Stochastic population models with interacting species.

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.

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.

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

Analysis of a stochastic hybrid population model with Allee effect

Complex dynamics in a stochastic internal HIV model

A machine‐learning‐assisted stochastic cloud population model is coupled with the Advanced Research Weather Research and Forecasting (WRF) model to represent fluctuations in the cloud‐base mass flux associated with the life cycles and interactions among cumulus convection cells. In this cloud population model, the size distribution and the associated cloud‐base mass flux of the convective cells are related to their previous state and to the change in the total convective area via a transition function. The convective area tendency in turn is assumed to depend on the cloud‐base mass flux that is resolved by the host WRF model. The transition function is represented by a single hidden‐layer neural network trained by the evolution of convective cell size distributions in a 1‐km grid‐spacing WRF simulation run over the Australian Monsoon region. At every grid point of the host model, the cloud population model predicts the cell size and cloud‐base mass flux distributions from which a random sample of cells is fed to an entraining parcel model that calculates precipitation as well as the associated liquid water potential temperature and total moisture tendencies. These tendencies are averaged over the cells and provided to the host model. Several regional simulations are performed over tropical and midlatitude domains to test this as a potential approach to scale‐aware parameterization. It is shown that such an approach could be a new promising path to simulating realistic precipitation statistics and propagation of precipitation associated with the Madden‐Julian Oscillation while maintaining realistic depictions of the diurnal cycle over both land and ocean.

Stochastic models are of increasing importance in ecology but are usually only applied to observational data. Here we use a stochastic population model to combine experimental and observational data to understand the colonization of old fields by monarch butterflies Danaus plexippus. We experimentally tested for density dependence in oviposition rates when predators were excluded, and we measured predation rates under natural conditions. Significance tests on the resulting data showed that both oviposition and predation were density dependent but could not show how oviposition and mortality combine to determine egg densities in nature. We therefore used our data to calculate the Akaike Information Criterion to choose between a nested suite of stochastic models that differed in their oviposition and mortality terms. When we simply fit the models to the observational data, the best model assumed density independence in both oviposition and predation. When we instead first estimated the oviposition rate at low density from experimental data, however, the best model included density dependence in oviposition, and a model that included density dependence in both oviposition and predation performed nearly as well. This result is consistent with our experiments and suggests that experiments can enhance the usefulness of stochastic models in ecology.

Newtonian mechanics and the theory of relativity, which describe planetary motion accurately, have enabled humans to land on the moon. There is a strict rule that mathematically connects gravity and motion. However, demographers have not identified such a connection in population dynamics thus far (or, it may not exist). Accordingly, we still do not have any theory that can predict the exact figures for future populations. Nevertheless, researchers have used mathematics and have systematized the theory of mathematical demography. One of the reasons is to not only identify the strict rule in demographic phenomena, but also to establish a universal index that determines the eventual state of a population. The Malthus equation is the simplest and most basic mathematical model in this field. This chapter discusses the difference between deterministic and stochastic demographic models through the addition of noise.

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.

On stochastic population model with the Allee effect

Stochastic population models and their relevance for the conservation of species

Application of stochastic models to a wildlife population

Here we review the use of stochastic models in population biology. We provide precise definitions of demographic and environmental variance, and present a new concept of demographic covariance. We suggest that population fluctuations in many cases can be approximated by a diffusion process. Many interesting properties of the process can be expressed by the Green function, including the expected time to extinction which is of great interest in population viability analysis. We then apply this approach on density-dependent models without age structure and give a summary of some results obtained for age-structured models without density regulation. We suggest that these results may be very useful in practical management because they also give us the possibility of estimating the uncertainties in our estimates of for instance the probability of extinction of small populations. Their application will, however, depend on the availability of long-term data sets, where we can estimate the order of magnitude of essential parameters and evaluate their predictive power.