Numerical study of diffusive fish farm system under time noise

In the current study, the fish farm model perturbed with time white noise is numerically examined. This model contains fish and mussel populations with external food supplied. The main aim of this work is to develop time-efficient numerical schemes for such models that preserve the dynamical properties. The stochastic backward Euler (SBE) and stochastic Implicit finite difference (SIFD) schemes are designed for the computational results. In the mean square sense, both schemes are consistent with the underlying model and schemes are von Neumann stable. The underlying model has various equilibria points and all these points are successfully gained by the SIFD scheme. The SIFD scheme showed positive and convergent behavior for the given values of the parameter. As the underlying model is a population model and its solution can attain minimum value zero, so a solution that can attain value less than zero is not biologically possible. So, the numerical solution obtained by the stochastic backward Euler is negative and divergent solution and it is not a biological phenomenon that is useless in such dynamical systems. The graphical behaviors of the system show that external nutrient supply is the important factor that controls the dynamics of the given model. The three-dimensional results are drawn for the various choices of the parameters.

disease on the human community 10 .Hanif et al. employed the numerical technique to find the solution of the Caputo-Fabrizio-fractional model of the coronavirus pandemic 11 .Authors considered the various epidemic models analyzed the disease dynamics [12][13][14] .
A great number of stochastic process models investigated comprise space and spatial variables 15 .Nevertheless, a couple of authors studied and observed fish farm dynamics models just as stochastic procedures due to undetermined stochastic terms or variables 16 .In 1998, Virtala et al. 17 constructed such a model, which is consistent and formulated the reason for dead fish due to inevitable accidents.In addition, from 1990 to 2003 Harris et al. 18 worked on stochastic models which are of fault damage zones that are generated by statistical properties for fault populations.Yoshioka et al. 19 showed the art of the state of modeling, computation, and analysis of stochastic control science in environmental engineering and research areas related to these fields.Gudmundsson et al. 20 gave their efforts on an age-structured system comprised of time and space and described the gap connecting biofuel and the age of the system.Furthermore, Sullivan et al. 21and Lewy et al. 22 appreciate stochastic procedure in their models and contemplate it emphatically reliable method.Schnute 23 narrated such models that accommodate a bit of error and then appraise both stochastic process and stochastic PDEs to estimate errors.Stochastic PDEs own such properties and characteristics that enhance unpredictability and errors.León-Santana et al. 24 figure out procedures to accomplish earthly and habitat models based on linear stochastic PDEs.Likewise, Nøstbakken 25 numerically works out on stochastic PDEs for ecological systems and wildlife like fish farms.Additionally, Reed and Clarke 26 collected optimum reaping rules for organic phenomena along with stochastic PDEs.As our model is substantial and earthly so, we convert the system of PDEs into stochastic PDEs because the substantial model can perform randomness in behavior at some particular stage and results may not be predicted 27 .In extension, stochastic has numerous applications in real life.The majority of the researchers and authors used the stochastic in their work and paper.So, we prefer stochastic PDEs instead of simple PDEs 28 .
The fish form model under the influence of time noise is given as, with initial conditions here u(x, t), v(x, t), and w(x, t) are the densities of nutrient, fish, and mussel population at a point (x, t) respectively.Also, there are four parameters present in the given model that is φ , µ , α , and ξ which describe the rate of external deposition of food, digestion rate of food of fish community and maximum digestion rate of food of mussel community respectively.These parameters have had a great effect on the ecosystem and its functioning.
Here, δ , γ , and β describe the death rate respectively, competition occurring within a species, and the proportion of food supply to biofuel of fish.In addition, ρ and η describe the death rate and transfer the ability of mussel.Thus, the external deposition of food does not be digested into fish biofuel so that it could reach oysters in the form of specific pesticide-free matter and mussels can easily absorb it.Moreover, d u , d v and d w be the coefficients of diffusion and Ḃi be the standard Wiener one-dimensional processes, with σ i that represent noisy strengths where i = 1, 2, 3 and it is a Borel functions 29 .
To work on stochastic is a challenging task especially when we have to deal with non-linear terms.Numerous researchers worked on SPDEs and their numerical solutions by two finite difference schemes and methods and proved the consistency and stability 30 .The stochastic procedure can be beneficial to demonstrate some of the unpredictable results in the accomplishment of the distinct objective because they handle the randomness in the model 31 .A stochastic method commonly used in various fields and has real-life applications in gaming theory, surveys, tracking location, and probability statistical analysis 32 .In extension, SPDEs are used in numerous models such as the substantial or physical system with time frame because it consists of a random variable that calls noise term calculated with the Wiener process which dominates the unpredicted behavior of random behavior 33 .Therefore, it is extensively in numerous mathematical models like echography, pictorial representation, ultrasonography, computational molecular biology, and financial markets like a trading floor that vary with time randomly 34 .
Chessari et al. considered the backward stochastic differential equations and employed various numerical methods 35 .Zheng et al. used the finite elements methods for the study of SPDEs 36 .Röckner et al. worked on the wellposedness of the SPDEs 37 .Gyöngy et.al. used the lattice approximation of the SPDEs perturbed by white noise on a bounded domain R d for d = 1, 2, 3 and gained the convergence rates of the approximation 38 .The authors gained the numerical approximation of Bagley-Trovik and fractional Painleve equations by using the cubic B-spline method 39 .Arqub et al. worked on the numerical computing of the Singular Lane-Emden type model by using the reproducing Kernal discretization method 40 .Sweilam et al. worked on the numerical solution of a stochastic extended Fisher-Kolmogorov equation perturbed by multiplicative noise 41 .
The main contribution of this work is given below: • The classical models cannot represent the true behavior of nature.So, need of the hour to consider the clas- sical model under the impact of some environmental noise.
• The fish farm model is considered under the temporal noise. (1) • The underlying is numerically investigated.
• Two schemes are constructed and used for the numerical study.
• Schemes are von Neumann stable and consistent.
• Equilibrium points are successfully gained.
• The graphical behavior of the state variables is explained from the biological point of view.
• The MATLAB 2015a is used for the graphical behavior of the test problem.

