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Ponosov, Arcady; Idels, Lev; Kadiev, Ramazan

Stochastic McKendrick-von Foerster models with applications. (English) Zbl 07571792

Physica A 537, Article ID 122641, 14 p. (2020).
Summary: A newly presented McKendrick-Von Foerster model with a stochastically perturbed mortality rate is examined. A transformation method converting the model with non-local boundary conditions into a system of stochastic functional differential equations is offered. The method could be viewed as analogous to the one which is widely used for such type of deterministic problems. The derived stochastic functional differential equations yield multiple classic population models with ‘naturally born’ stochasticity, including delayed Nicholson’s blowflies, general recruitment and models with cannibalism, which by itself could be objects of future analysis and applications.

Cited in 2 Documents

MSC:

82-XX Statistical mechanics, structure of matter
34K50 Stochastic functional-differential equations
60H30 Applications of stochastic analysis (to PDEs, etc.)
92Bxx Mathematical biology in general

Keywords:

stochastic PDE models with mortality; age-structured populations; stochastic functional differential equations; Doléans-Dade exponentials
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\textit{A. Ponosov} et al., Physica A 537, Article ID 122641, 14 p. (2020; Zbl 07571792)
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References:

[1] Cushing, J. M., Nonlinearity and stochasticity in population dynamics, (Mathematics for Ecology, and Environmental Sciences, Vol. 1 (2006), Springer: Springer Berlin)
[2] Gurtin, M. E.; MacCamy, R. C., Non-linear age-dependent population dynamics, Arch. Ration. Mech. Anal., 54, 281-300 (1974) · Zbl 0286.92005
[3] Robertson, S. L.; Henson, S. M.; Robertson, T.; Cushing, J. M., A matter of maturity: to delay or not to delay? continuous-time compartmental models of structured populations in the literature 2000-2016, Nat. Resour. Model., 31, 1, Article e12160 pp. (2017)
[4] Porporato, A.; Calabrese, S., On the probabilistic structure of water age, Water Resour. Res., 51, 5, 3588-3600 (2015)
[5] Cushing, J. M., (An Introduction to Structured Population Dynamics. An Introduction to Structured Population Dynamics, Conference Series in Applied Mathematics, vol. 71 (1998), SIAM: SIAM Philadelphia) · Zbl 0939.92026
[6] Avan, J.; Grosjean, N.; Huillet, T., Did the ever dead outnumber the living and when? A birth-and-death approach, Physica A, 419, 277-292 (2015) · Zbl 1402.91614
[7] Györi, I., Some mathematical aspects of modelling cell population dynamics, Comput. Math. Appl., 20, 4-6, 127-138 (1990) · Zbl 0729.92020
[8] De Falco, I.; Della Cioppa, A.; Tarantino, E., Effects of extreme environmental changes on population dynamics, Physica A, 41, 42, 619-631 (2006)
[9] Mukherjee, S.; Janssen, H. K.; Schmittmann, B., Universal properties of population dynamics with fluctuating resources, Physica A, 384, 1, 80-82 (2007)
[10] Han, Q.; Yang, T.; Zeng, C.; Wang, H.; Tian, D., Impact of time delays on stochastic resonance in an ecological system describing vegetation, Physica A, 408, 96-105 (2014) · Zbl 1395.92183
[11] Wang, W.; Wang, L.; Chen, W., Stochastic Nicholson’s blowflies delayed differential equations, Appl. Math. Lett., 87, 20-26 (2019) · Zbl 1408.34067
[12] Allen, E. J., Derivation of stochastic partial differential equations for size- and age-structured populations, J. Biol. Dyn., 3, 1, 73-86 (2009) · Zbl 1157.60331
[13] Alshanskiy, M. A., Model of age-structured population under stochastic perturbation of death and birth rates, Ural Math. J., 4, 1, 3-13 (2018) · Zbl 1453.60112
[14] Calabrese, S.; Porporato1, A.; Laio, F.; D’Odorico, P.; Ridolfi, L., Age distribution dynamics with stochastic jumps in mortality, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 473, Article 20170451 pp. (2017) · Zbl 1404.92145
