
Mathematical Biosciences and Engineering, 2018, 15(5): 10771098. doi: 10.3934/mbe.2018048.
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Stochastic dynamics and survival analysis of a cell population model with random perturbations
. Department of Mathematics and Statistics, Grant MacEwan University, Edmonton, AB T5J 4S2, Canada
Received: , Accepted: , Published:
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.
Keywords: Stochastic logistic equation; stationary distribution; ergodic property; extinction; stochastic permanence
Citation: Cristina Anton, Alan Yong. Stochastic dynamics and survival analysis of a cell population model with random perturbations. Mathematical Biosciences and Engineering, 2018, 15(5): 10771098. doi: 10.3934/mbe.2018048
References:
 [1] C. Anton,J. Deng,Y. Wong,Y. Zhang,W. Zhang,S. Gabos,D. Huang,C. Jin, Modeling and simulation for toxicity assessment, Math. BioSci. Eng., 14 (2017): 581606.
 [2] G. K. Basak,R. Bhattcharya, Stability in distribution for a class of singular diffusions, Ann. Prob., 20 (1992): 312321.
 [3] A. Friedman, Stochastic Differential Equations and Applications, Dover, New York, 2006.
 [4] A. Grey,D. Greenhalgh,L. Hu,X. Mao,J. Pan, A stochastic differential equation SIS epidemic model, SIAM. J. Appl. Math., 71 (2011): 876902.
 [5] T. Hallam,C. Clark,G. Jordan, Effects of toxicants on populations: A qualitative approach Ⅱ. First order kinetics, J. Math. Biology, 18 (1983): 2537.
 [6] R. Z. Hasminskii, Stochastic Stability of Differential Equations, Springer, Berlin, 2012, 2nd ed.
 [7] J. He,K. Wang, The survival analysis for a population in a polluted environment, Nonlinear Analysis: Real World Applications, 10 (2009): 15551571.
 [8] C. Ji,D. Jiang,N. Shi,D. O'Regan, Existence, uniqueness, stochastic persistence and global stability of positive solutions of the logistic equation with random perturbation, Math. Methods in the Appl. Sciences, 30 (2007): 7789.
 [9] D. Jiang,N. Shi, A note on nonautonomous logistic equation with random perturbation, J. Math. Anal. Appl., 303 (2005): 164172.
 [10] D. Jiang,N. Shi,X. Li, Global stability and stochastic permanence of a nonautonomous logistic equation with random perturbation, J. Math. Anal. Appl., 340 (2008): 588597.
 [11] J. Jiao,W. Long,L. Chen, A single stagestructured population model with mature individuals in a polluted environment and pulse input of environmental toxin, Nonlinear Analysis: Real World Applications, 10 (2009): 30733081.
 [12] P. Kloeden and E. Platen, Numerical Solutions of Stochastic Differential Equations, SpringerVerlag, Berlin, 1992.
 [13] Y. A. Kutoyants, Statistical Inference for Ergodic Diffusion Processes, Springer, London, 2004.
 [14] V. Lakshmikantham and S. Leela, Differential and Integral Inequalities, Vol. Ⅰ, Academic Press, New York, 1969.
 [15] M. Liu,K. Wang, Survival analysis of stochastic singlespecies population models in polluted environments, Ecological Modelling, 220 (2009): 13471357.
 [16] M. Liu,K. Wang,Q. Wu, Survival analysis of stochastic competitive models in a polluted environment and stochastic competitive exclusion principle, Bull. Math. Biol., 73 (2011): 19692012.
 [17] X. Mao, Stochastic Differential Equations and Applications, Woodhead Pubilshing, Philadelphia, 2011, 2nd ed.
 [18] X. Mao,G. Marion,E. Renshaw, Environmental brownian noise suppresses explosions in population dynamics, Markov Proc. and Their Appl., 97 (2002): 95110.
 [19] X. Mao,S. Sabanis,E. Renshaw, Asymptotic behaviour of the stochastic LotkaVolterra model, J. Math. Anal. Appl., 287 (2003): 141156.
 [20] T. Pan,B. Huang,W. Zhang,S. Gabos,D. Huang,V. Devendran, Cytotoxicity assessment based on the AUC_{5}0 using multiconcentration timedependent cellular response curves, Anal. Chim. Acta, 764 (2013): 4452.
 [21] S. Pinheiro, On a logistic growth model with predation and powertype diffusion coefficient: Ⅰ. Existence of solutions and extinction criteria, Math. Meth. Appl. Sci., 38 (2015): 49124930.
 [22] S. Resnik, A Probability Path, Birkhauser, Boston, 1999.
 [23] Z. Teng,L. Wang, Persistence and extinction for a class of stochastic SIS epidemic models with nonlinear incidence rate, Physica A, 451 (2016): 507518.
 [24] F. Wei,L. Chen, Psychological effect on singlespecies population models in a polluted environment, Math. Biosci., 290 (2017): 2230.
 [25] F. Wei,S. Geritz,J. Cai, A stochastic singlespecies population model with partial pollution tolerance in a polluted environment, Appl. Math. Letters, 63 (2017): 130136.
 [26] Z. Xi,S. Khare,A. Cheung,B. Huang,T. Pan,W. Zhang,F. Ibrahim,C. Jin,S. Gabos, Mode of action classification of chemicals using multiconcentration timedependent cellular response profiles, Comp. Biol. Chem., 49 (2014): 2335.
 [27] J. Xing,L. Zhu,S. Gabos,L. Xie, Microelectronic cell sensor assay for detection of cytotoxicity and prediction of acute toxicity, Toxicology in Vitro, 20 (2006): 9951004.
 [28] Q. Yang,D. Jiang,N. Shi,C. Ji, The ergodicity and extinction of stochastically perturbed SIR and SEIR epidemic models with saturated incidence, J. Math. Anal. Appl., 388 (2012): 248271.
 [29] Q. Yang,X. Mao, Stochastic dynamics of SIRS epidemic models with random perturbation, Math. BioSci. Eng., 11 (2014): 10031025.
 [30] Y. Zhang, Y. Wong, J. Deng, C. Anton, J. Deng, S. Gabos, W. Zhang, D. Huang and C. Jin, Machine learning algorithms for modeofaction classification in toxicity assessment, BioData Mining, 9 (2016), p19.
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