A linear model of population dynamics

2016 ◽  
Vol 30 (15) ◽  
pp. 1541008 ◽  
Author(s):  
A. A. Lushnikov ◽  
A. I. Kagan
Keyword(s):  
Population Dynamics ◽  
Asymptotic Analysis ◽  
Population Growth ◽  
Master Equation ◽  
Bessel Function ◽  
Population Size ◽  
Linear Model ◽  
Linear Functions ◽  
Modified Bessel Function ◽  
Death Rates

The Malthus process of population growth is reformulated in terms of the probability [Formula: see text] to find exactly [Formula: see text] individuals at time [Formula: see text] assuming that both the birth and the death rates are linear functions of the population size. The master equation for [Formula: see text] is solved exactly. It is shown that [Formula: see text] strongly deviates from the Poisson distribution and is expressed in terms either of Laguerre’s polynomials or a modified Bessel function. The latter expression allows for considerable simplifications of the asymptotic analysis of [Formula: see text].

scholarly journals Integrability of stochastic birth-death processes via differential Galois theory

2020 ◽  
Vol 15 ◽  
pp. 70
Author(s):  
Primitivo B. Acosta-Humánez ◽  
José A. Capitán ◽  
Juan J. Morales-Ruiz
Keyword(s):  
Master Equation ◽  
Continuous Time ◽  
Galois Theory ◽  
Linear Functions ◽  
System State ◽  
Differential Galois Theory ◽  
Death Rates ◽  
Number Of Individuals ◽  
Temporal Variable ◽  
Birth Death

Stochastic birth-death processes are described as continuous-time Markov processes in models of population dynamics. A system of infinite, coupled ordinary differential equations (the so-called master equation) describes the time-dependence of the probability of each system state. Using a generating function, the master equation can be transformed into a partial differential equation. In this contribution we analyze the integrability of two types of stochastic birth-death processes (with polynomial birth and death rates) using standard differential Galois theory. We discuss the integrability of the PDE via a Laplace transform acting over the temporal variable. We show that the PDE is not integrable except for the case in which rates are linear functions of the number of individuals.

scholarly journals Timing outweighs amount of rainfall in shaping population dynamics

2020 ◽  
Author(s):  
Guoliang Li ◽  
Xinrong Wan ◽  
Baofa Yin ◽  
Wanhong Wei ◽  
Xianglei Hou ◽  
...  
Keyword(s):  
Population Dynamics ◽  
Population Growth ◽  
Population Size ◽  
Vole Population ◽  
Bottom Up ◽  
Manipulative Experiments ◽  
Arid Steppe ◽  
Steppe Grassland ◽  
Lasiopodomys Brandtii ◽  
Manipulation Experiment

Abstract Climate variability has been widely documented to have bottom-up effects on the population dynamics of animals1,2, but the mechanisms underlying these effects have been rarely investigated through field manipulative experiments that control for confounding factors3. Here, we examined the effects of different rainfall patterns (i.e. timing and amount) on the population size of Brandt’s voles Lasiopodomys brandtii in semi-arid steppe grassland in Inner-Mongolia by conducting a 10-year (2010-2019) rainfall manipulation experiment in twelve 0.48 ha field enclosures. We found that moderate rainfall increase during the early rather than late growing season drove marked increases in population size through increasing the biomass of preferred plant species, whereas heavily increased rainfall produced no further increase in vole population growth. The increase in vole population size was more coupled with increased reproduction of overwintered voles and increased body mass of young-of-year than with better survival. Our results provide the first experimental evidence for the bottom-up effects of changing rainfall on the population growth of small mammals, and highlight the importance of rainfall timing on the population dynamics of wildlife in the steppe grassland environment.

QUALITATIVE PROPERTIES OF A POPULATION DYNAMICS SYSTEM DESCRIBING PREGNANCY

2005 ◽  
Vol 15 (04) ◽  
pp. 507-554 ◽  
Author(s):  
G. FRAGNELLI ◽  
P. MARTINEZ ◽  
J. VANCOSTENOBLE
Keyword(s):  
Population Dynamics ◽  
Asymptotic Behavior ◽  
Linear Model ◽  
Nonlinear Model ◽  
Total Population ◽  
Type A ◽  
Qualitative Properties

