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

A remark on stationary distribution of a stochastic SIR epidemic model with double saturated rates

A stochastic SIRS epidemic model with nonlinear incidence rate

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.

In this study, we investigate the dynamic mechanisms of tumor progression in response to fluctuations and uncertainties within the tumor-immune microenvironment. Utilizing temporal single-cell data, we develop a novel stochastic reaction-convection model that captures the spatiotemporal dynamics of macrophage responses to tumor cells subjected to both multiplicative and additive noise generated by non-homologous microenvironmental fluctuations. We prove the existence and uniqueness of a global positive solution for the proposed stochastic model. Then, by combining the stochastic Lyapunov analysis and the comparison theorem, we explore the moment boundaries for cell populations, as well as the asymptotic behavior at the boundary equilibrium points; sufficient conditions for driving sustained tumor growth and clearance are derived by employing the ergodicity theorem and are interestingly found to be only related to multiplicative noise. Furthermore, we employ an upwind finite difference scheme to simulate the effects of different noise types on a cell population distribution and the persistence of tumor growth. Results show that while additive noise influences the multimodal distribution of early tumor cell phenotypes, it has minimal impact on the mean density of tumor cells, indicating that additive noise acts primarily as a diffusion factor. In contrast, increasing multiplication noise effectively inhibits the development without altering the number of peaks in a phenotypic distribution. Interestingly, when additive and multiplicative noises are correlated, stronger additive noise can have dual effects on the steady-state distribution of tumor cells, with increased correlation positively influencing tumor cell elimination. These results provide novel insight into the tumor-immune microenvironment dynamics.

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.

Population viability analysis (PVA) and other stochastic population models are frequently built and often used, but rarely tested. Stochastic metapopulation models of the White-throated Treecreeper (Cormobates leucophaea) and the Red-browed Treecreeper (Climacteris erythrops) were developed in a system of 39 remnant patches of eucalypt forest in southeastern New South Wales, Australia. Parameters of the model were estimated using data obtained outside the fragmented system. Field surveys of the patches were conducted to test the predicted probabilities of patch occupancy, which is one of few instances where stochastic population models have been tested with empirical data. The initial models underestimated the occupancy of the patches, and the models were modified using the results of the tests in conjunction with further information on the biology of the species. A number of different modifications were made to determine changes that produced results that matched the observations. The best of these modifications made reasonable predictions, although this is not equivalent to a test with independent data because the data were known prior to the modifications. The best-fitting modified models were tested by comparing the observed number of extinction and colonization events to the predicted number. The models underestimated the observed number of events, although imperfect survey methods may have contributed to these differences. The tests of the stochastic models contributed to their development by highlighting the nature of the predictive error. The modified models predicted that the White-throated Treecreeper would be likely to persist over the next 100 years in most of the 39 patches. In contrast, the Red-browed Treecreeper was predicted to become extinct in most patches within about 50 years of fragmentation. This study illustrates how spatial patterns can be used to test the predictions of population dynamic models, although we note that the tests are limited by survey error and spatial correlation in occupancy data.

We consider a Richards growth model (modified logistic model) driven by correlated multiplicative and additive colored noises, and investigate the effects of noises on the eventual distribution of population size with the help of steady-state analysis. An approximative Fokker–Planck equation is first derived for the stochastic model. By performing detailed theoretical analysis and numerical simulation for the steady-state solution of the Fokker–Planck equation, i.e., stationary probability distribution (SPD) of the stochastic model, we find that the correlated noises have complex effects on the statistical property of the stochastic model. Specifically, the phenomenological bifurcation may be caused by the noises. The position of extrema of the SPD depends on the model parameter and the characters of noises in different ways.

Modeling and dynamic analysis of a stochastic mutualism model with distributed delays

A stochastic model concerning the evolution of a multi-species population is presented assuming species competition for a habitat. The model takes into account colonization, death and replacement for all individuals. Two cases are treated: (i) colonizations follow the hierarchic rule by which species of lower rank are always outcompeted by those of higher rank and (ii) there are no privileged species. In both cases, under suitable assumptions, a thorough description of the evolution of the population is obtained. The two models are finally compared and the corresponding evolutionary behaviors of the populations are discussed.

The parameter perturbation makes it difficult to analyze the dynamic mechanism of the brake chatter with the help of an ideal deterministic model. Hence, a stochastic dynamic model of brake chatter with the uncertainty of the torsional stiffness of the brake disc is established, the Stribeck model is used to describe the friction characteristic between the brake pad and brake disc. Then, the Itô stochastic differential equation is solved by means of the stochastic averaging method, the boundary type of the one-dimensional energy process is identified to discuss the stochastic stability of the brake system. Furthermore, the Fokker Planck Kolmogorov equation is derived, the dynamic bifurcation and phenomenological bifurcation are analytically proven, the influences of parameters, such as noise intensity, brake pressure, and friction coefficient, on bifurcation characteristics are revealed. The results show that the noise intensity, brake pressure, and friction coefficient difference need to be reduced to improve the system stability. Finally, the probability density function of the brake system is used to validate the analytical analysis. The relevant results provide a theoretical basis for better suppressing the brake chatter.

A stochastic SIS-type epidemic model with general nonlinear incidence and disease-induced mortality is investigated. It is proved that the dynamical behaviors of the model are determined by a certain threshold value . That is, when and together with an additional condition, the disease is extinct with probability one, and when , the disease is permanent in the mean in probability, and when there is not disease-related death, the disease oscillates stochastically about a positive number. Furthermore, when , the model admits positive recurrence and a unique stationary distribution. Particularly, the effects of the intensities of stochastic perturbation for the dynamical behaviors of the model are discussed in detail, and the dynamical behaviors for the stochastic SIS epidemic model with standard incidence are established. Finally, the numerical simulations are presented to illustrate the proposed open problems.

Noise-induced bifurcations in the stochastic chemostat model with general nutrient uptake functions

In this paper, a stochastic chemostat model with variable yield and Contois growth function is investigated. The yield coefficient depends on the limiting nutrient, and the environmental noises are given by independent standard Brownian motions. First, the existence and uniqueness of global positive solution are proved. Second, by using stochastic Lyapunov function, Itô’s formula, and some important inequalities, stochastic characteristics for the stochastic model are studied, including the extinction of micro-organism, the strong persistence in the mean of micro-organism and, the existence of a unique stationary distribution of the stochastic model. Third, the necessary condition of an optimal stochastic control for the stochastic model is established by Hamiltonian function. In addition, some numerical simulations are carried out to illustrate the theoretical results and the influence of the variable yield on the microorganism.

We consider a general class of Markov population models formulated as stochastic difference equations. The population density is shown to converge either to 0, to +∞, or to a unique stationary distribution concentrated on (0, +∞), depending on the signs of the mean log growth rates near 0 and +∞. These results are applied to the Watkinson-MacDonald “bottleneck” model of annual plants with a seedbank, extended to allow for random environmental fluctuations and competition among co-occurring species. We obtain criteria for long-term persistence of single-species populations, and for coexistence of two competing species, and the biological significance of the criteria is discussed. The lamentably few applications to the problem at hand of classical limit-theory for Markov chains are surveyed.