# Analysis of Structure-Preserving Discrete Models for Predator-Prey Systems with Anomalous Diffusion

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

**Definition**

**1**

**.**Assume that $f:\mathbb{R}\to \mathbb{R}$, and suppose that $n\in \mathbb{N}\cup \left\{0\right\}$ and $\alpha \in \mathbb{R}$ are such that $n-1<\alpha <n$. If it exists, we introduce the fractional derivative in the sense of Riesz of the function f of order α at the point $x\in \mathbb{R}$ as:

**Definition**

**2.**

**Definition**

**3**

**.**Let $f:\mathbb{R}\to \mathbb{R}$, and assume that $h>0$ and $\alpha >-1$. The centered difference of fractional order α of the function f at x is given (when it exists) as:

**Lemma**

**1**

- (a)
- The following iterative formulas hold:$$\begin{array}{cc}\hfill {g}_{0}^{(\alpha )}& {\displaystyle =\frac{\Gamma (\alpha +1)}{{[\Gamma (\frac{\alpha}{2}+1)]}^{2}},}\hfill \end{array}$$$$\begin{array}{cc}\hfill {g}_{k+1}^{(\alpha )}& {\displaystyle =\left(1-\frac{\alpha +1}{\alpha /2+k+1}\right){g}_{k},\phantom{\rule{2.em}{0ex}}\forall k\in \mathbb{N}\cup \left\{0\right\}.}\hfill \end{array}$$
- (b)
- ${g}_{0}^{(\alpha )}>0$.
- (c)
- ${g}_{k}^{(\alpha )}={g}_{-k}^{(\alpha )}<0$for all$k\ne 0$.
- (d)
- $\sum _{k=-\infty}^{\infty}{g}_{k}^{(\alpha )}=0$.

**Lemma**

**2**

**.**Let $0<\alpha \le 2$ and $\alpha \ne 1$, and suppose that $f\in {\mathcal{C}}^{5}(\mathbb{R})$ is a function whose derivatives up to order five are all integrable. For almost all $x\in \mathbb{R}$,

## 3. Numerical Models

**Definition**

**4.**

#### 3.1. Explicit Method

**Definition**

**5.**

#### 3.2. Implicit Method

**Definition**

**6.**

## 4. Structural Properties

**Definition**

**7.**

**Theorem**

**1**

**.**Let $k\in {I}_{K-1}$. Assume that F is a bounded function with domain $[0,1]$ and that ${G}^{u}$ and ${G}^{v}$ are positive and bounded on $[0,1]\times [0,1]$. If $0\le {u}^{k}\le 1$, $0\le {v}^{k}\le 1$, and:

**Proof.**

**Definition**

**8.**

**Definition**

**9.**

- (i)
- A is a Z-matrix,
- (ii)
- all the diagonal components of A are positive, and
- (iii)
- A is strictly diagonally dominant.

**Lemma**

**3.**

**Proof.**

**Theorem**

**2**

**.**Let $k\in {I}_{K-1}$, and suppose that ${u}^{k}>0$ and ${v}^{k}>0$. If $a\tau F({u}_{m,n}^{k})<1$, $c\tau {G}^{v}({u}_{m,n}^{k},{v}_{m,n}^{k})<1$, and ${G}^{u}({u}_{m,n}^{k},{v}_{m,n}^{k})\ge 0$ for each $m\in {\overline{I}}_{M}$ and $n\in {\overline{I}}_{N}$, then the recursive system (42) has a unique solution.

**Proof.**

**Theorem**

**3**

**.**Let $k\in {I}_{K-1}$. Suppose that F is a bounded function over $[0,1]$ and that ${G}^{u}$ and ${G}^{v}$ are positive and bounded on the set $[0,1]\times [0,1]$. If $0\le {u}^{k}\le 1$, $0\le {v}^{k}\le 1$, and

**Proof.**

## 5. Numerical Properties

**Definition**

**10.**

**Lemma**

**4**

**.**Let $M=({m}_{ij})$ be an M-matrix, and let $N=({n}_{ij})$ be a nonnegative matrix of the same size as M. If M is strictly diagonally dominant by rows, then $\rho ({M}^{-1}N)$ satisfies:

