Asymptotic behavior of some stochastic difference equation population models

Population growth models are abstract representation of the real world objects, systems or processes to illustrate the theoretical concepts that these days are increasingly being used in more applied situations such as predicting future outcomes or simulation experiments. In mathematical literature, many population models have been considered, from deterministic and stochastic population models where the population size is represented by a discrete random variable, to very complex continuous stochastic models. A nonrandom case, ignores natural variation and produces a single value result, whereas a stochastic model incorporates some natural variations in to model to state unpredictable situations such as weather or random fluctuations in resources and itgenerates a mean or most probable result. Nowadays, the well-known model like logistic isplaying a major role in modern ecological theory. In paper [1], the Laguerre-type derivatives and the Laguerre-type exponentials are introduced and then by using Laguerre-type exponentials the LExponential and L-Logistic population growth models are derived, and output of these models is given for world population growth in the period 1955-2005. The paper [2], develops a stochastic logistic population growth model with immigration and multiple births. The differential equation for the low-order cumulant functions (i.e., mean, variance, and skewness) of the single birth model is reviewed, and the corresponding equations for the multiple birth model are derived. Accurate approximate solutions for the cumulant functions are obtained using moment closure methods for two families of model parameterizations, one for badger and the other for fox population growth. For both model families, the equilibrium size distribution may be approximated well using the normal approximation, and even more accurately using the saddle point approximation and it is shown that in comparison with the corresponding single birth model, the

This issue's Research Spotlights section contains three papers. First, “Canards, Folded Nodes, and Mixed-Mode Oscillations in Piecewise-Linear Slow-Fast Systems" extends the theory of canards---a method commonly used for smooth systems with multiple time scales---to the case of piecewise linear systems. In two dimensions, a classical canard is a solution to a smooth system of differential equations such that one variable is moving much faster than the other. The result is a periodic solution with a graph that appears almost discontinuous, with the shape resembling a cartoon duck (canard being French for both duck and a small joke). In this paper, the theory is extended to include systems that are only piecewise smooth, such as for Filippov vector fields. In addition to new results, this paper presents a self-contained introduction to recent approaches in multiple time scale systems. It is worth a read for those interested in systems with multiple time scales, particularly those with an interest in neuronal modeling. The second article, “Communicability Angle and the Spatial Efficiency of Networks," considers communication processes in a connected network, with the goal of understanding whether the graphs are spatially efficient. The paper develops a new measure of spatial efficiency, called the communicability angle, relating this angle to more traditional graph measures, such as the level of clustering and the existence of holes. The methods are applied to a series of 120 real-world data networks, including networks of urban streets, the power grid, the Internet, and the brain. This compelling new idea would be worthwhile for anyone with interest in theoretical, computational, or data networks. The third paper in the section uses a stochastic birth-death population model for competitive growth between a population of healthy cells and a population of cancer cells. “An Evolutionary Model of Tumor Cell Kinetics and the Emergence of Molecular Heterogeneity Driving Gompertzian Growth" explores the formation of tumors under this model and investigates different treatment methods within the subclinical, clinical, and lethal regimes of tumor growth. This paper is an excellent example of the power of mathematical modeling: The model borrows ideas from stochastic population models which were developed in a completely different applied context. It allows for new insights on tumor control, as it is possible to systematize the finding of optimized strategies, and it is also possible to develop therapies for subclinical tumors, which with current methods would not be clinically detected.

Application of stochastic models to a wildlife population

Long-term persistence of amphibian populations in a restored wetland complex

We have used a stochastic population model with immigration to calculate extinction rates for two amphibian species: the common toad, Bufo bufo, and the crested newt, Triturus cristatus. Population models for amphibians are difficult to parameterize. It was not possible to measure all the relevant parameters for the model. The other parameters were obtained from other studies or estimated on the basis of similar species. Contour maps for persistence were generated using the model. The persistence of populations associated with a pond is affected both by the pond's internal dynamics and its proximity to a source pond. On a 20-generation time scale, toad ponds that were initially occupied were relatively unaffected by the proximity of a source. Persistence was almost certain ( > 95%) when the average carrying capacity of a pond was greater than 30 adult females. Initially unoccupied toad-ponds were equally likely to persist if they lay within 4 km of a source and could support more than 50 adult females. Initially occupied newt ponds were likely to persist if they supported more than 40 females or lay within 0.5 km of a typical source pond. Initially unoccupied newt ponds were mainly dependent upon immigration rates. Very small ponds (N k > 10) only needed to be within 1.5 km of a source pond. This model allows us to use the data available, to make predictions about the criteria which must be met by a landscape to ensure the survival of amphibian populations. It offers the possibility of better predictions when the data is refined. The model also suggests directions for further research such as the statistical study of environmental variation, the nature of minimum viable populations for toads and the statistics of long-range dispersal.