Numerical methods
In order to analyze the given system of equations, we discretized the whole domain of space and temporal variables.The grid points (x l , t m ) are explained as Here, M and N 1 are the integers, and x = h and t = k are stepsizes of space and temporal respectively.

Stability analysis
By Von-Neuman method of stability 42 , where Ân m is a Fourier transformation of A n m which is given below, Where θ is a real variable.By substituting this value in finite schemes, we may get The sufficient and necessary condition for Von Neumann the stability method is, x d =dh, d = 0, 1, 2, 3, . . ., M. t e =ek, e = 0, 1, 2, 3, . . ., N 1 .
(  Proof Von Neuman stability criteria are used for linear equations.So non-Linear term can be linearized by taking v e d = ψ 1 and w e d = ψ 2 where ψ 1 , ψ 2 and φ are local constants so it can be set equal to zero.Thus, finite difference scheme for Eqs.(1) in ( 5) can be written as, By amplification factor, Eq. ( 13) can be written as, Now by using independent of Wiener process increment and amplification factor, we reached at ≤ 1 , then (15) becomes, Here, = ̺ 1 .According to the stability definition, we deduced that Eq. ( 1)is von Neumann stable.
Similarly, Eq. ( 6) is linearized as follows, Now by using independent of Wiener process increment and amplification factor, we reached at �t. ≤ 1 , then equation (20) becomes, Here, According to the stability definition, we deduced that this scheme is von Neumann stable.
Similarly, Eq. ( 7) is linearized as follows, Now by using independent of Wiener process increment and amplification factor, we reached at ≤ 1 , then (25) becomes, Here, According to the stability definition, we deduced that this scheme is von Neumann stable.

