Analysis of the Stochastic Population Model with Random Parameters

Abstract

The population models allow for a better understanding of the dynamical interactions with the environment and hence can provide a way for understanding the population changes. They are helpful in studying the biological invasions, environmental conservation and many other applications. These models become more complicated when accounting for the stochastic and/or random variations due to different sources. In the current work, a spectral technique is suggested to analyze the stochastic population model with random parameters. The model contains mixed sources of uncertainties, noise and uncertain parameters. The suggested algorithm uses the spectral decompositions for both types of randomness. The spectral techniques have the advantages of high rates of convergence. A deterministic system is derived using the statistical properties of the random bases. The classical analytical and/or numerical techniques can be used to analyze the deterministic system and obtain the solution statistics. The technique presented in the current work is applicable to many complex systems with both stochastic and random parameters. It has the advantage of separating the contributions due to different sources of uncertainty. Hence, the sensitivity index of any uncertain parameter can be evaluated. This is a clear advantage compared with other techniques used in the literature.

Analysis of the Stochastic Quarter-Five Spot Problem Using Polynomial Chaos

“…Applying inner product with each polynomial and utilizing the orthogonality relation to get a set of deterministic equations of the form where which can be computed and stored priori [ 29 ]. We can apply the TPFA numerical scheme for this set of deterministic equations and obtain the pressure components .…”

“…where e ijk = ψ i ψ j ψ k , which can be computed and stored priori [29]. We can apply the TPFA numerical scheme for this set of deterministic equations and obtain the pressure components p i (x).…”

Most Probable Dynamics of the Single-Species with Allee Effect under Jump-Diffusion Noise

“…Outside of a few clear trends, the dynamics of biological phenomena, particularly of populations for living beings, are frequently influenced by unpredictable components due to the complexity and variability of environmental conditions [10]. Researchers have been extensively studying biological dynamical systems for a long time now, particularly in the context of modeling and analyzing random fluctuations [11,12].…”

Most Probable Dynamics of the Single-Species with Allee Effect under Jump-Diffusion Noise

“…The dynamics of biological phenomena, particularly that of populations for living beings, besides some clear trends, are frequently influenced by unpredictable components due to the complexity and variability of environmental conditions [35]. Extensive researchers in modeling and analysing of random fluctuations have been studying the biological dynamical systems for a long time now [5,39,40].…”