2 and λ 3 , we get the following: Again comparing (68) and (71), we get (71) as positive.
As a result, when ς f � ς − f , all eigenvalues turn positive, which assures the instability of hexagonal patterns.Now, the system can be simulated numerically in a 350 × 350 2D square lattice.

Numerical Computation
Analytical studies are never completed unless the results are numerically validated.System (2) is simulated by the ODE solver, and the results are displayed with parameter values.

International Journal of Diferential Equations
Various numerical results are shown to validate the analytical stability analysis presented in the preceding sections.Te analyses have been carried out with positive parameter values.As shown in the images below, we use a variety of parameter values to acquire a better understanding of the dynamics of system (2).Te numbers specifed for the parameters are not based on feld data, and they are purely hypothetical parameters meant to depict the system's dynamic behaviour.
Te density plot graph depicts the interplay of prey and predator populations at various geographical regions.Overlapping density plots of both species allow for the identifcation of locations with more strong predator-prey interactions.Places with high predator density and low prey density may indicate signifcant predation pressure, whereas places having high prey density and low predator density may indicate prey refuge or favourable breeding grounds.
Let us examine model ( 2) with the following parameter values g 1 � 1.3, K 1 � 5, σ � 0.027, ] � 0.08, g 2 � 0.856, K 2 � 5, φ � 0.002, η � 0.09, μ 1 � 0.1, μ 2 � 12, and ϱ � 1.2.From Figures 1 and 2, it is clear that both populations survive for the long term.In addition, Figure 1 demonstrates that the periodic solution emerges for the frst prey, second prey, and predator.Consider the same set of parameters as above and slightly change the values of the intrinsic growth rate such that g 1 � 1.5 and g 2 � 0.5.From Figure 2, it is noticeable that all three species oscillate at a certain time interval then it becomes stable.Figures 1 and 2 are qualitatively equivalent.Te outcomes of the remaining instances with instances II and III parameters are qualitatively equivalent.Te sole distinction is the duration of the fuctuation period.It appears that being close to steady states is critical, even if the steady state is unstable.Only then can the species cohabit in an oscillating fashion and prevent a large fuctuation (or blooming).It is concluded that prey 1, prey 2, and predator populations coexist simultaneously.Now, we assume the following parameters c 1 � 1.5, K 1 � 15, α � 0.027, c � 0.08, c 2 � 1.9, K 2 � 15, β � 0.002, δ � 0.09, w 1 � 0.1, w 2 � 1.2, and ρ � 1.2. Figure 3, shows that the population persists and are stable.
For Figures 4 and 5, let the initial conditions be x 0 � 1.2, y 0 � 1.8, and z 0 � 2.4, and all other constants are 1.From Figure 4, it is noticed that even when K 1 is set to a peak value, prey 1 persists at zero level.Tat is, the prey population vanishes for a large value of K 1 , while the other two populations oscillate at time t.Similarly, Figure 5 shows that for the carrying capacity K 1 � 20, prey 1 remains at zero level and vanishes.A periodic solution occurs between prey 2 and the predator.Te dynamics of the predator species hunting the prey species are also visible in Figure 5(c).Te solution consists of multiple layers within the support and generates periodic patterns.Now, if we decrease the value of the conversion coefcient μ 2 , it leads to the extinction of the predator, while the prey population remains constant, which is shown in Figure 6.Similarly, from Figure 7, when the growth rate is increased, the prey population disappears while the predator population endures and remains stable.Only circular patterns exist within the solution's support in Figure 7(c).It is also possible to observe that the predator population is chasing the prey population which leads to the extinction of prey.

