Stochastic population growth in spatially heterogeneous environments: The density-dependent case

This work is devoted to studying the dynamics of a structured population that is subject to the combined effects of environmental stochasticity, competition for resources, spatio-temporal heterogeneity and dispersal. The population is spread throughout $n$ patches whose population abundances are modelled as the solutions of a system of nonlinear stochastic differential equations living on $[0,\infty)^n$. We prove that $r$, the stochastic growth rate of the total population in the absence of competition, determines the long-term behaviour of the population. The parameter $r$ can be expressed as the Lyapunov exponent of an associated linearized system of stochastic differential equations. Detailed analysis shows that if $r>0$, the population abundances converge polynomially fast to a unique invariant probability measure on $(0,\infty)^n$, while when $r<0$, the population abundances of the patches converge almost surely to $0$ exponentially fast. This generalizes and extends the results of Evans et al (2014 J. Math. Biol.) and proves one of their conjectures. Compared to recent developments, our model incorporates very general density-dependent growth rates and competition terms. Furthermore, we prove that persistence is robust to small, possibly density dependent, perturbations of the growth rates, dispersal matrix and covariance matrix of the environmental noise. Our work allows the environmental noise driving our system to be degenerate. This is relevant from a biological point of view since, for example, the environments of the different patches can be perfectly correlated. As an example we fully analyze the two-patch case, $n=2$, and show that the stochastic growth rate is a decreasing function of the dispersion rate. In particular, coupling two sink patches can never yield persistence, in contrast to the results from the non-degenerate setting treated by Evans et al.Comment: 43 pages, 1 figure, edited according to the suggestion of the referees, to appear in Journal of Mathematical Biolog

DYNAMIC RELATIONS AND SHARIA STOCK MARKET INTEGRATION WITH OIL PRICES (Studies: Indonesia, Malaysia, USA, UK, Japan 2012-2016)

The purpose of this research is to analyze the relationship of dynamic and integration between world sharia stock market with world crude oil price. This research can find out the integration relationship between world sharia stock market with world crude oil price. The object of this research is sharia stock market in Indonesia, Malaysia, United States, UK, Japan during period 2012-2016. The research method is Dynamic Coditional Correlation Multivariate-GARCH method is used to test the hypothesis in order to know the relationship of sharia stock market integration in world with world oil price. In this case to test the conditional correlation multivariate-GARCH method, reasearcher have taken any steps is descriptive statistical testing, heteroskedasticity testing, stationary test, and GARCH univariate testing. The result of the research shows that there is a significant dynamic correlation in world sharia stock price (Indonesia, Malaysia, United States, United Kingdom, Japan) and significant dynamic relationship between world sharia stock market with world crude oil price. It can be explained indirectly proves the existence of integration relationship between world sharia stock market with world crude oil price. Keywords: sharia stocks integration, sharia stock price, world crude oil price, Dynamic Conditional Correlation Multivariate-GARCH (DCC-MGARCH)

The free path in a high velocity random flight process associated to a Lorentz gas in an external field

We investigate the asymptotic behavior of the free path of a variable density random flight model in an external field as the initial velocity of the particle goes to infinity. The random flight models we study arise naturally as the Boltzmann-Grad limit of a random Lorentz gas in the presence of an external field. By analyzing the time duration of the free path, we obtain exact forms for the asymptotic mean and variance of the free path in terms of the external field and the density of scatterers. As a consequence, we obtain a diffusion approximation for the joint process of the particle observed at reflection times and the amount of time spent in free flight.Comment: 30 page