Total population size in critical branching processes in a random environment

2012 ◽  
Vol 91 (1-2) ◽  
pp. 12-21 ◽  
Author(s):  
V. A. Vatutin
Keyword(s):  
Population Size ◽  
Random Environment ◽  
Total Population ◽  
Branching Processes ◽  
Total Population Size

Asymptotic growth of a class of size-and-age-dependent birth processes

1974 ◽  
Vol 11 (2) ◽  
pp. 248-254 ◽  
Author(s):  
W. A. O'N. Waugh
Keyword(s):  
Population Size ◽  
Branching Process ◽  
Total Population ◽  
Asymptotic Expression ◽  
Branching Processes ◽  
Total Population Size ◽  
Stochastic Population Models ◽  
Age Dependent ◽  
Birth Processes ◽  
Special Case

A class of binary fission stochastic population models is described, in which the fission probabilities may depend on the age of an individual and the total population size. Age-dependent binary branching processes with Erlangian lifelength distributions are a special case. An asymptotic expression for the growth of the population size is developed, which generalizes known theorems about the asymptotic exponential growth of a branching process.

Asymptotic growth of a class of size-and-age-dependent birth processes

1974 ◽  
Vol 11 (02) ◽  
pp. 248-254 ◽  
Author(s):  
W. A. O'N. Waugh
Keyword(s):  
Population Size ◽  
Branching Process ◽  
Total Population ◽  
Asymptotic Expression ◽  
Branching Processes ◽  
Total Population Size ◽  
Stochastic Population Models ◽  
Age Dependent ◽  
Birth Processes ◽  
Special Case

A class of binary fission stochastic population models is described, in which the fission probabilities may depend on the age of an individual and the total population size. Age-dependent binary branching processes with Erlangian lifelength distributions are a special case. An asymptotic expression for the growth of the population size is developed, which generalizes known theorems about the asymptotic exponential growth of a branching process.

Random environment branching processes with equal environmental extinction probabilities

1973 ◽  
Vol 10 (03) ◽  
pp. 659-665
Author(s):  
Donald C. Raffety
Keyword(s):  
Markov Chains ◽  
Population Size ◽  
Random Environment ◽  
Transition Matrix ◽  
Branching Processes ◽  
Random Variables ◽  
Conditional Probabilities ◽  
Left Eigenvector ◽  
Extinction Probabilities ◽  
Limiting Values

R-positivity theory for Markov chains is used to obtain results for random environment branching processes whose environment random variables are independent and identically distributed and whose environmental extinction probabilities are equal. For certain processes whose eventual extinction is almost sure, it is shown that the distribution of population size conditioned by non-extinction at time n tends to a left eigenvector of the transition matrix. Limiting values of other conditional probabilities are given in terms of this left eigenvector and it is shown that the probability of non-extinction at time n approaches zero geometrically as n approaches ∞. Analogous results are obtained for processes whose extinction is not almost sure.

scholarly journals Carrying Capacity of a Population Diffusing in a Heterogeneous Environment

Mathematics ◽  
2020 ◽  
Vol 8 (1) ◽  
pp. 49 ◽  
Author(s):  
D.L. DeAngelis ◽  
Bo Zhang ◽  
Wei-Ming Ni ◽  
Yuanshi Wang
Keyword(s):  
Population Size ◽  
Carrying Capacity ◽  
Heterogeneous System ◽  
Total Population ◽  
Reaction Diffusion ◽  
Maximum Growth ◽  
Reaction Diffusion Equations ◽  
Logistic Growth ◽  
Chemostat Model ◽  
Total Population Size

The carrying capacity of the environment for a population is one of the key concepts in ecology and it is incorporated in the growth term of reaction-diffusion equations describing populations in space. Analysis of reaction-diffusion models of populations in heterogeneous space have shown that, when the maximum growth rate and carrying capacity in a logistic growth function vary in space, conditions exist for which the total population size at equilibrium (i) exceeds the total population that which would occur in the absence of diffusion and (ii) exceeds that which would occur if the system were homogeneous and the total carrying capacity, computed as the integral over the local carrying capacities, was the same in the heterogeneous and homogeneous cases. We review here work over the past few years that has explained these apparently counter-intuitive results in terms of the way input of energy or another limiting resource (e.g., a nutrient) varies across the system. We report on both mathematical analysis and laboratory experiments confirming that total population size in a heterogeneous system with diffusion can exceed that in the system without diffusion. We further report, however, that when the resource of the population in question is explicitly modeled as a coupled variable, as in a reaction-diffusion chemostat model rather than a model with logistic growth, the total population in the heterogeneous system with diffusion cannot exceed the total population size in the corresponding homogeneous system in which the total carrying capacities are the same.

Bionomics of Dytiscus alaskanus J. Balfour-Browne (Coleoptera: Dytiscidae) in a central Alberta lake

1985 ◽  
Vol 63 (6) ◽  
pp. 1316-1323 ◽  
Author(s):  
R. B. Aiken ◽  
C. W. Wilkinson
Keyword(s):  
Life History ◽  
Life Cycle ◽  
North America ◽  
Population Growth ◽  
Population Size ◽  
Total Population ◽  
Eutrophic Lake ◽  
North Central ◽  
Total Population Size

There are few studies of life history and population growth of large dytiscid beetles in North America. We sampled populations of Dytiscus alaskanus in a eutrophic lake in north central Alberta weekly in the summers of 1982 and 1983. Like many other temperate zone dytiscids, D. alaskanus has a univoltine life cycle. Dytiscus alaskanus prefers the area at the limit of emergent vegetation in the lake and is most often associated with shoreline vegetation of cattail and sedge. Populations of adult D. alaskanus are at a peak in the late spring and decline throughout the summer. Mark–recapture experiments allowed determination of total population size and monitoring of movement patterns in the lake. Data are discussed with reference to the relatively short summer with which these beetles must cope.

Estimation of the parameters of a branching process from migrating binomial observations

1998 ◽  
Vol 30 (4) ◽  
pp. 948-967 ◽  
Author(s):  
C. Jacob ◽  
J. Peccoud
Keyword(s):  
Confidence Interval ◽  
Asymptotic Behaviour ◽  
Population Size ◽  
Branching Process ◽  
Total Population ◽  
Initial Population ◽  
Asymptotic Confidence Interval ◽  
Total Population Size ◽  
Initial Population Size ◽  
Watson Process

This paper considers a branching process generated by an offspring distribution F with mean m < ∞ and variance σ2 < ∞ and such that, at each generation n, there is an observed δ-migration, according to a binomial law Bpvn*Nnbef which depends on the total population size Nnbef. The δ-migration is defined as an emigration, an immigration or a null migration, depending on the value of δ, which is assumed constant throughout the different generations. The process with δ-migration is a generation-dependent Galton-Watson process, whereas the observed process is not in general a martingale. Under the assumption that the process with δ-migration is supercritical, we generalize for the observed migrating process the results relative to the Galton-Watson supercritical case that concern the asymptotic behaviour of the process and the estimation of m and σ2, as n → ∞. Moreover, an asymptotic confidence interval of the initial population size is given.

Sign in / Sign up

Export Citation Format

Share Document