Evolutionary equilibrium strategies.

This paper re-examines the concept of evolutionary stability proposed by Maynard Smith.For any finite population it is shown that a strategy which is stable in the sense of Maynard Smith may have a lower fitness than a mutant strategy regardless of the proportion of contestants using the latter.Two alternative concepts of evolutionary stability are then proposed. A strategy is described as being strongly stable if no mutant is able to invade because of its higher fitness and weakly stable if it has a higher fitness whenever the contestants using any particular mutant strategy become sufficiently numerous.For the “war of attrition” between contestants of a given species Maynard Smith and others have argued that the evolutionarily stable strategy is for a contestant to bid (for food or territory) by attempting to wait out its opponent according to an exponential mixed strategy. This paper establishes that such a strategy is only weakly stable and that for any number n there exists a mutant strategy with a higher fitness until the number of mutants in the population exceeds n.The final section reconsiders the stability issue when a natural informational asymmetry is introduced. Each contestant is assumed to be uncertain as to the value its opponent places on the object over which they are competing. In contrast to the symmetric case it is shown that there is a strategy which is strongly stable with respect to any feasible mutant as long as the population is sufficiently large.     

Density dependent selection 1: A stable feasible equilibrium may not be attainable

The joint evolution of gene frequency p, and population size N is studied. It is shown that when the genotypic fitnesses are logistic functions of the population size, sets of initial states exist which lead to bizarre behavior. Even though equilibria may be locally stable, these sets of initial conditions eventually produce negative fitnesses. Alternative models are discussed as are some general properties of the mean fitness.     

A general formula for the sensitivity of population growth rate to changes in life history parameters.

This paper considers the sensitivity of population growth to small changes in birth, growth, survival, and migration probabilities for an arbitrary population classification (i.e., age, instar, size, developmental stage, age, and spatial location, etc.). The stage-specific life history parameters are expressed in a discrete-time system of linear difference equations, the dominant eigenvalue of which defines the population growth rate. The sensitivity of this eigenvalue to production of class i by class j individuals is shown to be proportional to the product of the reproductive value of stage i and the abundance of stage j in the stable stage distribution. This formula is readily computable, and several examples are presented. For the special case of age-structured populations, this formula reduces to those derived by Hamilton, Emlen, and Goodman.     

Statistical properties of estimators for continuous time Markov branching processes with varying and random environments

Asymptotic properties are established for estimators of time dependent intensities in Markov branching processes with varying and random environments. For the varying environment model, the estimators are shown to be uniformly strongly consistent on bounded intervals as the initial population size X₀ → ∞, and, when considered as empirical stochastic processes, to converge weakly to Gaussian processes with independent increments. For random environments, the estimators are shown to be asymptotically normal as t → ∞, where t is the time parameter.     

Historical DNA reveals the demographic history of Atlantic cod (Gadus morhua) in medieval and early modern Iceland

Atlantic cod (Gadus morhua) vertebrae from archaeological sites were used to study the history of the Icelandic Atlantic cod population in the time period of 1500–1990. Specifically, we used coalescence modelling to estimate population size and fluctuations from the sequence diversity at the cytochrome b (cytb) and Pantophysin I (PanI) loci. The models are consistent with an expanding population during the warm medieval period, large historical effective population size (N_E), a marked bottleneck event at 1400–1500 and a decrease in N_E in early modern times. The model results are corroborated by the reduction of haplotype and nucleotide variation over time and pairwise population distance as a significant portion of nucleotide variation partitioned across the 1550 time mark. The mean age of the historical fished stock is high in medieval times with a truncation in age in early modern times. The population size crash coincides with a period of known cooling in the North Atlantic, and we conclude that the collapse may be related to climate or climate-induced ecosystem change.     