We study a model of population dynamics describing pregnancy: our model is composed by an equation describing the evolution of the total population, and an equation describing the evolution of pregnant individuals. These equations are of course coupled: one coupling expresses that the total population varies with the number of born people, and another coupling says that the number of fecundated individuals depends on the total population. We study three models of that type: a linear model without diffusion, a nonlinear model without diffusion and a linear model with diffusion. For these three models, we study precisely the qualitative properties and the asymptotic behavior of the solutions.

scholarly journals Bounds and algorithms for the -Bessel function of imaginary order

2013 ◽  
Vol 16 ◽  
pp. 78-108 ◽  
Author(s):  
Andrew R. Booker ◽  
Andreas Strömbergsson ◽  
Holger Then
Keyword(s):  
Bessel Function ◽  
Convergent Series ◽  
Steepest Descent ◽  
Modified Bessel Function ◽  
Software Library ◽  
Asymptotic Bounds ◽  
Mixed Derivatives ◽  
Rigorous Computation ◽  
Working Out ◽  
Precise Asymptotic

AbstractUsing the paths of steepest descent, we prove precise bounds with numerical implied constants for the modified Bessel function${K}_{ir} (x)$of imaginary order and its first two derivatives with respect to the order. We also prove precise asymptotic bounds on more general (mixed) derivatives without working out numerical implied constants. Moreover, we present an absolutely and rapidly convergent series for the computation of${K}_{ir} (x)$and its derivatives, as well as a formula based on Fourier interpolation for computing with many values of$r$. Finally, we have implemented a subset of these features in a software library for fast and rigorous computation of${K}_{ir} (x)$.

scholarly journals Population dynamics in northwestern China

2021 ◽  
Author(s):  
Xueyan Yang ◽  
Wanxin Li ◽  
Wen Jing ◽  
Chezhuo Gao ◽  
Rui Li ◽  
...  
Keyword(s):  
Population Dynamics ◽  
Population Structure ◽  
Population Size ◽  
Ethnic Groups ◽  
Northwestern China ◽  
Average Level ◽  
Han Population ◽  
Sex Structure ◽  
Biological Sex

AbstractThis article analyzes the population dynamics in northwestern China from roughly 2010 to 2020. The area’s dynamics showed a slow, stable increase in population size, a stable increase in the population of non-Han ethnic groups, which increased at a more rapidly than the Han population, and population rejuvenation coupled with a population structure that aged. The biological sex structure fluctuated within a balanced range in northwestern China. Urbanization advanced in northwestern China, throughout this period, but the area’s level of urbanization is still significantly lower than the average level of urbanization nationally.

scholarly journals The influence of microhabitat on the population dynamics of four herbaceous species in a semiarid area of northeastern Brazil

2016 ◽  
Vol 76 (1) ◽  
pp. 45-54 ◽  
Author(s):  
K. A. Silva ◽  
J. M. F. F. Santos ◽  
J. R. Andrade ◽  
E. N. Lima ◽  
U. P. Albuquerque ◽  
...  
Keyword(s):  
Population Dynamics ◽  
Population Size ◽  
Mortality Rates ◽  
Dry Forest ◽  
Northeastern Brazil ◽  
Annual Rainfall ◽  
Herbaceous Species ◽  
Forest Fragment ◽  
Experimental Station ◽  
Over Time

Abstract Variation in annual rainfall is considered the most important factor influencing population dynamics in dry environments. However, different factors may control population dynamics in different microhabitats. This study recognizes that microhabitat variation may attenuate the influence of climatic seasonality on the population dynamics of herbaceous species in dry forest (Caatinga) areas of Brazil. We evaluated the influence of three microhabitats (flat, rocky and riparian) on the population dynamics of four herbaceous species (Delilia biflora, Commelina obliqua, Phaseolus peduncularis and Euphorbia heterophylla) in a Caatinga (dry forest) fragment at the Experimental Station of the Agronomic Research Institute of Pernambuco in Brazil, over a period of three years. D. biflora, C. obliqua and P. peduncularis were found in all microhabitats, but they were present at low densities in the riparian microhabitat. There was no record of E. heterophylla in the riparian microhabitat. Population size, mortality rates and natality rates varied over time in each microhabitat. This study indicates that different establishment conditions influenced the population size and occurrence of the four species, and it confirms that microhabitat can attenuate the effect of drought stress on mortality during the dry season, but the strength of this attenuator role may vary with time and species.

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