Pattern Formation
Tis section examines simulations of system (2) computationally, proving that difusion creates spatial patterns.Turing patterns can appear in ecological systems as spatially diferent zones of high and low population densities, resulting in elaborate patterns that resemble stripes, dots, or other complex structures.Te appearance of Turing patterns in prey-predator models emphasises the relevance of spatial dynamics and reveals a more complete knowledge of ecological interactions.Te emergence of patterns for specifc parameter values and difusion coefcients of ( 19) is studied, demonstrating how difusion can cause a stable steady state to become unstable.Te simulation proceeds by randomly or predefned distributing predator and prey populations over the geographical region.Te system of diferential equations is then numerically solved, which advances the populations in time.Te difusion parameters contribute to the expansion of populations at each time step, allowing predators and prey to travel and distribute throughout the domain.
Patterns develop in the population distributions as the experiment proceeds.Depending on the model's unique characteristics, these patterns might include spatial clusters, waves, or spiral structures.Patterns occur as a result of the intricate interplay between predator-prey interactions, diffusion processes, and the system's underlying nonlinear dynamics.Contour plots, showing the number of predators and prey at diferent places, can be used to visualise the simulated patterns.Tis visualisation gives insights on the population's geographical organisation and structure throughout time.
Troughout the prey-predator concept, a pattern is formed when both prey and predator difuse at diferent times in three-dimensional space.Figure 11 depicts the coexistence of spot and stripe patterns at intervals of 0, 0.1, 0.2, and 0.8.
Figure 12 illustrates that predator death rates infuence the pattern of spatial distribution.We get a pattern for t � 500 and choose difusion coefcients d 1 � 5 and d 2 � 0.2.Tere are blue and brown spot patterns.In Figure 12    International Journal of Diferential Equations        International Journal of Diferential Equations

Conclusion
In the present work, we have constructed and explored a predator-prey model characterized by a system of diferential equations incorporating reaction and difusion.Our investigation encompassed an in-depth examination of Turing patterns spatial pattern formation within the model.We conducted an extensive numerical analysis of the Turing system, investigating how pattern generation is afected by varying parameters.Te outcomes included the creation of intricate spatiotemporal patterns, showcasing the spatial complexity attainable in reaction-difusion systems, including chaotic patterns.We further introduced an innovative sinc function interpolation method applicable to three-species predator-prey systems (PPS) with intricate behaviours.In addition, we have used Taylor's expansion method to obtain the amplitude equation for the reactiondifusion system, which accurately described Turing patterns  Te investigation of Turing patterns in difusive preymodels provides important insights into the spatial organisation and stability of ecological systems.Researchers can get a better grasp of the variables impacting species coexistence, population dynamics, and ecological stability by studying the underlying mechanisms and circumstances that give birth to these patterns.Furthermore, these discoveries have practical applications in sectors such as conservation biology, pest control, and biodiversity protection, where knowing spatial dynamics is critical for making ethical choices.Tus, studying Turing patterns in difusive preypredator models is an important area of study in ecological systems.Tese models improve our knowledge of complicated ecological processes and contribute to the development of sustainable management methods by accounting for spatial dynamics.Further research into the mechanics and consequences of Turing patterns in predator-prey dynamics will surely contribute to a better understanding of the complex interplay between species interactions and geographical variability.
(a), blue spots indicate high population density and brown spots indicate low population density, whereas in Figure 12(b), both populations are equal.

Figure 1 :
Figure 1: Time evolution of species in various dimensions (a) and (b).Also, (c) depicts the spatial distribution of the species.

Figure 2 :
Figure 2: Upper panels (a and b) depict the existence of periodic solution between two preys and one predator in both two-and three-dimensional spaces and the lower panel and (c) shows the density plot view.

Figure 3 :
Figure 3: (a) and (b) exhibits the stability behaviour of prey and predator population when g 1 � 1.5, g 2 � 1.9, K 1 � K 2 � 15, and μ 2 � 1.2.In (c), we depict the non-Turing dynamic pattern view of the preys and predator in the spatial plane.

Figure 4 :
Figure 4: In the upper panels (a, b), we illustrate the time series solution of (2) showing the extinction of frst prey at K 1 � 20 in diferent dimensions.In the lower panel (c), we depict the pattern formation.

Figure 5 :
Figure 5: We show the time series solution in (a) and (b) which depicts frst prey's extinction at K 2 � 20 in multiple dimensions.In (c), it displays the pattern formations.

Figure 12 :
Figure 12: Snapshots of the spatial distribution of prey 1 and 2, and predator time has been taken as t � 50 in the frst panel (a) and as t � 500 in the second panel (b), respectively.

Table 1 :
Description of parameters.