# Analysis of the Stochastic Population Model with Random Parameters

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## Abstract

**:**

## 1. Introduction

## 2. Stochastic Spectral Techniques

^{2}of second-order functionals in the random variables $\zeta \left(\omega \right)$. Many choices for the basis functionals ${\psi}_{k}$ including multiwavelets or the multivariate polynomials [12]. Any second-order random process $X\left(t,\hspace{0.17em}\omega \right)$ that depends on $U\left(\omega \right)$ can then be decomposed as follows [16]:

## 3. The Deterministic Population Model

## 4. The Stochastic Model

## 5. Analysis Using WHE

**and I**

_{1}_{2}compared with ${\left({x}_{i}^{\left(0\right)}\right)}^{2}$ to get

^{−6}, respectively. The nonGaussian part of the variance is small for the given parameters. The decay of the kernels is a known property and one of the known advantages of WHE.

## 6. Stochastic Model with Random Parameters

**a**is a random parameter that depends on a set of standard random variables with known distribution. This means we can write

## 7. Numerical Example with Combined Randomness

**a**fluctuates uniformly around the mean with deviation 2% of the mean value ${a}_{0}$, i.e., $a=0.5+0.01\hspace{0.17em}\hspace{0.17em}{\psi}_{1}$ with ${\psi}_{1}\sim U[-1,\hspace{0.17em}1]$. This is equivalent to a uniform distribution $a\sim U[0.49,\hspace{0.17em}\hspace{0.17em}0.51]$. Using $\lambda $ = 0.02, $\epsilon $ = 0.01 and zero initial conditions for all kernels except for ${x}_{0}^{(0)}(t=0)$ = 0.5. The total variance and its components, due to noise and due to random parameters, are shown Figure 8. The variance due to mixed contributions are very small compared with other variance components. It is four orders of magnitude smaller than other variance components and hence will be neglected in the analysis.

**a**will affect the system sensitivity indices. As the deviations in

**a**increase, the system response deviates further (i.e., variance increases) and becomes more sensitive toward the deviations in

**a**. This result is also shown in Figure 9, which compares the sensitivity indices for a wide range of a deviations.

## 8. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 2.**The stochastic population model; (

**a**) mean; (

**b**) total variance; (

**c**) variance 1 (Gaussian only); (

**d**) variance 2 (nonGaussian) for ($\Delta t$ = 0.05, $a$ = 0.5, $\epsilon $ = 0.01, $\lambda $ = 0.01).

**Figure 3.**(

**a**) Mean absolute error between EM (10,000 samples) and second-order WHE and (

**b**) variance using EM (10,000 samples) and second-order WHE.

**Figure 5.**Gaussian variance and total variance (Gaussian with nonGaussian) (

**a**) for $\lambda $ = 0.01, (

**b**) for $\lambda $ = 0.02, ($\Delta t$ = 0.05, $a$ = 0.5, $\epsilon $ = 0.01).

**Figure 7.**(

**a**) Steady-state total variance with $\lambda $ and (

**b**) the steady-state nonGaussian part of variance with $\lambda $.

**Figure 8.**Variance components for $\lambda $ = 0.02, $\epsilon $ = 0.01, (

**a**) in case of $a\left(\omega \right)=0.5+0.01\hspace{0.17em}\hspace{0.17em}{\psi}_{1}$ and (

**b**) in case of $a\left(\omega \right)=0.5+0.015\hspace{0.17em}\hspace{0.17em}{\psi}_{1}$.

**Figure 10.**Case of $a<{\lambda}^{2}/2$ ($a\left(\omega \right)=0.0001+0.00004\hspace{0.17em}\hspace{0.17em}{\psi}_{1}$, $\lambda $ = 0.02, $\epsilon $ = 0.01, (

**a**) mean solution; (

**b**) total variance.

$\mathit{\lambda}$ | Total Var. | NonGaussian Var. |
---|---|---|

0.0000 | 0.000 | 0 |

0.0100 | 0.257 | 4.59 × 10^{−6} |

0.0150 | 0.593 | 2.50 × 10^{−5} |

0.0175 | 0.835 | 5.10 × 10^{−5} |

0.0200 | 1.220 | 1.21 × 10^{−4} |

Parameter a | Noise Sensitivity Index | Parameter a Sensitivity Index |
---|---|---|

$a\left(\omega \right)=0.5$ (deterministic) | 100% | 0.0% |

$a\left(\omega \right)=0.5+0.005\hspace{0.17em}\hspace{0.17em}{\psi}_{1}$ | 79.8% | 20.2% |

$a\left(\omega \right)=0.5+0.010\hspace{0.17em}\hspace{0.17em}{\psi}_{1}$ | 49.1% | 49.9% |

$a\left(\omega \right)=0.5+0.015\hspace{0.17em}\hspace{0.17em}{\psi}_{1}$ | 29.7% | 70.3% |

$a\left(\omega \right)=0.5+0.020\hspace{0.17em}\hspace{0.17em}{\psi}_{1}$ | 18.9% | 81.1% |

$a\left(\omega \right)=0.5+0.025\hspace{0.17em}\hspace{0.17em}{\psi}_{1}$ | 12.8% | 87.2% |

$a\left(\omega \right)=0.5+0.030\hspace{0.17em}\hspace{0.17em}{\psi}_{1}$ | 9.1% | 91.9% |

$a\left(\omega \right)=0.5+0.035\hspace{0.17em}\hspace{0.17em}{\psi}_{1}$ | 6.7% | 93.3% |

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**MDPI and ACS Style**

Noor, A.; Barnawi, A.; Nour, R.; Assiri, A.; El-Beltagy, M.
Analysis of the Stochastic Population Model with Random Parameters. *Entropy* **2020**, *22*, 562.
https://doi.org/10.3390/e22050562

**AMA Style**

Noor A, Barnawi A, Nour R, Assiri A, El-Beltagy M.
Analysis of the Stochastic Population Model with Random Parameters. *Entropy*. 2020; 22(5):562.
https://doi.org/10.3390/e22050562

**Chicago/Turabian Style**

Noor, Adeeb, Ahmed Barnawi, Redhwan Nour, Abdullah Assiri, and Mohamed El-Beltagy.
2020. "Analysis of the Stochastic Population Model with Random Parameters" *Entropy* 22, no. 5: 562.
https://doi.org/10.3390/e22050562