Bifurcation dynamics in a stochastic population model with Beverton–Holt growth

Event Abstract Back to Event Stochastic modeling of neuronal signaling Jukka Intosalmi1*, Tiina Manninen1, Keijo Ruohonen1 and Marja-Leena Linne1 1 Tampere University of Technology, Finland The time evolution of biochemical systems in neurons is traditionally modeled using deterministic ordinary differential equations (ODEs). Chemical reactions, however, are random in nature, and the deterministic approach is valid only for a restricted class of systems. Stochastic models take random fluctuations into account and are thus more realistic. Biochemical reactions can be modeled stochastically using numerous different methods. An ideal model would have the following three important properties. First, the model should be as realistic as possible, second, the mathematical method should be easily implementable as a computer algorithm, and third, the algorithm should be computationally effective. Naturally, all these three conditions cannot be fulfilled at the same time. Some realistic modeling approaches can be derived directly from chemical kinetics without making any approximations. Such approaches are called exact. A good example of an exact modeling approach is the stochastic simulation algorithm (SSA) developed by Gillespie. The SSA is applicable when the molecular populations in the system are small, but it becomes computationally inefficient when the numbers of molecules increase. In order to construct stochastic models that can be effectively simulated, new mathematical approaches have to be explored. As an approximate method also stochastic differential equations (SDEs) have been considered a promising way to model biochemical reactions stochastically. The SDE approach is attractive especially if we consider a system for which the SSA is computationally inefficient and the traditional deterministic ODE approach cannot be used as a good approximation. SDE models treat the chemical populations as real numbers and the construction of the model is based on the law of mass action, similarly as the construction of the traditional deterministic modeling approach. The SDE model, however, takes the random fluctuations into account by describing the time evolution of the system with the stochastic Itô process instead of the deterministic set of ODEs. The Itô process is basically a set of coupled stochastic differential equations. Although SDE modeling usually leads to equations that cannot be solved analytically, the solutions can be approximated numerically using different numerical integration methods. Thus, SDEs provide a mathematically rigorous way to construct models that can be simulated effectively. In order to investigate how well the SDE approach models biochemical systems, the results have to be compared with experimental data or with the simulation results from some exact simulation procedure. In this study, we use the methodology of spectral analysis to compare the simulation results of the SDE model and the SSA. As case studies, we consider e.g. calcium binding and protein kinase C (PKC) signal transduction pathway. We investigate the nature of noise in different modeling approaches and try to interpret the meaning of the noise in real biological systems. The main goal of our study is to find the most essential components and parameters in the SDE model and to adjust them so that the model is capable of giving similar results as the SSA. The simulations are carried out using the distributed computing resources (GRID) provided by Techila Technologies Ltd. Conference: Neuroinformatics 2008, Stockholm, Sweden, 7 Sep - 9 Sep, 2008. Presentation Type: Poster Presentation Topic: Computational Neuroscience Citation: Intosalmi J, Manninen T, Ruohonen K and Linne M (2008). Stochastic modeling of neuronal signaling. Front. Neuroinform. Conference Abstract: Neuroinformatics 2008. doi: 10.3389/conf.neuro.11.2008.01.023 Copyright: The abstracts in this collection have not been subject to any Frontiers peer review or checks, and are not endorsed by Frontiers. They are made available through the Frontiers publishing platform as a service to conference organizers and presenters. The copyright in the individual abstracts is owned by the author of each abstract or his/her employer unless otherwise stated. Each abstract, as well as the collection of abstracts, are published under a Creative Commons CC-BY 4.0 (attribution) licence (https://creativecommons.org/licenses/by/4.0/) and may thus be reproduced, translated, adapted and be the subject of derivative works provided the authors and Frontiers are attributed. For Frontiers’ terms and conditions please see https://www.frontiersin.org/legal/terms-and-conditions. Received: 28 Jul 2008; Published Online: 28 Jul 2008. * Correspondence: Jukka Intosalmi, Tampere University of Technology, Tampere, Finland, jukka.intosalmi@tut.fi Login Required This action requires you to be registered with Frontiers and logged in. To register or login click here. Abstract Info Abstract The Authors in Frontiers Jukka Intosalmi Tiina Manninen Keijo Ruohonen Marja-Leena Linne Google Jukka Intosalmi Tiina Manninen Keijo Ruohonen Marja-Leena Linne Google Scholar Jukka Intosalmi Tiina Manninen Keijo Ruohonen Marja-Leena Linne PubMed Jukka Intosalmi Tiina Manninen Keijo Ruohonen Marja-Leena Linne Related Article in Frontiers Google Scholar PubMed Abstract Close Back to top Javascript is disabled. Please enable Javascript in your browser settings in order to see all the content on this page.