Effects of density dependent sex allocation on the dynamics of a simultaneous hermaphroditic population: Modelling and analysis

In this work we present a mathematical model describing the dynamics of a population where sex allocation remains flexible throughout adult life and so can be adjusted to current environmental conditions. We consider that the fractions of immature individuals acquiring male and female sexual roles are density dependent through nonlinear functions of a weighted total population size. The main goal of this work is to understand the role of life-history parameters on the stabilization or destabilization of the population dynamics.The model turns out to be a nonlinear discrete model which is analysed by studying the existence of fixed points as well as their stability conditions in terms of model parameters. The existence of more complex asymptotic behaviours of system solutions is shown by means of numerical simulations.Females have larger fertility rate than males. On the other hand, increasing population density favours immature individuals adopting the male role. A positive equilibrium of the system exists whenever fertility and survival rates of one of the sexual roles, if shared by all adults, allow population growing while the opposite happens with the other sexual role. In terms of the female inherent net reproductive number, η_F, it is shown that the positive equilibria are stable when η_F is larger and closed to 1 while for larger values of η_F a certain asymptotic assumption on the investment rate in the female function implies that the population density is permanent. Depending on the other parameters values, the asymptotic behaviour of solutions becomes more complex, even chaotic. In this setting the stabilization/destabilization effects of the abruptness rate in density dependence, of the survival rates and of the competition coefficients are analysed.     

Not-so-social learning strategies

Social learning strategies (SLSs) are rules specifying the conditions in which it would be adaptive for animals to copy the behaviour of others rather than to persist with a previously established behaviour or to acquire a new behaviour through asocial learning. In behavioural ecology, cultural evolutionary theory and economics, SLSs are studied using a ‘phenotypic gambit’—from a purely functional perspective, without reference to their underlying psychological mechanisms. However, SLSs are described in these fields as if they were implemented by complex, domain-specific, genetically inherited mechanisms of decision-making. In this article, we suggest that it is time to begin investigating the psychology of SLSs, and we initiate this process by examining recent experimental work relating to three groups of strategies: copy when alternative unsuccessful, copy when model successful and copy the majority. In each case, we argue that the reported behaviour could have been mediated by domain-general and taxonomically general psychological mechanisms; specifically, by mechanisms, identified through conditioning experiments, that make associative learning selective. We also suggest experimental manipulations that could be used in future research to resolve more fully the question whether, in non-human animals, SLSs are mediated by domain-general or domain-specific psychological mechanisms.     

Population extinction and quasi-stationary behavior in stochastic density-dependent structured models

Density-independent and density-dependent, stochastic and deterministic, discrete-time, structured models are formulated, analysed and numerically simulated. A special case of the deterministic, density-independent, structured model is the well-known Leslie age-structured model. The stochastic, density-independent model is a multitype branching process. A review of linear, density-independent models is given first, then nonlinear, density-dependent models are discussed. In the linear, density-independent structured models, transitions between states are independent of time and state. Population extinction is determined by the dominant eigenvalue λ of the transition matrix. If λ ≤ 1, then extinction occurs with probability one in the stochastic and deterministic models. However, if λ > 1, then the deterministic model has exponential growth, but in the stochastic model there is a positive probability of extinction which depends on the fixed point of the system of probability generating functions. The linear, density-independent, stochastic model is generalized to a nonlinear, density-dependent one. The dependence on state is in terms of a weighted total population size. It is shown for small initial population sizes that the density-dependent, stochastic model can be approximated by the density-independent, stochastic model and thus, the extinction behavior exhibited by the linear model occurs in the nonlinear model. In the deterministic models there is a unique stable equilibrium. Given the population does not go extinct, it is shown that the stochastic model has a quasi-stationary distribution with mean close to the stable equilibrium, provided the population size is sufficiently large. For small values of the population size, complete extinction can be observed in the simulations. However, the persistence time increases rapidly with the population size. This author received partial support by the National Science Foundation grant # DMS-9626417.     

Harvesting strategies for conserving minimum viable populations based on World Conservation Union criteria: brown bears in Norway

The World Conservation Union (IUCN) defines populations as vulnerable if the probability of extinction is larger than 10\% within the next 100 years. With the objective of minimizing problems with predation on domestic livestock and, at the same time, conserving a viable population, we consider different threshold harvesting strategies for a small population of brown bear, based on a population dynamics model with growth rate and demographic and environmental variances estimated from the present Swedish population. Taking into account uncertainties in present estimates of the demographic parameters and in population size, we show that the population can be harvested when the population size exceeds 34 female bears,aged one year and older, if the entire population exceeding the threshold is harvested. To minimize the expected long-term population size, however, we show that it is optimal to harvest only a proportion equal to 35% of the population exceeding a lower threshold of 12 female bears. This strategy gives an expected long-term population size of around 20 female bears. If the growth rate of the population is reduced by ca. 3%, the threshold must, under some conditions, be doubled. We argue that the small thresholds are mainly a result of the high intrinsic growth rate of the population considered in the present paper. However, the analysis also suggests that IUCN''s criterion might allow a rate of extinction that is too high.     