#### 5.1. Explicit Method

**Theorem**

**4**

**.**Let $u,v\in {\mathcal{C}}^{5}({\overline{\Omega}}_{T})$. If $\tau <1$, then there are constants $C>0$ and ${C}^{\prime}>0$ that are independent of τ, ${h}_{{x}_{1}}$, and ${h}_{{x}_{2}}$, with the property that for all $m\in {\overline{I}}_{M}$, $n\in {\overline{I}}_{N}$, and $k\in {I}_{K-1}$,

**Proof.**

**Theorem**

**5**

**.**Let ${({u}^{k})}_{k=0}^{K}$, ${({v}^{k})}_{k=0}^{K}$ and ${({\mathfrak{u}}^{k})}_{k=0}^{K}$, ${({\mathfrak{v}}^{k})}_{k=0}^{K}$ be two sets of positive and bounded solutions of (20) corresponding to the initial conditions $({\varphi}_{u}^{0},{\varphi}_{v}^{0})$ and $({\varphi}_{\mathfrak{u}}^{0},{\varphi}_{\mathfrak{v}}^{0})$, respectively. If the matrices ${A}_{u}^{k}$ and ${A}_{\mathfrak{u}}^{k}$ are identical to some constant matrix ${A}_{1}^{k}$ and the matrices ${A}_{v}^{k}$ and ${A}_{\mathfrak{v}}^{k}$ are identical to some constant matrix ${A}_{2}^{k}$ for each $k\in {\overline{I}}_{K}$, then there exist constants ${C}_{1},{C}_{2}>0$ such that:

**Proof.**

**Theorem**

**6**

**Proof.**

#### 5.2. Implicit Method

**Theorem**

**7**

**.**Let $u,v\in {\mathcal{C}}^{5}({\overline{\Omega}}_{T})$. If $\tau <1$, then there exist constants $C>0$ and ${C}^{\prime}>0$ that are independent of τ, ${h}_{{x}_{1}}$ and ${h}_{{x}_{2}}$ such that for all $m\in {\overline{I}}_{M}$, $n\in {\overline{I}}_{N}$, and $k\in {I}_{K-1}$,

**Proof.**

**Lemma**

**5**

**.**Suppose that A is a matrix of size $m\times m$ and real components, which satisfies:

**Lemma**

**6.**

**Proof.**

**Lemma**

**7.**

**Proof.**

**Theorem**

**8**

**.**Let ${({u}^{k})}_{k=0}^{K}$, ${({v}^{k})}_{k=0}^{K}$, and ${({\mathfrak{u}}^{k})}_{k=0}^{K}$, ${({\mathfrak{v}}^{k})}_{k=0}^{K}$ be positive and bounded solutions of (42) corresponding to the initial conditions $({\varphi}_{u}^{0},{\varphi}_{u}^{1},{\varphi}_{v}^{0},{\varphi}_{v}^{1})$ and $({\varphi}_{\mathfrak{u}}^{0},{\varphi}_{\mathfrak{u}}^{1},{\varphi}_{\mathfrak{v}}^{0},{\varphi}_{\mathfrak{v}}^{1})$, respectively. Suppose that the inequalities (71)–(74) hold for each $k\in {\overline{I}}_{K}$. If the matrices ${A}_{1}^{k}={A}_{u}^{k}={A}_{\mathfrak{u}}^{k}$, ${E}_{1}^{k}={E}_{u}^{k}={E}_{\mathfrak{u}}^{k}$, ${A}_{2}^{k}={A}_{v}^{k}={A}_{\mathfrak{v}}^{k}$, and ${E}_{2}^{k}={E}_{v}^{k}={E}_{\mathfrak{v}}^{k}$ for each $k\in {\overline{I}}_{K}$, then:

**Proof.**

**Theorem**

**9**

**Proof.**

**Theorem**

**10**

**.**Suppose that $\mathfrak{u},\mathfrak{v}\in {\mathcal{C}}^{5}({\overline{\Omega}}_{T})$ are positive and bounded solutions of (14). Let $\tau <1$, and suppose that ${({u}^{k})}_{k=0}^{K}$ and ${({v}^{k})}_{k=0}^{K}$ are positive and bounded solutions of (43). If (71)–(74) are satisfied for all $k\in {\overline{I}}_{K}$, then there are constants ${\kappa}_{u},{\kappa}_{v}\in {\mathbb{R}}^{+}$ that are independent of τ and h, such that:

