Discounting initial population sizes for prediction of extinction probabilities in patchy environments

1993 ◽  
Vol 70 (1-2) ◽  
pp. 51-61 ◽  
Author(s):  
Jianguo Liu
Keyword(s):  
Initial Population ◽  
Patchy Environments ◽  
Extinction Probabilities ◽  
Population Sizes

Extinction probabilities in predator–prey models

1986 ◽  
Vol 23 (01) ◽  
pp. 1-13
Author(s):  
S. E. Hitchcock
Keyword(s):  
Stochastic Models ◽  
Birth Rate ◽  
Exact Expression ◽  
Initial Population ◽  
Predator Prey ◽  
Extinction Probabilities ◽  
Population Sizes ◽  
The Mean ◽  
Predator Prey Models ◽  
Two Populations

Two stochastic models are developed for the predator-prey process. In each case it is shown that ultimate extinction of one of the two populations is certain to occur in finite time. For each model an exact expression is derived for the probability that the predators eventually become extinct when the prey birth rate is 0. These probabilities are used to derive power series approximations to extinction probabilities when the prey birth rate is not 0. On comparison with values obtained by numerical analysis the approximations are shown to be very satisfactory when initial population sizes and prey birth rate are all small. An approximation to the mean number of changes before extinction occurs is also obtained for one of the models.

Extinction probabilities in predator–prey models

1986 ◽  
Vol 23 (1) ◽  
pp. 1-13 ◽  
Author(s):  
S. E. Hitchcock
Keyword(s):  
Stochastic Models ◽  
Birth Rate ◽  
Exact Expression ◽  
Initial Population ◽  
Predator Prey ◽  
Extinction Probabilities ◽  
Population Sizes ◽  
The Mean ◽  
Predator Prey Models ◽  
Two Populations

Two stochastic models are developed for the predator-prey process. In each case it is shown that ultimate extinction of one of the two populations is certain to occur in finite time. For each model an exact expression is derived for the probability that the predators eventually become extinct when the prey birth rate is 0. These probabilities are used to derive power series approximations to extinction probabilities when the prey birth rate is not 0. On comparison with values obtained by numerical analysis the approximations are shown to be very satisfactory when initial population sizes and prey birth rate are all small. An approximation to the mean number of changes before extinction occurs is also obtained for one of the models.

Forward and backward processes in bisexual models with fixed population sizes

1994 ◽  
Vol 31 (02) ◽  
pp. 309-332 ◽  
Author(s):  
M. Möhle
Keyword(s):  
Overlapping Generations ◽  
Weak Limit ◽  
Fixed Number ◽  
Uhlenbeck Process ◽  
Fisher Model ◽  
Extinction Probabilities ◽  
Ornstein Uhlenbeck Process ◽  
Population Sizes ◽  
Watson Process ◽  
Fixed Population

This paper introduces exchangeable bisexual models with fixed population sizes and non-overlapping generations. In each generation there are N pairs of individuals consisting of a female and a male. The N pairs of a generation produce N daughters and N sons altogether, and these 2N children form the N pairs of the next generation at random. First the extinction of the lines of descendants of a fixed number of pairs is studied, when the population size becomes large. Under suitable conditions this structure can be approximately treated in the framework of a Galton-Watson process. In particular it is shown for the Wright-Fisher model that the discrepancy between the extinction probabilities in the model and in the approximating Galton-Watson process is of order N. Next, the process of the number of ancestor-pairs of all pairs of a generation is analysed. Under suitable conditions this process, properly normed, has a weak limit as N becomes large. For the Wright-Fisher model this limit is an Ornstein–Uhlenbeck process (restricted to a discrete time-set). The corresponding stationary distributions of the backward processes converge to the normal distribution, as expected.

scholarly journals Cost and benefits of clustered regularly interspaced short palindromic repeats spacer acquisition

2019 ◽  
Vol 374 (1772) ◽  
pp. 20180095 ◽  
Author(s):  
Serena Bradde ◽  
Thierry Mora ◽  
Aleksandra M. Walczak
Keyword(s):  
Optimal Strategies ◽  
Initial Population ◽  
Acquisition Rate ◽  
Bacterial Populations ◽  
Bacterial Colony ◽  
Immune Systems ◽  
Adaptive Immune ◽  
Population Sizes ◽  
Dynamical Balance ◽  
Adaptive Immune Systems

