Spatio-temporal numerical modeling of stochastic predator-prey model

In this article, the ratio-dependent prey-predator system perturbed with time noise is numerically investigated. It relates to the population densities of the prey and predator in an ecological system. The initial prey-predator models only depend on the time and a couple of the differential equations. We are considering a model where the prey-predator interaction is influenced by both space and time and the need for a coupled nonlinear partial differential equation with the effect of the random behavior of the environment. The existence of the solutions is guaranteed by using Schauder’s fixed point theorem. The computation of the underlying model is carried out by two schemes. The proposed stochastic forward Euler scheme is conditionally stable and consistent with the system of the equations. The proposed stochastic non-standard finite difference scheme is unconditionally stable and consistent with the system of the equations. The graphical behavior of a test problem for different values of the parameters is shown which depicts the efficacy of the schemes. Our numerical results will help the researchers to consider the effect of the noise on the prey-predator model.

where y(t) and x(t) represent the population density of the predator and prey at any time t.They discuss the various aspects of the model such as uniform boundedness, stability, Hopf bifurcation, etc.In this model population densities only depends on time and a couple of the differential equations.In a natural system, either the predator or the prey may move from one area to another for a variety of reasons.In this scenario, the prey-predator interaction is influenced by both space and time and the need for a coupled nonlinear partial differential equation.If such models are under investigation and also considering the effect of the fluctuation of the environment then stochastic models are preferable.
The stochastic version of the ratio-dependent prey-predator model is taken as having initial and homogeneous Neumann boundary conditions.Where P(x, t) and Q(x, t) represent the population densities of the prey and predator at any point (x, t) respectively.Here d > 0 is the death rate of the predator, a b > 0 is the carrying capacity of the prey, and m, f, c, a are positive constants that denote half capturing saturation constant, conversion rate, capturing rate, and prey intrinsic growth rate respectively.The diffusion coefficients are σ 1 > 0 and σ 2 > 0 .The B 1 (t) and B 2 (t) are the one-dimensional standard Wiener processes such that the Ḃ1 (t) and Ḃ2 (t) are the Gaussian distribution with zero mean 10 ,η 1 and η 2 are the noisy strengths which are the Borel functions.
Dealing with stochastic partial differential equations numerically is not a simple task and it becomes more difficult when it has nonlinear terms.Various researcher is working on the numerical solution of the SPDEs.Iqbal et al. considered the stochastic Newell-Whitehead-Segel equation.They discussed the existence results, derived the numerical approximation by two finite difference schemes, and proved the consistency and stability of the schemes in man square sense 11 .The authors used the multiple scale method for the numerical solutions of the SPDEs having quadratic nonlinearities 12 .Allen et al. worked numerically on the linear elliptic and parabolic SPDEs under the influence of white noise by finite difference and element methods.They showed that both methods have the same order of accuracy but the different method is not as computationally efficient as the finite element method 13 .The authors obtained the approximation of the linear SPDEs with special additive noise.The error analysis and convergence analysis of the standard finite difference and element methods are discussed.The impacts of noise on approximation accuracy are explained 14 .
Some researchers worked on the consistency and stability of the schemes as well.Namjoo et al worked on the numerical approximation of the linear SPDEs of the Itô type.They showed the consistency, stability, and convergence of the finite difference scheme 15 .In 16 , the authors found the numerical computing of hyperbolic SPDEs with finite difference methods and discussed the stability, consistency, and convergence of the scheme.Kruse worked on the computational approximation of semi-linear SPDEs by using the Milstein-Galerkin finite element scheme and discussed the error analysis of the scheme 17 .Sohalay worked on the numerical solutions of parabolic SPDEs with finite difference methods and derived the condition of convergence in mean square sense 18 .
Belabbas et al. worked on the stochastic prey-predator model under the influence of multiplicative noise with a protection zone for the prey.They discuss the different aspects of the models such as the existence, uniqueness of the global positive solutions, and boundedness.The conditions for the extinction and persistence of two species are derived 19 .The authors worked on the existence of the solutions for the class of the stochastic differential equation 20 .Souna et al. considered the prey-predator model and applied the linear stability analysis to gain the conditions for the Turing-driven instability and Hopf bifurcation 21 .More work on prey-predator, one may see [22][23][24] .
The prey-predator models are population dynamical models and necessarily the solutions must be positive.We have applied two techniques for the numerical solutions of the prey-predator model.One technique fails to preserve the convergent and positive behavior while the other preserves the positivity and convergent toward the steady states.One of the strongest motivations for considering this model is to ensure the application and construction of the numerical scheme which provides positive solutions as per the requirement of the underlying model because the under consideration model is a population dynamical model and the population may attain minimum value zero and can never be negative.So many solutions are preferred which preserve the positivity and bounded behavior for the whole domain.The results of the stochastic non-standard finite difference scheme are aligned with the actual steady states which are the positive steady states.The diffusion process is hardly considered for the prey-predator models and diffusion is a basic phenomenon in the population dynamical models because they interact with each other and diffuse with a certain diffusion rate.If we are considering the very smallest organism population models, they are not continuous in the usual sense.The microorganisms can mix and produce random behavior.So, it is quite better to consider the continuous model with a random effect.Such random behavior is observed in every physical phenomenon at a certain level.So we incorporate diffusion as well as random behavior in the prey-predator model. (