千岛湖姥山马尾松种群结构和分布格局研究

为研究千岛湖马尾松(Pinus massoniana)种群动态变化,在姥山岛上设置面积为5.76 hm2的固定样地,进行群落学调查。根据野外调查的数据,对马尾松种群的结构和分布格局进行分析。结果表明:马尾松种群1、2级个体数极少,仅占样地中马尾松个体数的0.06%和1.01%,幼苗、幼树储备严重不足,5级个体的比例最高,达到28.28%,马尾松种群径级结构为纺锤型;马尾松种群的存活曲线为凸型。这些均表明马尾松种群的年龄结构趋向衰退型。但马尾松种群中树个体很多,在较长的时期内马尾松还不会退出群落。马尾松种群静态生命表也可看出中一些小型径级的死亡率为负值,也说明马尾松种群幼苗严重缺乏,种群呈衰退趋势。马尾松种群各径级和总体的分布格局均呈聚集分布,从小树→中树→大树聚集指数逐渐减小,种群呈扩散趋势。     

Evolutionarily stable strategies for a finite population and a variable contest size

This paper presents a generalization of Maynard Smith's concept of an evolutionarily stable strategy (ESS) to cover the cases of a finite population and a variable contest size. Both equilibrium and stability conditions are analysed. The standard Maynard Smith ESS with an infinite population and a contest size of two (pairwise contests) is shown to be a special case of this generalized ESS. An important implication of the generalized ESS is that in finite populations the behaviour of an ESS player is "spiteful", in the sense that an ESS player acts not only to increase his payoff but also to decrease the payoffs of his competitors. The degree of this "spiteful" behaviour is shown to increase with a decrease in the population size, and so is most likely to be observed in small populations. The paper concludes with an extended example: a symmetric two-pure-strategies two-player game for a finite population. It is shown that a mixed strategy ESS is globally stable against invasion by any one type of mutant strategist. The condition for the start of simultaneous invasion by two types of mutant is also given.     

Global mutations and local mutations have very different effects on evolution, illustrated by mixed strategies of asymmetric binary games

We study the evolutionary effect of rare mutations causing global changes in traits. We consider asymmetric binary games between two players. The first player takes two alternative options with probability x and 1−x; and the second player takes options with probability y and 1−y. Due to natural selection and recurrent mutation, the population evolves to have broad distributions of x and y. We analyze three cases showing qualitatively different dynamics, exemplified by (1) vigilance-intrusion game, (2) asymmetric hawk-dove game and (3) cleaner-client game. We found that the evolutionary outcome is strongly dependent upon the distribution of mutants’ traits, more than the mutation rates. For example in the vigilance-intrusion game, the evolutionary dynamics show a perpetual stable oscillation if mutants are always close to the parent (local-mutation mode), whilst the population converges to a stable equilibrium distribution if mutants can be quite different from the parent (global-mutation mode), even for extremely low mutation rate. When common local mutations and rare global mutations occur simultaneously, the evolutionary outcome is controlled by the latter.     

The Effects of Overdominance on Linkage in a Multilocus System

Computer simulations were performed with overdominant multiple alleles among tightly linked multiple loci under a multiplicative fitness model. The quantity X²/N(n — 1) was introduced as a new measure of linkage disequilibrium which, unlike previously available measures, can be applied to multiple allele models, where N is the sample size, and n is the number of alleles at the locus possessing fewest alleles. Simulations showed that (1) With multiple (three or four) alleles, the approach to stable disequilibrium is slower and the amount of disequilibrium established is weaker than in a two allele system. (2) The number of complementary chromosomes is a function of number of alleles and of population size. (3) As population size increases, the rate of the approach to stable disequilibrium is slower. (4) There is an optimum selection coefficient which minimizes the transient fixation probability of alleles when linkage is present. (5) The absence of linkage disequilibrium is in most cases not a practical method of testing the hypothesis of balancing selection of genetic polymorphisms because it depends strongly on population size in determining linkage disequilibria.     