Clustered regularly interspaced short palindromic repeats (CRISPR)-Cas-mediated immunity in bacteria allows bacterial populations to protect themselves against pathogens. However, it also exposes them to the dangers of auto-immunity by developing protection that targets its own genome. Using a simple model of the coupled dynamics of phage and bacterial populations, we explore how acquisition rates affect the probability of the bacterial colony going extinct. We find that the optimal strategy depends on the initial population sizes of both viruses and bacteria. Additionally, certain combinations of acquisition and dynamical rates and initial population sizes guarantee protection, owing to a dynamical balance between the evolving population sizes, without relying on acquisition of viral spacers. Outside this regime, the high cost of auto-immunity limits the acquisition rate. We discuss these optimal strategies that minimize the probability of the colony going extinct in terms of recent experiments. This article is part of a discussion meeting issue ‘The ecology and evolution of prokaryotic CRISPR-Cas adaptive immune systems’.

scholarly journals Martingales and fixation probabilities of evolutionary graphs

2014 ◽  
Vol 470 (2165) ◽  
pp. 20130730 ◽  
Author(s):  
T. Monk ◽  
P. Green ◽  
M. Paulin
Keyword(s):  
Graph Theory ◽  
Random Walk ◽  
Large Population ◽  
Initial Population ◽  
Weak Selection ◽  
Simplifying Assumptions ◽  
Population Sizes ◽  
Fixation Probabilities ◽  
Evolutionary Graph Theory ◽  
Complete Bipartite Graphs

Evolutionary graph theory is the study of birth–death processes that are constrained by population structure. A principal problem in evolutionary graph theory is to obtain the probability that some initial population of mutants will fixate on a graph, and to determine how that fixation probability depends on the structure of that graph. A fluctuating mutant population on a graph can be considered as a random walk. Martingales exploit symmetry in the steps of a random walk to yield exact analytical expressions for fixation probabilities. They do not require simplifying assumptions such as large population sizes or weak selection. In this paper, we show how martingales can be used to obtain fixation probabilities for symmetric evolutionary graphs. We obtain simpler expressions for the fixation probabilities of star graphs and complete bipartite graphs than have been previously reported and show that these graphs do not amplify selection for advantageous mutations under all conditions.

Forward and backward processes in bisexual models with fixed population sizes

1994 ◽  
Vol 31 (2) ◽  
pp. 309-332 ◽  
Author(s):  
M. Möhle
Keyword(s):  
Overlapping Generations ◽  
Weak Limit ◽  
Fixed Number ◽  
Uhlenbeck Process ◽  
Fisher Model ◽  
Extinction Probabilities ◽  
Ornstein Uhlenbeck Process ◽  
Population Sizes ◽  
Watson Process ◽  
Fixed Population

This paper introduces exchangeable bisexual models with fixed population sizes and non-overlapping generations. In each generation there are N pairs of individuals consisting of a female and a male. The N pairs of a generation produce N daughters and N sons altogether, and these 2N children form the N pairs of the next generation at random.First the extinction of the lines of descendants of a fixed number of pairs is studied, when the population size becomes large. Under suitable conditions this structure can be approximately treated in the framework of a Galton-Watson process. In particular it is shown for the Wright-Fisher model that the discrepancy between the extinction probabilities in the model and in the approximating Galton-Watson process is of order N.Next, the process of the number of ancestor-pairs of all pairs of a generation is analysed. Under suitable conditions this process, properly normed, has a weak limit as N becomes large. For the Wright-Fisher model this limit is an Ornstein–Uhlenbeck process (restricted to a discrete time-set). The corresponding stationary distributions of the backward processes converge to the normal distribution, as expected.

scholarly journals On sampling error in genetic programming

2021 ◽  
Author(s):  
Dirk Schweim ◽  
David Wittenberg ◽  
Franz Rothlauf
Keyword(s):  
Genetic Programming ◽  
Population Size ◽  
Sampling Error ◽  
Search Space ◽  
Initial Population ◽  
Frequency Model ◽  
Severe Problem ◽  
Large Populations ◽  
Population Sizes ◽  
Problem Instances