Existence and regularity analysis
The coupled system (2) and (3) can be inverted in the form of the following Volterra type integral equations; where P and Q are positive, being the population densities and they are the function of space and time i.e., P(x, t) and Q(x, t).
The goal of the current section is to guarantee the existence of the vector (P * , Q * ) being at least one solution of the system (2) and (3).The Eqs. ( 5) and ( 6) are assumed as the fixed point operator equations.For existence, we choose the space of continuous functions C equipped with supremum norm.Subsequently, we have to see the existence of the solution in a closed, convex, and bounded subset of Banach space defined as where is the zero elements of the function space C. The Schauder fixed point theorem will be used leading to the following two conditions to be verified.(i) P, Q : B r (�) → B r (�) , (ii) P(B r (�)) and Q(B r (�)) relatively compact.To ensure (i), we take the norm of an Eqs.(5), and (6)   where || 1 mQ+P || ≤ k , is true when P and Q are positive functions.Suppose || Ḃ1 (t)|| = || Ḃ2 (t)|| = H , are the bounded noise, also condition (10) is important as it serves for the length of the interval [0, ρ].Now, it is remaining to show that (P, Q) : B r (�) is relatively compact.For that we have two families P i , Q i as images for pre-images P i , Q i and we see that the difference ||P i (t) − P i (t * )|| and ||Q i (t) − Q i (t * )|| approaches to zero as t → t * i.e., easy calculation will show that P i , and Q i are equi-continuous family of operators by the well known Arzela-Ascoli theorem there exist two uniformly convergent subfamilies P i , and Q i .So, P(B r (�)) and Q(B r (�)) are relatively compact.Thus there exist at least one fixed point vector (P * , Q * ) which is the solution of the system (2) and (3).We have the following result, Theorem 1 If P(x,t) and Q(x,t) are twice continuously differentiable function and α(x), β(x), σ 1 , σ 2 , a, b, cd, f , η 1 , η 2 are bounded function then the coupled system (2) and (3) has solutions by Schauder fixed point theorem in (7) and the obtained solutionis continuous in [0, ρ * ] , where ρ * is defined in (10).

Numerical schemes
The proposed stochastic forward Euler scheme (proposed scheme-I) for Eqs. ( 2) and ( 3) is given below where �τ and h are time and space stepsizes and D 1 = �τ σ 1 h 2 and D 2 = �τ σ 2 h 2 .Mickens proposed a finite difference scheme that preserves the positivity of the solution.The proposed stochastic non-standard finite difference scheme (proposed scheme-II) for Eqs. ( 2) and ( 3) is given below www.nature.com/scientificreports/Consistency of a scheme.Definition 1 [25][26][27] .A stochastic finite difference (SFD) scheme L| r,s U| r,s = G| r,s is consistent with stochastic partial differential equation LU = G at a point (x, t), if there is any continuously differentiable function � = �(x, t) then as x → 0, t → 0 and (r�x, (s + 1)�t) → (x, t).