Optimal Control of Stochastic Systems

When studying deterministic and stochastic population models, the actual problems are the formalization of processes, taking into account new effects caused by the interaction of species, and the development of computer research methods. Computer research methods make it possible to analyze the trajectories of multidimensional population systems. We consider the “two competitors - two migration areas” model, which takes into account intraspecific and interspecific competition in two populations, as well as bidirectional migration of both populations. For this model, we take into account the variability of the reproduction rates of species. A formalized description of the four-dimensional model “two competitors - two migration areas” and its modifications is proposed. Using the implementation of the evolutionary algorithm, a set of parameters is obtained that ensure the coexistence of populations under conditions of competition between two species in the main area, taking into account the migration of these species. Taking into account the obtained set of parameters, a positive stationary state is found. Two-dimensional and three-dimensional projections of phase portraits are constructed. Stochastization of the model “two competitors - two migration areas” is carried out based on the method of self-consistent one-step models constructing. The Fokker-Planck equations are used to describe the structure of the model. A transition to a four-dimensional stochastic differential equation in the Langevin form is performed. To carry out numerical experiments, a specialized software package is used to construct and study stochastic models, and a computer program based on differential evolution is developed. Algorithms for generating trajectories of the Wiener process and multipoint distributions and modifications of the Runge-Kutta method are used. In the deterministic and stochastic cases, the dynamics of the trajectories of populationmigration systems is studied. A comparative analysis of deterministic and stochastic models is carried out. The results can be used in modeling of different classes of dynamic systems.

Fractional Brownian Motions, Fractional Noises and Applications

Analysis of a stochastic hybrid population model with Allee effect

In this work, we generalize the current theory of strong convergence rates for the backward Euler–Maruyama scheme for highly non-linear stochastic differential equations, which appear in both mathematical finance and bio-mathematics. More precisely, we show that under a dissipative condition on the drift coefficient and super-linear growth condition on the diffusion coefficient the BEM scheme converges with strong order of a half. This type of convergence gives theoretical foundations for efficient variance reduction techniques for Monte Carlo simulations. We support our theoretical results with relevant examples, such as stochastic population models and stochastic volatility models.

Extinction-time for stochastic population models

In this study, a glucose-insulin model with the Michaelis-Menten function as the rate of insulin degradation is analyzed using stochastic differential equations. Further, we solve the stochastic glucose-insulin model using the Milstein method, which is based on truncated Ito-Taylor expansions. Comparison of the approximation solution of a stochastic and deterministic model is illustrated by comparing the approximation solution with the deterministic model. A stochastic model allows random fluctuations in glucose-insulin diseases. Furthermore, the stochastic glucose-insulin model's numerical solution provides insight into its variability. The model predicts glucose-insulin dynamics accurately, which is a powerful tool for managing diabetes. Analytical and simulation results are consistent. Improved treatment strategies and personalized medical interventions could result. Treatments and insulin injections are sensitive to these parameters. Numerical simulation corroborates theoretical results.

One-dimensional Ito stochastic differential equations and stochastic differential equations with local time that depend on a small parameter and have irregular coefficients (for example, the unlimited drift coefficient or the non-Lipschitz diffusion coefficient) have been considered. The available results concerning the conditions for a mutually univocal correspondence between stochastic Ito equations and stochastic equations with local time were generalized. The weak convergence of the solutions of those equations at $\varepsilon \to 0$ were analyzed. The form of the coefficients for the limiting process was obtained. The necessary and sufficient conditions for the weak convergence of the solutions of those equations to the limit random process were proved.