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Stochastic Models in Mathematical Biology, 2nd Edition
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Stochastic Models in Mathematical Biology, 2nd Edition

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Mathematical Biology".

Deadline for manuscript submissions: 30 June 2025 | Viewed by 2249

Special Issue Editor

Dr. Ivo Siekmann

E-Mail Website
Guest Editor
School of Computer Science and Mathematics, Liverpool John Moores University, Liverpool L3 3AF, UK
Interests: mathematical physiology; mathematical ecology; mathematical epidemiology; stochastic modelling; Markov Chain Monte Carlo (MCMC)
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Is biology deterministic or stochastic? Independent from the answer to this question, stochastic models have been applied with great success in all areas of mathematical biology. Applications extend beyond examples such as evolutionary processes or dynamics of biomolecules where the underlying dynamics themselves are considered stochastic.

For example, in population dynamics, stochastic models are used for describing population-level effects of processes occurring at the level of individuals. Examples include the description of stochastic extinction via branching processes and models for dispersal based on diffusion models.

Many biological systems, including ecosystems or neural networks in the brain, are influenced by a variety of processes—each of which, by itself, only has a small impact. The collective impact of these perturbations can be modelled by stochastic differential equations (SDEs), which enable us to study the dynamics of a deterministic process under stochastic fluctuations.

Finally, stochastic models enable us to represent parameter uncertainties in data-driven models obtained using a Bayesian statistics approach or to account for incomplete knowledge, for example, when considering a heterogeneous population of patients.

For this Special Issue, we are particularly interested in new approaches for applying stochastic models in biology as well as original applications of existing frameworks. We also invite manuscripts from the emerging field of parameterising stochastic models with experimental data.

Dr. Ivo Siekmann
Guest Editor

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Mathematics is an international peer-reviewed open access semimonthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2600 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • population dynamics
  • mathematical physiology
  • biophysics
  • signalling
  • Markov models
  • Markov decision processes
  • stochastic differential equations
  • branching processes
  • parametrisation of stochastic models
  • Bayesian analysis

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Related Special Issue

Published Papers (2 papers)

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Research

8 pages, 259 KiB  
Article
Persistence and Stochastic Extinction in a Lotka–Volterra Predator–Prey Stochastically Perturbed Model
by Leonid Shaikhet and Andrei Korobeinikov
Mathematics 2024, 12(10), 1588; https://doi.org/10.3390/math12101588 - 19 May 2024
Cited by 1 | Viewed by 815
Abstract
The classical Lotka–Volterra predator–prey model is globally stable and uniformly persistent. However, in real-life biosystems, the extinction of species due to stochastic effects is possible and may occur if the magnitudes of the stochastic effects are large enough. In this paper, we consider [...] Read more.
The classical Lotka–Volterra predator–prey model is globally stable and uniformly persistent. However, in real-life biosystems, the extinction of species due to stochastic effects is possible and may occur if the magnitudes of the stochastic effects are large enough. In this paper, we consider the classical Lotka–Volterra predator–prey model under stochastic perturbations. For this model, using an analytical technique based on the direct Lyapunov method and a development of the ideas of R.Z. Khasminskii, we find the precise sufficient conditions for the stochastic extinction of one and both species and, thus, the precise necessary conditions for the stochastic system’s persistence. The stochastic extinction occurs via a process known as the stabilization by noise of the Khasminskii type. Therefore, in order to establish the sufficient conditions for extinction, we found the conditions for this stabilization. The analytical results are illustrated by numerical simulations. Full article
(This article belongs to the Special Issue Stochastic Models in Mathematical Biology, 2nd Edition)
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18 pages, 1049 KiB  
Article
Most Probable Dynamics of the Single-Species with Allee Effect under Jump-Diffusion Noise
by Almaz T. Abebe, Shenglan Yuan, Daniel Tesfay and James Brannan
Mathematics 2024, 12(9), 1377; https://doi.org/10.3390/math12091377 - 30 Apr 2024
Viewed by 957
Abstract
We explore the most probable phase portrait (MPPP) of a stochastic single-species model incorporating the Allee effect by utilizing the nonlocal Fokker–Planck equation (FPE). This stochastic model incorporates both non-Gaussian and Gaussian noise sources. It has three fixed points in the deterministic case. [...] Read more.
We explore the most probable phase portrait (MPPP) of a stochastic single-species model incorporating the Allee effect by utilizing the nonlocal Fokker–Planck equation (FPE). This stochastic model incorporates both non-Gaussian and Gaussian noise sources. It has three fixed points in the deterministic case. One is the unstable state, which lies between the two stable equilibria. Our primary focus is on elucidating the transition pathways from extinction to the upper stable state in this single-species model, particularly under the influence of jump-diffusion noise. This helps us to study the biological behavior of species. The identification of the most probable path relies on solving the nonlocal FPE tailored to the population dynamics of the single-species model. This enables us to pinpoint the corresponding maximum possible stable equilibrium state. Additionally, we derive the Onsager–Machlup function for the stochastic model and employ it to determine the corresponding most probable paths. Numerical simulations manifest three key insights: (i) when non-Gaussian noise is present in the system, the peak of the stationary density function aligns with the most probable stable equilibrium state; (ii) if the initial value rises from extinction to the upper stable state, then the most probable trajectory converges towards the maximally probable equilibrium state, situated approximately between 9 and 10; and (iii) the most probable paths exhibit a rapid ascent towards the stable state, then maintain a sustained near-constant level, gradually approaching the upper stable equilibrium as time goes on. These numerical findings pave the way for further experimental investigations aiming to deepen our comprehension of dynamical systems within the context of biological modeling. Full article
(This article belongs to the Special Issue Stochastic Models in Mathematical Biology, 2nd Edition)
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