Abstract
We are concerned with a nonlinear nonautonomous model represented by an equation describing the dynamics of an age-structured population diffusing in a space habitat O, governed by local Lipschitz vital factors and by a stochastic behavior of the demographic rates possibly representing emigration, immigration and fortuitous mortality. The model is completed by a random initial condition, a flux type boundary conditions on \(\partial O\) with a random jump in the population density and a nonlocal nonlinear boundary condition given at age zero. The stochastic influence is expressed by a linear multiplicative Gaussian noise perturbation in the equation. The main result proves that the stochastic model is well-posed, the solution being in the class of path-wise continuous functions and satisfying some particular regularities with respect to the age and space. The approach is based on a rescaling transformation of the stochastic equation into a random deterministic time dependent hyperbolic-parabolic equation with local Lipschitz nonlinearities. The existence and uniqueness of a strong solution to the random deterministic equation is proved by combined semigroup, variational and approximation techniques. The information given by these results is transported back via the rescaling transformation towards the stochastic equation and enables the proof of its well-posedness.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Barbu, V.: Nonlinear Differential Equations of Monotone Types in Banach Spaces. Springer, New York (2010)
Barbu, V., Röckner, M.: Stochastic variational inequalities and applications to the total variation flow perturbed by linear multiplicative noise. Arch. Ration. Mech. Anal. 209, 797–834 (2013)
Barbu, V., Röckner, M.: An operatorial approach to stochastic partial differential equations driven by linear multiplicative noise. J. Eur. Math. Soc. 17, 1789–1815 (2015)
Crandall, M.G., Pazy, A.: Nonlinear evolution equations in Banach spaces. Israel J. Math. 11, 57–94 (1971)
Cusulin, C., Iannelli, M., Marinoschi, G.: Age-structured diffusion in a multi-layer environment. Nonlinear Anal. Real World Appl. 6, 207–223 (2005)
Cusulin, C., Iannelli, M., Marinoschi, G.: Convergence in a multi-layer population model with age-structure. Nonlinear Anal. Real World Appl. 8, 887–902 (2007)
Da Prato, G., Zabcyk, J.: Stochastic Equations in Infinite Dimensions. Encyclopedia of Mathematics and its Applications, vol. 152, 2nd edn. Cambridge University Press, Cambridge (2014)
Kato, T.: Nonlinear semigroups and evolution equations. J. Math. Soc. Japan 19, 508–520 (1967)
Iannelli, M., Milner, F.: The Basic Approach to Age-structured Population Dynamics. Models, Methods and Numerics. Springer, Dordrecht (2017)
Lions, J.L.: Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires. Dunod, Paris (1969)
Prévôt, C., Röckner, M.: A Concise Course on Stochastic Partial Differential Equations. Lecture Notes in Mathematics. Springer, Berlin (2007)
Acknowledgements
The author would like to thank the anonymous reviewer for the extremely careful lecture of the paper and for the pertinent observations he/she made. This work was supported by a Grant of Ministry of Research and Innovation, CNCS—UEFISCDI, Project Number PN-III-P4-ID-PCE-2016-0011, within PNCDI III.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Marinoschi, G. Rescaling Approach for a Stochastic Population Dynamics Equation Perturbed by a Linear Multiplicative Gaussian Noise. Appl Math Optim 81, 511–544 (2020). https://doi.org/10.1007/s00245-018-9507-8
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00245-018-9507-8
Keywords
- Stochastic nonlinear equation
- Noise induced phenomena
- Multiplicative noise
- Brownian motion
- Population dynamics
- Semigroup approach