Theorem 2 The SIFD scheme for u(x, t), v(x, t), and w(x, t) is von Neumann stable in the mean square sense.
Proof Similarly, Eq. ( 8) is linearized as follows, (1 www.nature.com/scientificreports/Now by using independent of Wiener process increment and amplification factor, we reached at ≤ 1 , then (30) becomes, Here, According to the stability definition, we deduced that ( 8) is von Neumann stable.
Similarly, Eq. ( 9) is linearized as follows, Now by using independent of Wiener process increment and amplification factor, we reached at ≤ 1 , then Eq. ( 35) becomes, Here, According to the stability definition, we deduced that ( 9)is von Neumann stable.

Consistency
A finite difference scheme L| (l,m) u| (l,m) = G| (l,m) is consistent with SPDE LU = G at a point (x, t), if for any continuously differentiable function φ(x, t) = φ in mean square that is

Theorem 3 The SBE scheme given in Eqs. (5)-(7) is consistent in the mean square with SPDE Eqs. (1)-(3).
Proof The consistency of the proposed SBE scheme is given as By using the operator as F(u) = (1 (1 )) ŵe (θ).So, the given scheme for state variable u is consistent.Similarly, Consistency for Eq. ( 6) is as follows, Hence, we conclude that  (47)  So, the given scheme for state variable v is consistent.Similarly consistency for Eq. ( 7) is , Hence, we conclude that E|F(w)| e d − F| e d (w)| 2 → 0 as (d, e) → ∞ .So, the given scheme for state variable w is consistent.

Theorem 4
The proposed SIFD scheme given in Eqs. ( 8)-( 10) is consistent in the mean square with SPDE Eqs.
Proof Now, the consistency of the proposed SIFD scheme for Eq. ( 8) is, Hence, we conclude that  (53) So, the given scheme for state variable u is consistent.Similarly, consistency for the Implicit finite scheme for equation (9).Hence, we conclude that . So, the given scheme for state variable v is consistent.Similarly, consistency for the Implicit finite scheme for equation ( 10) is Hence, we conclude that . So, the given scheme for state variable w is consistent.

Convergence
The convergence of the stochastic implicit scheme is analyzed in the means square sense.

Theorem 5
The stochastic implicit scheme is given by Eqs.(8)-( 10) is convergent in the mean square sense.

Proof
as the scheme is consistent in the mean square sense i.e., L e d u e d → L e d u as x → 0, t → 0 and (d�x, e�t, ) → (x, t), also, scheme is stable, then (L e d ) −1 is bounded.So, E u e d − u 2 → 0 .Hence proposed scheme for u is convergent in the mean square sense.By doing the same practice, we can show that the scheme for v, w is convergent.The convergence of another scheme can be deduced in a similar way.

