Exponential growth of a branching process usually implies stable age distribution

1979 ◽  
Vol 16 (3) ◽  
pp. 651-656 ◽  
Author(s):  
Bo Berndtsson ◽  
Peter Jagers
Keyword(s):  
Population Size ◽  
Exponential Function ◽  
Exponential Growth ◽  
Branching Process ◽  
Age Distribution ◽  
Length Distribution ◽  
Random Variables ◽  
Stable Age Distribution

Start a Bellman–Harris branching process from one or several ancestors, whose ages are identically distributed random variables. Assume that the life-length distribution decays more quickly than exponentially and that the distribution of ages at start does not give too much mass to high ages (in a sense to be made precise). Then, if the expected population size is an exponential function of time, the ages must follow the stable age distribution of the process.

Exponential growth of a branching process usually implies stable age distribution

1979 ◽  
Vol 16 (03) ◽  
pp. 651-656
Author(s):  
Bo Berndtsson ◽  
Peter Jagers
Keyword(s):  
Population Size ◽  
Exponential Function ◽  
Exponential Growth ◽  
Branching Process ◽  
Age Distribution ◽  
Length Distribution ◽  
Random Variables ◽  
Stable Age Distribution

Start a Bellman–Harris branching process from one or several ancestors, whose ages are identically distributed random variables. Assume that the life-length distribution decays more quickly than exponentially and that the distribution of ages at start does not give too much mass to high ages (in a sense to be made precise). Then, if the expected population size is an exponential function of time, the ages must follow the stable age distribution of the process.

The growth and composition of branching populations

1984 ◽  
Vol 16 (2) ◽  
pp. 221-259 ◽  
Author(s):  
Peter Jagers ◽  
Olle Nerman
Keyword(s):  
Exponential Growth ◽  
Probability Space ◽  
Point Processes ◽  
Age Distribution ◽  
Single Type ◽  
Survey Paper ◽  
Stable Age Distribution ◽  
Definition Of ◽  
Counting Yield ◽  
Strict Definition

A single-type general branching population develops by individuals reproducing according to i.i.d. point processes on R+, interpreted as the individuals' ages. Such a population can be measured or counted in many different ways: those born, those alive or in some sub-phase of life, for example. Special choices of reproduction point process and counting yield the classical Galton–Watson or Bellman–Harris process. This reasonably self-contained survey paper discusses the exponential growth of such populations, in the supercritical case, and the asymptotic stability of composition according to very general ways of counting. The outcome is a strict definition of a stable population in exponential growth, as a probability space, a margin of which is the well-known stable age distribution.

Limit theorems for a branching process with disasters

1976 ◽  
Vol 13 (03) ◽  
pp. 466-475
Author(s):  
K. B. Athreya ◽  
N. Kaplan
Keyword(s):  
Population Size ◽  
Limit Theorems ◽  
Branching Process ◽  
Stable Distribution ◽  
Age Distribution ◽  
Random Times

A Bellman–Harris process is considered where the population is subjected to disasters which occur at random times. Each particle alive at the time of a disaster survives it with probability p. In the situation when explosion can occur, several limit theorems are proven. In particular, we prove that the age-distribution converges to the same stable distribution as the Bellman-Harris process and that the population size continues to be asymptotically exponential.

The growth and composition of branching populations

1984 ◽  
Vol 16 (02) ◽  
pp. 221-259 ◽  
Author(s):  
Peter Jagers ◽  
Olle Nerman
Keyword(s):  
Exponential Growth ◽  
Probability Space ◽  
Point Processes ◽  
Age Distribution ◽  
Single Type ◽  
Survey Paper ◽  
Stable Age Distribution ◽  
Definition Of ◽  
Counting Yield ◽  
Strict Definition

A single-type general branching population develops by individuals reproducing according to i.i.d. point processes on R +, interpreted as the individuals' ages. Such a population can be measured or counted in many different ways: those born, those alive or in some sub-phase of life, for example. Special choices of reproduction point process and counting yield the classical Galton–Watson or Bellman–Harris process. This reasonably self-contained survey paper discusses the exponential growth of such populations, in the supercritical case, and the asymptotic stability of composition according to very general ways of counting. The outcome is a strict definition of a stable population in exponential growth, as a probability space, a margin of which is the well-known stable age distribution.