Von-Neumann analysis. In this technique P k
m is taken as follow and Pk m is defined as Here, η is a variable, and by putting values in the given PDE The following is the necessary and sufficient condition for this method 28 .
where χ is a constant.

Consistency of proposed scheme-I.
The consistent results of the schemes are established in the mean square sense.

Consistency of proposed scheme-II.
The consistent results of the schemes are established in the mean square sense.
Proof Suppose that P(x, t) and Q(x, t) are smooth functions and by using the L(f ) = (k+1)�τ k�τ fds on Eq. ( 2).We get By using the proposed stochastic NSFD scheme on Eq. ( 2) Equations ( 23) and ( 24) takes the form   ( L| m,k (P) =P(mh, (k + 1)�τ www.nature.com/scientificreports/By using the symmetry property of the Itô's integral , so the proposed scheme for P is consistent with stochastic PDE (2).Now, to check the consistency of 14 with SPDE (3).

Stability of the scheme-I.
Theorem 4 If |1 + a�τ − 4D 1 sin 2 ( �xη 2 )| 2 ≤ 1 , and |1 − d�τ − 4D 2 sin 2 ( �xη 2 )| 2 ≤ 1 , then the proposed SFE for P(x, t) and Q(x, t) is stable with (k + 1)�τ = T. Proof As the given technique is used on linear equations, Eq. ( 11) is linearized as follows by using Eq. ( 16), the above equation takes the following form, So, the amplification factor takes the form As the Wiener process is independent of the from the state of state variable P(x, t), the amplification factor takes the form    3), ( 12) is linearized as follow As the given technique is used on linear equations, Eq. ( 12) is linearized as follows by using Eq. ( 16), the above equation takes the following form, So, the amplification factor takes the form As the Wiener process is independent of the from the state of state variable P(x, t), the amplification factor takes the form where, |η 2 | 2 = χ .So, the given scheme for (3) is stable.

Stability of the scheme-II.
Theorem 5 The proposed stochastic NSFD scheme for P(x, t) and Q(x, t) is unconditionally stable in the mean square sense.
Proof As the given technique is used on linear equations, Eq. ( 13) is linearized as follows By using Eq. ( 16), the above equation takes the following form So, the amplification factor takes the form As the Wiener process is independent of the from the state of state variable P(x, t), the amplification factor takes the form Vol:.( 1234567890 = χ .So, given for (2) is stable.Now, for the stability of Eq. (3), Eq. ( 14) is linearized as follow By using Eq. ( 16), the above equation takes the following form So, the amplification factor takes the form As the Wiener process is independent of the from the state of state variable P(x, t), the amplification factor takes the form as 1+2D 2 +d�τ | 2 = χ .So, the given scheme is unconditionally stable for (14).