The general biology of Verruca stroemia (O.F. Müller). II. Reproductive cycle,population structure,and factors affecting release of nauplii

The reproductive cycle of Verruca stroemia (O. F. Müller) has been investigated. From an examination of the percentage of animals with egg masses it is shown that there is a major brood, virtually synchronous throughout the whole population, and which is released at about the time of the spring diatom increase: subsequently, smaller broods are produced during the summer, ≈ 20 % of the population carrying egg masses at any one time. The relation of these broods to the size — frequency composition of the oocytes throughout the year has been examined.Sexual maturity is reached at a size of ≈ 2.0 mm basal diameter.Animals maintained on a raft and kept free from other organisms have a greater breeding potential; this is ascribed to the better nutrient conditions.No egg masses are present during the autumn. It is suggested that the poor nutrient conditions allow all the animals in the population to bring the female gonads to the same state by late winter and so give marked synchrony of the spring brood.Most size-classes are present throughout the year.Release of nauplii is stimulated by extracts of V. stroemia itself, Balanus balanoides, B. balanus, and Chthamalus stellatus.     

Mutation-selection equilibrium in games with mixed strategies

We develop a new method for studying stochastic evolutionary game dynamics of mixed strategies. We consider the general situation: there are n pure strategies whose interactions are described by an n×n payoff matrix. Players can use mixed strategies, which are given by the vector (p₁,…,p_n). Each entry specifies the probability to use the corresponding pure strategy. The sum over all entries is one. Therefore, a mixed strategy is a point in the simplex S_n. We study evolutionary dynamics in a well-mixed population of finite size. Individuals reproduce proportional to payoff. We consider the case of weak selection, which means the payoff from the game is only a small contribution to overall fitness. Reproduction can be subject to mutation; a mutant adopts a randomly chosen mixed strategy. We calculate the average abundance of every mixed strategy in the stationary distribution of the mutation-selection process. We find the crucial conditions that specify if a strategy is favored or opposed by selection. One condition holds for low mutation rate, another for high mutation rate. The result for any mutation rate is a linear combination of those two. As a specific example we study the Hawk-Dove game. We prove general statements about the relationship between games with pure and with mixed strategies.     

Is the Relationship between Body Size and Trophic Niche Position Time-Invariant in a Predatory Fish? First Stable Isotope Evidence

Characterizing relationships between individual body size and trophic niche position is essential for understanding how population and food-web dynamics are mediated by size-dependent trophic interactions. However, whether (and how) intraspecific size-trophic relationships (i.e., trophic ontogeny pattern at the population level) vary with time remains poorly understood. Using archival specimens of a freshwater predatory fish Gymnogobius isaza (Tanaka 1916) from Lake Biwa, Japan, we assembled a long-term (>40 years) time-series of the size-dependence of trophic niche position by examining nitrogen stable isotope ratios (δ ¹⁵N) of the fish specimens. The size-dependence of trophic niche position was defined as the slope of the relationship between δ ¹⁵N and log body size. Our analyses showed that the slope was significantly positive in about 60% of years and null in other years, changing through time. This is the first quantitative (i.e., stable isotope) evidence of long-term variability in the size-trophic relationship in a predatory fish. This finding had implications for the fish trophic dynamics, despite that about 60% of the yearly values were not statistically different from the long-term average. We proposed hypotheses for the underlying mechanism of the time-varying size-trophic relationship.     

Multiple coping strategies maintain stability of a small mammal population in a resource‐restricted environment

In semi‐arid environments, aperiodic rainfall pulses determine plant production and resource availability for higher trophic levels, creating strong bottom‐up regulation. The influence of climatic factors on population vital rates often shapes the dynamics of small mammal populations in such resource‐restricted environments. Using a 21‐year biannual capture–recapture dataset (1993 to 2014), we examined the impacts of climatic factors on the population dynamics of the brush mouse (Peromyscus boylii) in semi‐arid oak woodland of coastal‐central California. We applied Pradel''s temporal symmetry model to estimate capture probability (p), apparent survival (φ), recruitment (f), and realized population growth rate (λ) of the brush mouse and examined the effects of temperature, rainfall, and El Niño on these demographic parameters. The population was stable during the study period with a monthly realized population growth rate of 0.993 ± SE 0.032, but growth varied over time from 0.680 ± 0.054 to 1.450 ± 0.083. Monthly survival estimates averaged 0.789 ± 0.005 and monthly recruitment estimates averaged 0.175 ± 0.038. Survival probability and realized population growth rate were positively correlated with rainfall and negatively correlated with temperature. In contrast, recruitment was negatively correlated with rainfall and positively correlated with temperature. Brush mice maintained their population through multiple coping strategies, with high recruitment during warmer and drier periods and higher survival during cooler and wetter conditions. Although climatic change in coastal‐central California will likely favor recruitment over survival, varying strategies may serve as a mechanism by which brush mice maintain resilience in the face of climate change. Our results indicate that rainfall and temperature are both important drivers of brush mouse population dynamics and will play a significant role in predicting the future viability of brush mice under a changing climate.     