Limit theorems for a branching process with disasters

1976 ◽  
Vol 13 (3) ◽  
pp. 466-475 ◽  
Author(s):  
K. B. Athreya ◽  
N. Kaplan
Keyword(s):  
Population Size ◽  
Limit Theorems ◽  
Branching Process ◽  
Stable Distribution ◽  
Age Distribution ◽  
Random Times

A Bellman–Harris process is considered where the population is subjected to disasters which occur at random times. Each particle alive at the time of a disaster survives it with probability p. In the situation when explosion can occur, several limit theorems are proven. In particular, we prove that the age-distribution converges to the same stable distribution as the Bellman-Harris process and that the population size continues to be asymptotically exponential.

Bias in Using an Age–Length Key to Estimate Age-Frequency Distributions

1978 ◽  
Vol 35 (2) ◽  
pp. 184-189 ◽  
Author(s):  
S. J. Westrheim ◽  
W. E. Ricker
Keyword(s):  
Frequency Distribution ◽  
Age Distribution ◽  
Length Distribution ◽  
Fish Population ◽  
Systematic Bias ◽  
Parental Age ◽  
Frequency Distributions ◽  
Class Strength ◽  
Key Words Age ◽  
Age Frequency Distribution

Consider two representative samples of fish taken in different years from the same fish population, this being a population in which year-class strength varies. For the "parental" sample the length and age of the fish are determined and are used to construct an "age–length key," the fractions of the fish in each (short) length interval that are of each age. For the "filial" sample only the length is measured, and the parental age–length key is used to compute the corresponding age distribution. Trials show that the age–length key will reproduce the age-frequency distribution of the filial sample without systematic bias only if there is no overlap in length between successive ages. Where there is much overlap, the age–length key will compute from the filial length-frequency distribution approximately the parental age distribution. Additional bias arises if the rate of growth if a year-class is affected by its abundance, or if the survival rate in the population changes. The length of the fish present in any given part of a population's range can vary with environmental factors such as depth of the water; nevertheless, a sample taken in any part of that range can be used to compute age from the length distribution of a sample taken at the same time in any other part of the range, without systematic bias. But this of course is not likely to be true of samples taken from different populations of the species. Key words: age–length key, bias, Pacific ocean perch, Sebastes alutus

Continuity for multi-type branching processes with varying environments

1999 ◽  
Vol 36 (01) ◽  
pp. 139-145 ◽  
Author(s):  
Owen Dafydd Jones
Keyword(s):  
Linear Combination ◽  
Branching Process ◽  
Branching Processes ◽  
Random Variables ◽  
Main Tool ◽  
Concentration Function ◽  
Independent Random Variables ◽  
Varying Environments

Conditions are derived for the components of the normed limit of a multi-type branching process with varying environments, to be continuous on (0, ∞). The main tool is an inequality for the concentration function of sums of independent random variables, due originally to Petrov. Using this, we show that if there is a discontinuity present, then a particular linear combination of the population types must converge to a non-random constant (Equation (1)). Ensuring this can not happen provides the desired continuity conditions.

