# Symmetry Analysis of the Stochastic Logistic Equation

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## Abstract

**:**

## 1. Introduction

## 2. Symmetry of Deterministic Equations

#### 2.1. The Jet Space

#### 2.2. Geometry of Differential Equations, Contact Structure, Prolongation

#### 2.3. Symmetry

#### 2.4. Using the Symmetry of Deterministic Equations

## 3. Symmetry of Stochastic Equations

**Remark**

**1.**

#### 3.1. Admissible Maps

#### 3.2. Classification of Symmetries

- If $\tau =0$, we have a simple symmetry;
- If $R=0$, we have a standard symmetry; standard symmetries can be deterministic if ${\phi}^{i}$ do not depend on w, or random if (at least one of) the ${\phi}^{i}$ does depend on (at least one of) the ${w}^{k}$;
- If $R\ne 0$, hence X acts on the ${w}^{k}$ variables, then we have a W-symmetry.

**Remark**

**2.**

**Remark**

**3.**

**Remark**

**4.**

#### 3.3. Determination of Symmetries

**Remark**

**5.**

**Remark**

**6.**

**Remark**

**7.**

#### 3.4. Symmetry and Symmetry Adapted Variables

#### 3.4.1. Standard Symmetries

**Proposition**

**1.**

**Lemma**

**1.**

**Remark**

**8.**

**Remark**

**9.**

#### 3.4.2. W Symmetries

## 4. The Logistic Equation

#### 4.1. Stochastic Logistic Equations

- In the first case, one can have a SLE with environmental noise, i.e.,$$dx\phantom{\rule{4pt}{0ex}}=\phantom{\rule{4pt}{0ex}}\left(\alpha \phantom{\rule{0.166667em}{0ex}}x\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{4pt}{0ex}}-\phantom{\rule{4pt}{0ex}}\beta \phantom{\rule{0.166667em}{0ex}}{x}^{2}\right)\phantom{\rule{0.166667em}{0ex}}d\phantom{\rule{3.33333pt}{0ex}}t\phantom{\rule{4pt}{0ex}}+\phantom{\rule{4pt}{0ex}}\sigma \left(t\right)\phantom{\rule{0.166667em}{0ex}}x\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{0.166667em}{0ex}}d\phantom{\rule{3.33333pt}{0ex}}w\phantom{\rule{4pt}{0ex}};$$$$\sigma \left(t\right)\phantom{\rule{4pt}{0ex}}=\phantom{\rule{4pt}{0ex}}{\sigma}_{0}\phantom{\rule{0.166667em}{0ex}}$$
- A different case is obtained if one considers a SLE with demographical noise; this reads$$dx\phantom{\rule{4pt}{0ex}}=\phantom{\rule{4pt}{0ex}}\left(\alpha \phantom{\rule{0.166667em}{0ex}}x\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{4pt}{0ex}}-\phantom{\rule{4pt}{0ex}}\beta \phantom{\rule{0.166667em}{0ex}}{x}^{2}\right)\phantom{\rule{0.166667em}{0ex}}d\phantom{\rule{3.33333pt}{0ex}}t\phantom{\rule{4pt}{0ex}}+\phantom{\rule{4pt}{0ex}}\sigma \left(t\right)\phantom{\rule{0.166667em}{0ex}}\sqrt{x}\phantom{\rule{0.166667em}{0ex}}d\phantom{\rule{3.33333pt}{0ex}}w\phantom{\rule{4pt}{0ex}};$$
- Finally, one can consider the so called complete model, in which both environmental and demographical noises are taken into account. This reads$$dx\phantom{\rule{4pt}{0ex}}=\phantom{\rule{4pt}{0ex}}\left(\alpha \phantom{\rule{0.166667em}{0ex}}x\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{4pt}{0ex}}-\phantom{\rule{4pt}{0ex}}\beta \phantom{\rule{0.166667em}{0ex}}{x}^{2}\right)\phantom{\rule{0.166667em}{0ex}}d\phantom{\rule{3.33333pt}{0ex}}t\phantom{\rule{4pt}{0ex}}+\phantom{\rule{4pt}{0ex}}\rho \left(t\right)\phantom{\rule{0.166667em}{0ex}}x\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{0.166667em}{0ex}}d\phantom{\rule{3.33333pt}{0ex}}{w}_{1}\phantom{\rule{4pt}{0ex}}+\phantom{\rule{4pt}{0ex}}\mu \left(t\right)\phantom{\rule{0.166667em}{0ex}}\sqrt{x}\phantom{\rule{0.166667em}{0ex}}d\phantom{\rule{3.33333pt}{0ex}}{w}_{2}\phantom{\rule{4pt}{0ex}};$$

