Stochastic Dynamical Theta-Logistic Population Growth Model

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.

We consider the optimal harvesting and planting control problem to maximize the expected total net benefits in the stochastic logistic population model. The variational inequality associated with this problem is given by the degenerate form of elliptic type with quadratic coefficients. Using the viscosity solutions technique, we solve the corresponding penalty equation and show the existence of a solution to the variational inequality. The optimal harvesting and planting policy is characterized in terms of two thresholds for the variational inequality.

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.

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.

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

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.

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.

Analysis of a stochastic hybrid population model with Allee effect

Optimal harvesting policy for stochastic Logistic population model

Time-dependent probability density function for general stochastic logistic population model with harvesting effort

We propose a stochastic logistic model with mate limitation and stochastic immigration. Incorporating stochastic immigration into a continuous time Markov chain model, we derive and analyze the associated master equation. By a standard result, there exists a unique globally stable positive stationary distribution. We show that such stationary distribution admits a bimodal profile which implies that a strong Allee effect exists in the stochastic model. Such strong Allee effect disappears and threshold phenomenon emerges as the total population size goes to infinity. Stochasticity vanishes and the model becomes deterministic as the total population size goes to infinity. This implies that there is only one possible fate (either to die out or survive) for a species constrained to a specific community and whether population eventually goes extinct or persists does not depend on initial population density but on a critical inherent constant determined by birth, death and mate limitation. Such a conclusion interprets differently from the classical ordinary differential equation model and thus a paradox on strong Allee effect occurs. Such paradox illustrates the diffusion theory's dilemma.

SUMMARY In this paper we present stochastic models to describe the changes that take place with time in the numbers of females and males in the various age-intervals in the population. We first consider the case where females, or males, are marriage dominant, i.e. where the joint distribution of the numbers of boys and girls born at time t depends upon the numbers of females, or males, in the various age-intervals in the immediately preceding time-period. We then consider the case where neither females nor males are marriage-dominant, i.e. where the joint distribution of the numbers of boys and girls born at time t depends upon both the numbers of females and males in the various age-intervals in the immediately preceding time-period. Methods are given for calculating the expected values, variances, and covariances of the number of females and males in each age-interval at time t, and asymptotic results are presented for the case where t -* so. The application of these results is illustrated with data on the fertility and mortality conditions in the U.S. in 1965. The literature on two-sex stochastic models of population growth starts with articles by Kendall (1949) and Goodman (1953). Kendall outlined some of the analytic difficulties of a study of the stochastic aspects of two-sex population growth, and for simplicity limited his discussion of two-sex stochastic models to the special case where (a) each birth is equally likely to add a new female or a new male to the population, (b) the death-rate for females is equal to that for males, and (c) the birth- and death-rates per person are constants that are independent of the age of the person, and also independent of other relevant variables. Although Goodman's analysis was not limited by (a) and (b), it was limited by (c), as was all subsequent analytic work by other authors. One of the models of Goodman (1953) was a stochastic model where one sex is marriage-dominant, i.e. where the joint distribution of the numbers of girls and boys born at time t depends only upon the number of the dominant sex in the population in the immediately preceding time-period; he calculated the means, variances, and covariances of the numbers of females and males in the population at time t. A similar analysis of this model was made by Joshi (1954); see also Barucha-Reid (1960, pp. 175-9), Bailey (1964, pp. 119-20), and Keyfitz (1968). Lamens (1957) supposed that the birth- and death-rates may be functions of time, but he did not allow for possible dependence of these rates on the age-composition of the population, nor for the possibility that neither females nor males are marriage-dominant. Here we shall deal with stochastic models where the birth- and death-rates for females and for males may depend upon the age-composition of the population. First, say, the females will be assumed marriage-dominant, and then the more general situation will be considered. The above articles dealt mainly with some of the stochastic aspects of two-sex populationgrowth. Some of the deterministic aspects have been discussed in the demographic and bio

Applications of computerized stochastic models of human reproduction and population growth in family planning evaluation

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.

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.