Population-size-dependent branching process with linear rate of growth

1983 ◽  
Vol 20 (02) ◽  
pp. 242-250 ◽  
Author(s):  
F. C. Klebaner
Keyword(s):  
Population Size ◽  
Branching Process ◽  
Positive Probability ◽  
Linear Rate ◽  
Size Dependent ◽  
Rate Of Growth ◽  
Binary Splitting

The process we consider is a binary splitting, where the probability of division, , depends on the population size, 2i. We show that Zn converges to ∞ almost surely on a set of positive probability. Zn /n converges in distribution to a proper limit, diverges almost surely on converges almost surely on and there are no constants cn such that Zn /cn converges in probability to a non-degenerate limit.

scholarly journals Estimating population size of the cave shrimp Troglocaris anophthalmus (Crustacea, Decapoda, Caridea) using mark–release–recapture data

2015 ◽  
Vol 38 (1) ◽  
pp. 77-86
Author(s):  
J. Jugovic ◽  
◽  
E. Praprotnik ◽  
E. V. Buzan ◽  
M. Luznik ◽  
...  
Keyword(s):  
Population Size ◽  
Aquatic Invertebrates ◽  
Anthropogenic Activities ◽  
Age Distribution ◽  
Mark Release Recapture ◽  
Population Size Estimates ◽  
Cave Ecosystems ◽  
Size Estimates ◽  
Abundant Population ◽  
Population Numbers

Population size estimates are lacking for many small cave–dwelling aquatic invertebrates that are vulnerable to groundwater contamination from anthropogenic activities. Here we estimated the population size of freshwater shrimp Troglocaris anophthalmus sontica (Crustacea, Decapoda, Caridea) based on mark–release–recapture techniques. The subspecies was investigated in Vipavska jama (Vipava cave), Slovenia, with estimates of sex ratio and age distribution. A high abundance of shrimps was found even after considering the lower limit of the confidence intervals. However, we found no evidence of differences in shrimp abundances between summer and winter. The population was dominated by females. Ease of capture and abundant population numbers indicate that these cave shrimps may be useful as a bioindicator in cave ecosystems.

scholarly journals STUDI LALAT PENGOROK DAUN LIRIOMYZA SPP. PADA PERTANAMAN BAWANG DAUN, DAN PARASITOID OPIUS CHROMATOMYIAE BELOKOBYLSKIJ & WHARTON (HYMENOPTERA: BRACONIDAE)

2009 ◽  
Vol 9 (1) ◽  
pp. 22-31
Author(s):  
Rusli Rustam ◽  
Aunu Rauf ◽  
Nina Maryana ◽  
Pudjianto Pudjianto ◽  
Dadang Dadang
Keyword(s):  
Age Distribution ◽  
Host Density ◽  
Field Studies ◽  
Reproductive Value ◽  
Laboratory Studies ◽  
Demographic Parameter ◽  
Liriomyza Huidobrensis ◽  
Stable Age Distribution ◽  
Net Reproduction ◽  
Density Results

Studies on Leafminer Liriomyza spp. in Green Onion Fields, and Parasitoid Opius chromatomyiae Belokobylskij & Wharton (Hymenoptera: Braconidae).  Field studies were conducted to determine population abundance of leafminers and their parasitoids in green onion fields in Puncak, West Java. In addition to that, laboratory studies were carried out to determine demographic parameter of Opius chromatomyiae as well as response of parasitoid to increasing host density. Results revealed that green onions were infested by two species of leafminers, Liriomyza huidobrensis and Liriomyza chinensis.  Leafminer flies emerged from Erwor leaves (54.5) were significantly higher than those of RP leaves (18.65) (P = 0.0005). However, number of  leafminer flies caught on sticky traps was not statistically different (P = 0.297).  Two species of parasitoid, Hemiptarsenus varicornis and O. chromatomyiae, were associated with leafminers in green onion fields. Higher number of parasitoids emerged from Erwor leaves (13.68) as compared to RP (6.90) (P =0.0007 ). However, level of parasitization were 24.36% on Erwor and 28.45% on RP, and was not significantly different (P = 0.387). Laboratory studies indicated that net reproduction (Ro) of O. chromatomyiae was 28.55, generation time (T) 15.96 days, intrinsic growth rate 0.21, and total of reproductive value 223.64.  The stable age distribution of parasitoid were 37.93% eggs, 24.92% larvae, 20.36% pupae and 16.78% adults.  The parasitoid showed functional response type II to increasing host density, with a = 0.08 and Th = 2.58.

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