#### 4.2. The Diffusion Coefficient

## 5. Symmetry Analysis of the Stochastic Logistic Equation with Environmental Noise

#### 5.1. Constant Environmental Noise

#### 5.2. Standard Symmetries

#### 5.3. W-Symmetries

#### 5.4. Non-Constant Environmental Noise

## 6. Symmetries of the Stochastic Logistic Equation with Demographical Noise

#### 6.1. Constant Noise

#### 6.2. Non-Constant Noise

## 7. Symmetries of the Complete Model for the Stochastic Logistic Equation

#### 7.1. Constant Amplitude Fluctuations

#### 7.2. Non-Constant Amplitude Fluctuations

#### 7.3. Summary of Obtained Results

**Lemma**

**2.**

## 8. Integration of the Stochastic Logistic Equation with Constant Environmental Noise

#### 8.1. Integration via Kozlov Theory

**Remark**

**11.**

#### 8.2. Numerical Experiments

- Generate, by means of a random number generator, a sequence of normally distributed $\left(\delta w\right)\left(k\right)$ (for $k=0,...,{k}_{\mathrm{max}}={k}_{M}$); store these.
- Using the stored values of $\left(\delta w\right)\left(k\right)$, build a discrete-time Wiener process (time step $\delta t$) setting $w\left[0\right]=0$, $w\left[\right(k+1\left)\right]=w\left[k\right]+\left(\delta w\right)\left(k\right)\delta t$; here $w\left[k\right]$ represents the value taken by $w\left(t\right)$ at time ${t}_{k}=k\left(\delta t\right)$. Store this.
- Numerically integrate (28), again using the stored $\left(\delta w\right)\left(k\right)$, by setting $x\left(0\right)={x}_{0}$, $x\left[(k+1)\right]=x\left[k\right]+(\alpha x\left[k\right]-\beta x{\left[k\right]}^{2})dt+\gamma x\left[k\right]\left(\delta w\right)\left[k\right]$, and store these. Here $x\left[k\right]$ represents the value taken by $x\left(t\right)$ at time ${t}_{k}=k\left(\delta t\right)$, for the given realization of $w\left(t\right)$. The values $x\left[k\right]$ represent a bona fide direct numerical solution of our stochastic equation, with the approximation resulting from the finite size of the time step $\left(\delta t\right)$. (We stress we are here using a very basic Euler first order integration scheme, thus have to expect rather poor precision in our numerical results. One could use more refined integration schemes—see e.g., [56] —but the point here is just to have a reference numerical solution to compare our exact solution with.)
- Use the map (72) to determine $y\left(0\right)={y}_{0}$ corresponding to the assigned initial value ${x}_{0}$. Using the stored values of $w\left[k\right]$ (i.e., of $w\left(t\right)$ for the given realization), build $y\left(t\right)$ by means of (75), i.e., setting $y\left[0\right]={y}_{0}$ and $y[k+1]=y\left[0\right]-\beta exp(A\phantom{\rule{0.166667em}{0ex}}{t}_{k}\phantom{\rule{4pt}{0ex}}+\phantom{\rule{4pt}{0ex}}\gamma \phantom{\rule{0.166667em}{0ex}}w\left[k\right])\delta t$; here again $y\left[k\right]$ represents the value taken by $y\left(t\right)$ at time ${t}_{k}$. The values $y\left[k\right]$ are stored and represent a bona fide direct numerical solution of our equivalent stochastic Equation (75), i.e., of (77); again with the to approximation due to finite size of $\delta t$.
- Use now the inverse map (73) to generate from the values stored in Y—i.e., from $y\left(t\right)$—values $\widehat{x}\left[k\right]$ which represent the values taken by a stochastic process $\widehat{x}\left(t\right)$ at time ${t}_{k}$.
- If our procedure is correct, the stochastic process $\widehat{x}\left(t\right)$ is the solution to the Equation (28) for the given realization of $w\left(t\right)$, i.e., should correspond to $x\left(t\right)$ computed directly before. Thus we compare the strings $x\left[k\right]$ and $\widehat{x}\left[k\right]$.

## 9. Conclusions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**For $\alpha =\beta =1$, $\gamma =0.01$, ${x}_{0}=\alpha /\beta $ (the equilibrium solution for zero noise) and a specific realization of $w\left(t\right)$ we show: (

**a**) the direct numerical solution $x\left(t\right)$; (

**b**) the solution $\widehat{x}\left(t\right)$ obtained by our procedure; (

**c**) the realization of the driving stochastic process $w\left(t\right)$; (

**d**) the relative error $|\widehat{x}\left(t\right)-x\left(t\right)|/x\left(t\right)$. The numerical integrations are performed over 10,000 steps.

**Figure 2.**For the same parameters values as in Figure 1 but a different realization of the driving Wiener process, we show the same quantities: (

**a**) the direct numerical solution $x\left(t\right)$; (

**b**) the solution $\widehat{x}\left(t\right)$ obtained by our procedure; (

**c**) the realization of the driving stochastic process $w\left(t\right)$; (

**d**) the relative error $|\widehat{x}\left(t\right)-x\left(t\right)|/x\left(t\right)$. The numerical integrations are performed over 10,000 steps.

**Figure 3.**For the same parameters values as in Figure 1 but a different realization of the driving Wiener process, we show the same quantities: (

**a**) the direct numerical solution $x\left(t\right)$; (

**b**) the solution $\widehat{x}\left(t\right)$ obtained by our procedure; (

**c**) the realization of the driving stochastic process $w\left(t\right)$; (

**d**) the relative error $|\widehat{x}\left(t\right)-x\left(t\right)|/x\left(t\right)$. The numerical integrations are performed over 10,000 steps.

**Figure 4.**For the same parameters values as in Figure 1 but a different realization of the driving Wiener process, we show the same quantities: (

**a**) the direct numerical solution $x\left(t\right)$; (

**b**) the solution $\widehat{x}\left(t\right)$ obtained by our procedure; (

**c**) the realization of the driving stochastic process $w\left(t\right)$; (

**d**) the relative error $|\widehat{x}\left(t\right)-x\left(t\right)|/x\left(t\right)$. The numerical integrations are performed over 10,000 steps.

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**MDPI and ACS Style**

Gaeta, G.
Symmetry Analysis of the Stochastic Logistic Equation. *Symmetry* **2020**, *12*, 973.
https://doi.org/10.3390/sym12060973

**AMA Style**

Gaeta G.
Symmetry Analysis of the Stochastic Logistic Equation. *Symmetry*. 2020; 12(6):973.
https://doi.org/10.3390/sym12060973

**Chicago/Turabian Style**

Gaeta, Giuseppe.
2020. "Symmetry Analysis of the Stochastic Logistic Equation" *Symmetry* 12, no. 6: 973.
https://doi.org/10.3390/sym12060973