Effective Population Size,Genetic Variation,and Their Relevance for Conservation: The Bighorn Sheep in Tiburon Island and Comparisons with Managed Artiodactyls

The amount of genetic diversity in a finite biological population mostly depends on the interactions among evolutionary forces and the effective population size (N _e) as well as the time since population establishment. Because the N _e estimation helps to explore population demographic history, and allows one to predict the behavior of genetic diversity through time, N _e is a key parameter for the genetic management of small and isolated populations. Here, we explored an N _e-based approach using a bighorn sheep population on Tiburon Island, Mexico (TI) as a model. We estimated the current (N _crnt) and ancestral stable (N _stbl) inbreeding effective population sizes as well as summary statistics to assess genetic diversity and the demographic scenarios that could explain such diversity. Then, we evaluated the feasibility of using TI as a source population for reintroduction programs. We also included data from other bighorn sheep and artiodactyl populations in the analysis to compare their inbreeding effective size estimates. The TI population showed high levels of genetic diversity with respect to other managed populations. However, our analysis suggested that TI has been under a genetic bottleneck, indicating that using individuals from this population as the only source for reintroduction could lead to a severe genetic diversity reduction. Analyses of the published data did not show a strict correlation between H _E and N _crnt estimates. Moreover, we detected that ancient anthropogenic and climatic pressures affected all studied populations. We conclude that the estimation of N _crnt and N _stbl are informative genetic diversity estimators and should be used in addition to summary statistics for conservation and population management planning.     

Response of population size to changing vital rates in random environments

Population size and population growth rate respond to changes in vital rates like survival and fertility. In deterministic environments change in population growth rate alone determines change in population size. In random environments, population size at any time t is a random variable so that change in population size obeys a probability distribution. We analytically show that, in a density-independent population, the proportional change in population size with respect to a small proportional change in a vital rate has an asymptotic normal distribution. Its mean grows linearly at a rate equal to the elasticity of the long-term stochastic growth rate λ _S while the standard deviation scales as $\sqrt t$ . Consequently, a vital rate with a larger elasticity of λ _S may produce a larger mean change in population size compared to one with a smaller elasticity of λ _S. But a given percentage change in population size may be more likely when the vital rate with smaller elasticity is perturbed. Hence, the response of population size to perturbation of a vital rate depends not only on the elasticity of the population growth rate but also on the variance in change in population size. Our results provide a formula to calculate the probability that population size changes by a given percentage that works well even for short time periods.     

Persistence of Common Alleles in Two Related Populations or Species

Mathematical studies are conducted on three problems that arise in molecular population genetics. (1) The time required for a particular allele to become extinct in a population under the effects of mutation, selection, and random genetic drift is studied. In the absence of selection, the mean extinction time of an allele with an initial frequency close to 1 is of the order of the reciprocal of the mutation rate when 4Nv << 1, where N is the effective population size and v is the mutation rate per generation. Advantageous mutations reduce the extinction time considerably, whereas deleterious mutations increase it tremendously even if the effect on fitness is very slight. (2) Mathematical formulae are derived for the distribution and the moments of extinction time of a particular allele from one or both of two related populations or species under the assumption of no selection. When 4Nv << 1, the mean extinction time is about half that for a single population, if the two populations are descended from a common original stock. (3) The expected number as well as the proportion of common neutral alleles shared by two related species at the tth generation after their separation are studied. It is shown that if 4Nv is small, the two species are expected to share a high proportion of common alleles even 4N generations after separation. In addition to the above mathematical studies, the implications of our results for the common alleles at protein loci in related Drosophila species and for the degeneration of unused characters in cave animals are discussed.