**Proof.**

## 6. Applications

**Example**

**1.**

**Example**

**2.**

**Example**

**3.**

## 7. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. MATLAB Code

function [U,V]=fpde a=0.8; c=0.3; d=0.1; D1=0.01; D2=0.6; alpha=1.2; beta=1.2; T=2000; L=100; h=1/3; tau=0.02; Ru=tau*D1/h^alpha; Rv=tau*D2/h^beta; x=0:h:L; M=length(x); N=floor(T/tau); ga=zeros(1,M); gb=zeros(1,M); ga(1)=gamma(alpha+1)/gamma((alpha/2)+1)^2; gb(1)=gamma(beta+1)/gamma((beta/2)+1)^2; for k=1:(M-1) ga(k+1)=(1-(1+alpha)/((alpha/2)+k))*ga(k); gb(k+1)=(1-(1+beta)/((beta/2)+k))*gb(k); end Ha=zeros(M); Hb=zeros(M); for j=1:M for i=1:M Ha(j,i)=ga(abs(i-j)+1); Hb(j,i)=gb(abs(i-j)+1); end end x0=(a*c-c+d)/a/c; y0=(c-d)*x0/d; U=zeros(M); U(130:170,130:170)=0.2*ones(length(130:170)); V=normrnd(y0,0.01,M,M); for i=1:N W=a.*U.*(1-U)-U.*V./(U+V); Z=c.*U.*V./(U+V)-d*V; U=U+tau.*W-Ru.*Ha*U-Ru.*U*Ha; V=V+tau.*Z-Rv.*Hb*V-Rv.*V*Hb; end end

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**Figure 1.**Snapshots of the approximate solution u in (118) versus x and y. The parameters employed are $a=0.8$, $c=0.3$, $d=0.1$, ${D}_{1}=0.01$, ${D}_{2}=0.6$, and $\alpha =\beta =2$. Meanwhile, we considered the times (

**a**) $t=0$, (

**b**) $t=160$, (

**c**) $t=290$, (

**d**) $t=500$, (

**e**) $t=1010$, and (

**f**) $t=3000$. We let ${\varphi}_{v}(x)$ be a random sample from a normally distributed random variable with the mean equal to ${v}^{\ast}$ and the standard deviation equal to $0.01$, and ${\varphi}_{u}$ is the function depicted in (a). The approximations were calculated using our implementation of (20) shown in Appendix A, with $\tau =0.02$ and ${h}_{{x}_{1}}={h}_{{x}_{2}}=1/3$.

**Figure 2.**Snapshots of the approximate solution u in (118) versus x and y. The parameters employed are $a=0.8$, $c=0.3$, $d=0.1$, ${D}_{1}=0.01$, ${D}_{2}=0.6$, and $\alpha =\beta =1.6$. Meanwhile, we considered the times (

**a**) $t=0$, (

**b**) $t=160$, (

**c**) $t=290$, (

**d**) $t=500$, (

**e**) $t=1010$, and (

**f**) $t=3000$. We let ${\varphi}_{v}(x)$ be a random sample from a normally distributed random variable with the mean equal to ${v}^{\ast}$ and the standard deviation equal to $0.01$, and ${\varphi}_{u}$ is the function depicted in (a). The approximations were calculated using our implementation of (20) shown in Appendix A, with $\tau =0.02$ and ${h}_{{x}_{1}}={h}_{{x}_{2}}=1/3$.

**Figure 3.**Snapshots of the approximate solution u in (118) versus x and y. The parameters employed are $a=0.8$, $c=0.3$, $d=0.1$, ${D}_{1}=0.01$, ${D}_{2}=0.6$, and $\alpha =\beta =1.2$. Meanwhile, we considered the times (

**a**) $t=0$, (

**b**) $t=160$, (

**c**) $t=290$, (

**d**) $t=500$, (

**e**) $t=1010$, and (

**f**) $t=3000$. We let ${\varphi}_{v}(x)$ be a random sample from a normally distributed random variable with the mean equal to ${v}^{\ast}$ and the standard deviation equal to $0.01$, and ${\varphi}_{u}$ is the function depicted in (a). The approximations were calculated using our implementation of (20) shown in Appendix A, with $\tau =0.02$ and ${h}_{{x}_{1}}={h}_{{x}_{2}}=1/3$.

**Figure 4.**Snapshots of the approximate solution u in (118) versus x, y, and z. The parameters employed are $a=0.8$, $c=0.3$, $d=0.1$, ${D}_{1}=0.01$, ${D}_{2}=0.6$, and $\alpha =\beta =1.6$. Meanwhile, we considered the times (

**a**) $t=0$, (

**b**) $t=160$, (

**c**) $t=290$, (

**d**) $t=500$, (

**e**) $t=1010$, and (

**f**) $t=3000$. The initial data are random samples of a uniform distribution on $[0,1]$. The approximations were calculated using our implementation of (20), with $\tau =0.02$ and ${h}_{{x}_{1}}={h}_{{x}_{2}}={h}_{{x}_{3}}=1$.