[15] Zhang, Q.; Li, X., Existence and uniqueness for stochastic age-dependent population with fractional Brownian motion, Math. Probl. Eng. (2012), Article ID 813535 · Zbl 1264.60041
[16] Ma, W.; Ding, B.; Zhang, Q., The existence and asymptotic behaviour of energy solutions to stochastic age-dependent population equations driven by Levy processes, Appl. Math. Comput., 256, 656-665 (2015) · Zbl 1338.92108
[17] Lin, Q.-F.; Wang, C.-J.; Yang, K.-L.; Tian, M.-Y.; Dai, J.-L., Cross-correlated bounded noises induced the population extinction and enhancement of stability in a population growth model, Physica A, 525, 1046-1057 (2019) · Zbl 07565847
[18] Métivier, M., Semimartingales: A course on stochastic processes, (De Gruyter Studies in Mathematics, vol. 2 (2011))
[19] Protter, P. E., Stochastic Integration and Differential Equations (2004), Springer · Zbl 1041.60005
[20] Buckwar, E.; Pikovsky, A.; Scheutzow, M., Stochastic dynamics with delay and memory. Preface, Stoch. Dyn., 5, 2, III-IV (2005) · Zbl 1080.60500
[21] Mao, X., Stochastic Differential Equations & Applications (1997), Horwood Publishing ltd.: Horwood Publishing ltd. Chichester · Zbl 0874.60050
[22] Mohammed, S.-E. A., Stochastic differential systems with memory: Theory, examples and applications, (Decreusefond, L.; Gjerde, J.; Øksendal, B.; Üstünel, A. S., Stochastic Analysis and Related Topics VI, The Geilo Workshop. Stochastic Analysis and Related Topics VI, The Geilo Workshop, Progress in Probability (1998), Birkhäuser), 1-77 · Zbl 0901.60030
[23] Ponosov, A., Reducible stochastic functional differential equations, J. Funct. Spaces (2017), Ch. 7.7 in: N.V. Azvelev, V.P. Maksimov, L.F. Rakhmatullina, Introduction to the Theory of Functional Differential Equations: Methods and Applications, Contemporary Mathematics and its Applications, 3, Hindawi 264-280
[24] Shaikhet, L., Lyapunov Functionals and Stability of Stochastic Functional Differential Equations (2013), Springer: Springer Dordrecht, Heidelberg, New York, London · Zbl 1277.34003
[25] Cushing, J. M., The dynamics of hierarchical age-structured populations, J. Math. Biol., 32, 705-729 (1994) · Zbl 0823.92018
[26] Kurtz, T. G.; Pardoux, É.; Protter, P., Stratonovich stochastic differential equations driven by general semimartingales, Ann. Inst. H. Poincaré B, 31, 2, 351-377 (1995) · Zbl 0823.60046
[27] Jacod, J.; Memin, J., Weak and strong solutions of stochastic differential equations: Existence and stability, (Stochastic Integrals. Stochastic Integrals, Lecture Notes in Mathematics, vol. 851 (1980), Springer-Verlag: Springer-Verlag Berlin, Heidelberg, New York), 169-213
[28] Freedman, H.; Gopalsamy, K., Global stability in time-delayed single-species dynamics, Bull. Math. Biol., 48, 5-6, 485-492 (1986) · Zbl 0606.92020
[29] Berezansky, L.; Braverman, E.; Idels, L., Nicholson’s blowflies differential equations revisited: Main results and open problems, Appl. Math. Model., 34, 6, 1405-1417 (2010) · Zbl 1193.34149
[30] Berezansky, L.; Idels, L.; Kipnis, M., Mathematical model of marine protected areas, IMA J. Appl. Math., 1-14 (2010) · Zbl 1219.35321
[31] Diekmann, O.; Nisbet, R. M.; Gurney, W. S.C.; van den Bosch, F., Simple mathematical models for cannibalism: a critique and a new approach, Math. Biosci., 78, 21-46 (1986) · Zbl 0587.92020
[32] Claessen, D.; de Roos, A. M.; Persson, L., Population dynamic theory of size-dependent cannibalism, Proc. R. Soc. B, 271, 1537, 333-340 (2004)
[33] Zhou, Q., Modelling Walleye Population and its Cannibalism Effect (2017), 4809, https://ir.lib.uwo.ca/etd/4809
[34] Deng, M., Stability of a stochastic delay commensalism model with Lévy jumps, Physica A, 527, Article 121061 pp. (2019) · Zbl 07568242
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