Stochastic dynamics and survival analysis of a cell population model with random perturbations

Cristina Anton; Alan Yong

doi:10.3934/mbe.2018048

Article Contents

2018, Volume 15, Issue 5: 1077-1098. Doi: 10.3934/mbe.2018048

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Stochastic dynamics and survival analysis of a cell population model with random perturbations

Cristina Anton^, and
Alan Yong

Department of Mathematics and Statistics, Grant MacEwan University, Edmonton, AB T5J 4S2, Canada

* Corresponding author: Cristina Anton
* Corresponding author: Cristina Anton

Received: March 22, 2017

Accepted: April 6, 2018

Published online: October 2018

The first author is supported by NSRC grant DDG-2015-00041.

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Abstract

We consider a model based on the logistic equation and linear kinetics to study the effect of toxicants with various initial concentrations on a cell population. To account for parameter uncertainties, in our model the coefficients of the linear and the quadratic terms of the logistic equation are affected by noise. We show that the stochastic model has a unique positive solution and we find conditions for extinction and persistence of the cell population. In case of persistence we find the stationary distribution. The analytical results are confirmed by Monte Carlo simulations.

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Mathematics Subject Classification: Primary: 37H10, 60J70; Secondary: 60H10.

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Figure 1. TCRCs for monastrol

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Figure 3. Trajectories corresponding to initial values $n(0) = 2.5$, $C_o(0) = 0$, $\sigma_1 = 0$, $\sigma_2 = 0.002$: blue "- -" line deterministic model, $C_e(0) = 380$; red "-" line stochastic model, $C_e(0) = 380$; green "-.-" line stochastic model, $C_e(0) = 379$

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Figure 2. Trajectories corresponding to initial values $n(0) = 2.5$, $C_o(0) = 0$, $\sigma_1 = 0.01$, $\sigma_2 = 0$: blue "- -" line deterministic model, $C_e(0) = 380$; red "-" line stochastic model, $C_e(0) = 380$; green "-.-" line stochastic model, $C_e(0) = 375$

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Figure 4. Histograms of the values of n(t) for the last iteration from 10 000 runs (a) and (c) and for the last 4 000 000 samples out of 5 000 000 sample of a single run (b) and (d)

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Figure 5. Histograms for the last 4 000 000 samples of a single run of 5 000 000 iterations and corresponding density functions a. $\sigma_2 = 0.001$ b. $\sigma_2 = 0.01$ c. $\sigma_2 = 0.1$ d. $\sigma_2 = 0.15$

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Figure 6. Histograms for the last 4 000 000 samples of a single run of 5 000 000 iterations and corresponding Gamma density functions a. $\sigma_1 = 0.001$ b. $\sigma_1 = 0.01$ c. $\sigma_1 = 0.1$ d. $\sigma_1 = 0.5$

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