Graphically representation
The initial conditions are u(x, 0) = v(x, 0) = w(x, 0) = 1 43 .The equilibria of given equations are Also, assume that ς 1 = µδ β , ς 2 = µρ η , ς 3 = ς 2 1 + αδ γ µ ς 2 ς 1 − 1 In 44 , Gazi et al. analyzed the different dynamics of the Fish farm model.They worked on the local behavior of the Fish farm system of the equations and summarized the existence of equilibria as follows Equilibrium point feasibility condition stability condition always By concluding this brief discussion, equilibrium E 1 is sable if φ < ς 1 , φ < ς 2 and becomes unstable if exterior nutrients exceed the value of ς 1 .The equilibrium E 2 goes stable if exterior nutrient between the value ofς 1 and ς 3 with (ς 1 < ς 3 ) .The equilibrium point E 3 is stable only if the feasibility condition, as well as stability condition, exists A. The given model is around the coexistence equilibrium if the exterior nutrient supply exceeds values ς 3 with ς 1 < ς 2 .Thus, the given Fish form model is stable in various levels of exterior nutrient supply.
The values of the parameters that are given in Table 1 showed by the Figs. 1, 2, 3, 4, 5, 6, 7 and 8) .The Figs. 1, 2, 3 and 4) have noise strength 0.25 and the Figs. 4, 5, 6, 7 and 8 have noise strength 0.025.The BFD Scheme for external nutrient φ = 4 is discussed in the Fig. 1 and E 1 is a stable point nevertheless the graphical representation displays that state variable u(x, t) and v(x, t) have negative values and it is an imperfection in SBE scheme.Furthermore, the SIFD scheme for external nutrient φ = 4 discussed in the Fig. 2 and E 1 is a stable point nevertheless the graphical representation displays that state variable u, v, and w have positive values for the entire domain and this showed that this scheme is stable as well as keep the entire behavior.The SBE scheme for external nutrient φ = 10 is discussed in the Fig. 3 and E 2 is the stable point but the graphical representation displays that the state variable u(x, t) has negative values and it is insignificant for the population dynamics.Similarly, The IFD scheme for external nutrient φ = 10 discussed in the Fig. 4 and E 2 is a stable point neverthe- less the graphical representation displays that entire state variable u, v, and w have positive values.Moreover, The BFD scheme for external nutrient φ = 100 is discussed in Fig. 5 and η = 10.75 , is not stable and the state variables have negative values along with divergent behavior.In addition, The IFD scheme for external nutrient φ = 100 discussed in the Fig. 6 and η = 10.75 , the equilibrium point E 3 is stable as well as its graphical behavior display that all state variables own the positivity.The BFD scheme for external nutrient φ = 100 is discussed in Fig. 7 E 4 is gained and the state variables keep the positivity.Similarly, The IFD scheme for external nutrient φ = 100 discussed in the Fig. 8 and the E 4 point is stable nevertheless the graphical representation displays that the entire state variable has positive values.MATLAB 2015a is manipulated for this discussion and analysis of the stated Fish Form model.The fish farm models are population dynamics and necessarily their; solutions must be positive.We have used 2 schemes for the numerical approximation of the governing model.One technique fails to preserve the  www.nature.com/scientificreports/positive behavior while the other has positive and converges towards the true steady states.One of the most compelling reasons to consider this model with a numerical scheme is to construct and apply the scheme in a way that yields positive solutions.As the governing model is population dynamics and its minimum values can be zero but can never be negative.So its solution must preserve the positivity.The results of the stochastic implicit finite difference scheme resemble the positive steady states.As population dynamics have random behavior, it

Conclusions
The fish farm model under the influence of the Wiener process is considered.The literature on numerical study for the stochastic partial differential equations is required.We have designed two novel and time-efficient numerical techniques for the computational study of the underlying model.The schemes are proposed stochastic backward Euler(SBE) and proposed stochastic Implicit finite difference(SIFD) scheme.The analysis of schemes is analyzed in the mean square sense.Both schemes are compatible with the given system of equations and the linear analysis of schemes is analyzed.The underlying model has four equilibrium points.When the SBE scheme is used for the study, equilibrium E 1 is successfully gained but it also shows negative behavior, E 2 is also obtained with  www.nature.com/scientificreports/negative behavior, E 3 is not gained because it has divergent behavior, and E 4 is obtained with positive behavior.On the other hand, when the SIFD scheme is applied, equilibrium point E 1 is successfully attained with positive behavior E 2 is attained with convergent and positivity, E 3 also gained with positive behavior, and E 4 is success- fully obtained with positive behavior for the given values of the parameter.From the graphical behavior of the system, it is observed that the effect of external food is the main factor that controls the dynamics of the model.The simulations are drawn for various values of the parameters.The study of the SPDEs and their dynamics is the need of the hour.So, we will try to analyze its various aspects such as qualitative analysis of the model and use of optimal strategies to control the SPDEs.

Table 1 .
The values of parameters.