Results and discussion
Problem 1 with initial conditions 10 , and having homogeneous Neumann boundary conditions.The system of Eqs. ( 2) and (3) have two equilibriums, one is predator free point (1, 0) and second is coexistence point (P * , Q * ) , where P * = . T h e c o e x i s te n c e e qu i l i br iu m p oi nt i s s t abl e or u ns t abl e i f The Fig. 1a-e are drawn by the proposed scheme-I.The Fig. 2a-e are drawn by the proposed scheme-II.We choose the values of the parameters as follows for Figs.1a-e) and 2a-e) a = 1.1 , 0.01 , and k = 500/N , then coexistence equi- librium point (P * , Q * ) has value (0.4699, 0.2719) by proposed scheme-I and scheme-II .The solution is stable and converging to (P * , Q * ) and it is depicted graphically in Figs.1e and 2e.
The Fig. 3a-e are drawn by the proposed scheme-I.The Fig. 4a-e are drawn by the proposed scheme-II.We have taken the values of the parameters as a = 1.1 , σ 1 = σ 2 = 1 , m = 1 , c = 2.1 , b = 0.7 , f = 0.80 , d = 0.5 ,n = 100,h = 30/n , N = 5000 , η 1 = η 2 = 0.01 , and k = 500/N .If we increase the value of f gradually and it passes through the bifurcation values, the behavior of our system changes and it becomes unstable in Figs.3a-e and 4a-e the population densities P and Q have a periodic orbit around (P * , Q * ) .Both prey and predator have oscillations behavior, which can be observed in 2D graphs of Figs.3b, d, and 4b, d. (1 (1  9a-e and 10a-e.For given values of the parameters, scheme-I showed divergence and negative behavior but scheme-II has convergent and positive behavior.The proposed scheme II preserves the true traits of the underlying model.Such models cannot possess negative values.So, scheme II can be recommended for the solution of such models.
In Fig. 9a-e the numerical solution is provided by the stochastic forward Euler scheme and it has divergent as well as negative behavior for the given values of the parameters.Such solutions are not permitted for the papulation dynamical model because the population may attain a minimum value of zero and it can never attain negative values.A solution that preserves the positive behavior for the whole domain is preferable because the positivity of the solutions is a basic property for the population dynamical models.The proposed stochastic non-standard finite difference scheme is applied for the underlying model and we gave obtained the convergent as well positive solutions for the whole domain.

Conclusion
The ratio-dependent prey-predator model under the influence of time noise has been numerically investigated by two novel schemes.The existence of the solutions is guaranteed by using the fixed point theory with a priori estimates.Both schemes are consistent with the systems of the equations in the mean square sense.The stability is shown by Von-Neumann criteria.The proposed scheme-I is conditionally stable and conditions are derived.The proposed scheme II is stable for the whole domain.We have gained a coexistence equilibrium point for the different values of the parameters.By increasing the values of the conversion rate f, the systems change the behavior from stable to unstable.When we further increased the values of the parameter f, the population densities become extinct.So, conversion rate f played a key role in the system obtaining desired results.For specified values of the parameters, we have also gained a predator-free point.The graphical behavior of a test problem for different values of the parameters is drawn which depicts the efficacy of the schemes.Our numerical solutions are well accurate to the solutions available in the literature.As population densities have positivity, so there must be a scheme that possesses such properties.So, the proposed stochastic non-standard finite difference scheme is preferred which preserves all properties.Hopefully, these results will motivate the researchers to consider the stochastic prey-predator model and analyze them.

Figure 1 .
Figure 1.The subfig (a and c) and (b and d) represents the numerical approximation of P(x, t), and Q(x, t) respectively by proposed scheme-I.The subfig (e) represents the periodic orbit around (P * , Q * ).

Figure 2 .
Figure 2. The subfig (a and c) and (b and d) represents the numerical approximation of P(x, t), and Q(x, t) respectively by proposed scheme-II.The subfig (e) represents the periodic orbit around (P * , Q * ).

Figure 3 .
Figure 3.The subfig (a and c) and (b and d) represents the numerical approximation of P(x, t), and Q(x, t) respectively by proposed scheme-I.The subfig (e) represents the periodic orbit around (P * , Q * ).

Figure 4 .Figure 5 .
Figure 4.The subfig (a and c) and (b and d) represents the numerical approximation of P(x, t), and Q(x, t) respectively by proposed scheme-II.The subfig (e) represents the periodic orbit around (P * , Q * ).

Figure 6 .Figure 7 .Figure 8 .Figure 9 .
Figure 6.The subfig (a and c) and (b and d) represents the numerical approximation of P(x, t), and Q(x, t) respectively by proposed scheme-II.The subfig (e) represents the periodic orbit around (P * , Q * ).

Figure 10 .
Figure 10.The subfig (a and c) and (b and d) represents the numerical approximation of P(x, t), and Q(x, t) respectively by proposed scheme-II.The subfig (e) represents the periodic orbit around (P * , Q * ).