AbstractThe initial population in genetic programming (GP) should form a representative sample of all possible solutions (the search space). While large populations accurately approximate the distribution of possible solutions, small populations tend to incorporate a sampling error. This paper analyzes how the size of a GP population affects the sampling error and contributes to answering the question of how to size initial GP populations. First, we present a probabilistic model of the expected number of subtrees for GP populations initialized with full, grow, or ramped half-and-half. Second, based on our frequency model, we present a model that estimates the sampling error for a given GP population size. We validate our models empirically and show that, compared to smaller population sizes, our recommended population sizes largely reduce the sampling error of measured fitness values. Increasing the population sizes even more, however, does not considerably reduce the sampling error of fitness values. Last, we recommend population sizes for some widely used benchmark problem instances that result in a low sampling error. A low sampling error at initialization is necessary (but not sufficient) for a reliable search since lowering the sampling error means that the overall random variations in a random sample are reduced. Our results indicate that sampling error is a severe problem for GP, making large initial population sizes necessary to obtain a low sampling error. Our model allows practitioners of GP to determine a minimum initial population size so that the sampling error is lower than a threshold, given a confidence level.

scholarly journals A Hybrid Machine Learning and Population Knowledge Mining Method to Minimize Makespan and Total Tardiness of Multi-Variety Products

2019 ◽  
Vol 9 (24) ◽  
pp. 5286 ◽  
Author(s):  
Yongtao Qiu ◽  
Weixi Ji ◽  
Chaoyang Zhang
Keyword(s):  
Machine Learning ◽  
Total Tardiness ◽  
Production Model ◽  
Initial Population ◽  
Mining Method ◽  
Pareto Solutions ◽  
Nsga Ii ◽  
Knowledge Mining ◽  
Population Sizes ◽  
Hybrid Machine

Nowadays, the production model of many enterprises is multi-variety customized production, and the makespan and total tardiness are the main metrics for enterprises to make production plans. This requires us to develop a more effective production plan promptly with limited resources. Previous research focuses on dispatching rules and algorithms, but the application of the knowledge mining method for multi-variety products is limited. In this paper, a hybrid machine learning and population knowledge mining method to minimize makespan and total tardiness for multi-variety products is proposed. First, through offline machine learning and data mining, attributes of operations are selected to mine the initial population knowledge. Second, an addition–deletion sorting method (ADSM) is proposed to reprioritize operations and then form the rule-based initial population. Finally, the nondominated sorting genetic algorithm II (NSGA-II) hybrid with simulated annealing is used to obtain the Pareto solutions. To evaluate the effectiveness of the proposed method, three other types of initial populations were considered under different iterations and population sizes. The experimental results demonstrate that the new approach has a good performance in solving the multi-variety production planning problems, whether it is the function value or the performance metric of the acquired Pareto solutions.

scholarly journals Long-term dynamics of rodent populations in arid Australia: the influence of rainfall

1999 ◽  
Vol 26 (4) ◽  
pp. 389 ◽  
Author(s):  
C. R. Dickman ◽  
P. S. Mahon ◽  
P. Masters ◽  
D. F. Gibson
Keyword(s):  
Native Species ◽  
Field Experiments ◽  
Reproductive Potential ◽  
Initial Population ◽  
Time Lags ◽  
Strong Correlations ◽  
Population Declines ◽  
Population Sizes ◽  
Rodent Populations

Populations of rodents were studied for periods of 7−9 years at each of three sites in arid Australia. All species fluctuated dramatically in abundance, being absent or in low numbers during droughts but erupting after significant rainfall. Strong correlations were obtained between capture rates and cumulative monthly rainfall residuals, with time lags, that had been modified by an exponential decay function to model the post-rain depletion of resources. The introduced Mus domesticus erupted within only two months of exceptional rainfall at one site, whereas the native rodents Notomys alexis, Pseudomys hermannsburgensis and P. desertor generally responded 3Œ10 months after rain. The faster response of M. domesticus reflects its high reproductive potential compared with the native rodents, and perhaps greater immigration. Differences in magnitudes and delays in response to rainfall among the native species were due probably to differences in initial population sizes and times available for response and, for P. desertor, to between-site differences in the quality or quantity of food resources. Population declines in all species during dry periods probably followed resource shortages. Despite the usually strong linkage between rainfall and population dynamics, rain failed to trigger population responses in rodents for prolonged periods at two of the sites, and was not clearly associated with an eruption of P. hermannsburgensis at the third. The lack of a population response at one site was attributed to predation, but events at the other two remain unexplained. We conclude that long-term studies should play an important role in describing temporal changes in rodent populations in arid Australia and, with field experiments, in evaluating how rainfall and other factors combine to effect the changes.

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