**Figure 5.**Snapshots of x-, y-, and z-cross-sections of the approximate solution u of (118) versus x, y, and z. The parameters employed are $a=0.8$, $c=0.3$, $d=0.1$, ${D}_{1}=0.01$, ${D}_{2}=0.6$, and $\alpha =\beta =1.6$. Meanwhile, we considered the times (

**a**) $t=0$, (

**b**) $t=160$, (

**c**) $t=290$, (

**d**) $t=500$, (

**e**) $t=1010$, and (

**f**) $t=3000$. The initial data are random samples of a uniform distribution on $[0,1]$. The approximations were calculated using our implementation of (20), with $\tau =0.02$ and ${h}_{{x}_{1}}={h}_{{x}_{2}}={h}_{{x}_{3}}=1$.

**Figure 6.**Snapshots of the approximate solution u of (118) versus x, y, and z. The parameters employed are $a=0.8$, $c=0.3$, $d=0.1$, ${D}_{1}=0.01$, ${D}_{2}=0.6$, and $\alpha =\beta =1.2$. Meanwhile, we considered the times (

**a**) $t=0$, (

**b**) $t=160$, (

**c**) $t=290$, (

**d**) $t=500$, (

**e**) $t=1010$, and (

**f**) $t=3000$. The initial data are random samples of a uniform distribution on $[0,1]$. The approximations were calculated using our implementation of (20), with $\tau =0.02$ and ${h}_{{x}_{1}}={h}_{{x}_{2}}={h}_{{x}_{3}}=1$.

**Figure 7.**Snapshots of x-, y-, and z-cross-sections of the approximate solution u of (118) versus x, y, and z. The parameters employed are $a=0.8$, $c=0.3$, $d=0.1$, ${D}_{1}=0.01$, ${D}_{2}=0.6$, and $\alpha =\beta =1.2$. Meanwhile, we considered the times (

**a**) $t=0$, (

**b**) $t=160$, (

**c**) $t=290$, (

**d**) $t=500$, (

**e**) $t=1010$, and (

**f**) $t=3000$. The initial data are random samples of a uniform distribution on $[0,1]$. The approximations were calculated using our implementation of (20), with $\tau =0.02$ and ${h}_{{x}_{1}}={h}_{{x}_{2}}={h}_{{x}_{3}}=1$.

**Figure 8.**Snapshots of the approximate solution u in (118) versus x and y, at the time $T=500$. The parameters employed are $a=0.8$, $c=0.3$, $d=0.1$, ${D}_{1}=0.01$, ${D}_{2}=0.6$, and $\alpha =\beta =1.6$. We let ${\varphi}_{v}(x)$ be a random sample from a normally distributed random variable with the mean equal to ${v}^{\ast}$ and the standard deviation equal to $0.01$, and ${\varphi}_{u}$ is the function depicted in (

**a**). The approximations were calculated using our implementation of (20) shown in Appendix A, with $\tau =0.02$ and $h={h}_{{x}_{1}}={h}_{{x}_{2}}$ satisfying (

**a**) $h=1/3$, (

**b**) $h=1/4$, (

**c**) $h=1/5$, and (

**d**) $h=1/8$.

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**MDPI and ACS Style**

Alba-Pérez, J.; Macías-Díaz, J.E.
Analysis of Structure-Preserving Discrete Models for Predator-Prey Systems with Anomalous Diffusion. *Mathematics* **2019**, *7*, 1172.
https://doi.org/10.3390/math7121172

**AMA Style**

Alba-Pérez J, Macías-Díaz JE.
Analysis of Structure-Preserving Discrete Models for Predator-Prey Systems with Anomalous Diffusion. *Mathematics*. 2019; 7(12):1172.
https://doi.org/10.3390/math7121172

**Chicago/Turabian Style**

Alba-Pérez, Joel, and Jorge E. Macías-Díaz.
2019. "Analysis of Structure-Preserving Discrete Models for Predator-Prey Systems with Anomalous Diffusion" *Mathematics* 7, no. 12: 1172.
https://doi.org/10.3390/math7121172