Dynamics of age- and size-structured populations in fluctuating environments: Applications of stochastic matrix models to natural populations

Recent developments of the theory of stochastic matrix modeling have made it possible to estimate general properties of age- and size-structured populations in fluctuating environments. However, applications of the theory to natural populations are still few. The empirical studies which have used stochastic matrix models are reviewed here to examine whether predictions made by the theory can be generally found in wild populations. The organisms studied include terrestrial grasses and herbs, a seaweed, a fish, a reptile, a deer and some marine invertebrates. In all the studies, the stochastic population growth rate (ln λ _s ) was no greater than the deterministic population growth rate determined using average vital rates, suggesting that the model based only on average vital rates may overestimate growth rates of populations in fluctuating environments. Factors affecting ln λ _s include the magnitude of variation in vital rates, probability distribution of random environments, fluctuation in different types of vital rates, covariances between vital rates, and autocorrelation between successive environments. However, comprehensive rules were hardly found through the comparisons of the empirical studies. Based on shortcomings of previous studies, I address some important subjects which should be examined in future studies.     

A probabilistic interpretation of the United Nations’ 1995–2005 population projections

I develop probabilistic interpretations for the United Nations’ 10-year population forecasts by comparing 1995 projections for 212 countries to the population sizes reported for 2005. Errors in the estimation of the intrinsic rate of increase, presumably caused by erroneous assumptions about birth, death and/or immigration rates, appear to be more consequential than errors based on inaccurate estimation of the starting, or ‘jump-off’, population size. For only about 20% of the countries did the ‘actual’ 2005 population size fall between the United Nations’ low- and high-variant projections. I propose prediction intervals for country-specific population sizes 10 years in the future of the form [ N_i^￠ (t+10) / k ,  k ·N_i^￠ (t+10) ],[ N_i^{\prime} (t+10) / k , \, k \cdot N_i^{\prime} (t+10) ], where N _i ′(t + 10) is the medium-variant prediction for year t + 10 made in year t, and k is a number that varies with starting population size. Based on the 1995–2005 United Nations’ data, values of k giving 95% coverage range from 1.11 for countries with a population on the order of 10⁹, to 1.45 for countries with a population of 10⁵.     

A distance measure of spatial pattern in a biological population

Summary Suppose thatn individuals locate independently and randomly on a segment of line of finite length (habitat). Let the theoretical and observed ranges of the sites of the individuals on the segment be μ _n-1 andr _n-1, respectively. Then, the degree of dispersion of the individual sites is measured by the ratio, T _n =n _n-1/μ _n-1, as follows: A random spatial pattern forI _r−1 =1 An aggregated spatial pattern for 0≤I _r <1 A uniform spatial pattern for (n+1)/(n−1)≥I _r >1. Another method was derived. Let the probability that an observed range is less thanr _n−1 beI _p , under the hypothesis of a Beta distribution. Then indicates A random spatial pattern forI _p =1/2 An aggregated spatial pattern forI _p <1/2 A uniform spatial pattern forI _p >1/2. The first index can be used for comparing populations having the same number of individuals, whereas the second one can be used for comparing populations with different numbers of individuals.     

Some strange properties of the logistic equation defined withr andK: Inherent defects or artifacts?

Summary In some situations the logistic equation in the usual expression, dN/dt=r(1−N/K)N, exhibits properties that are biologically unrealistic. For example, whenr≦0 the population can no longer show any normal, negative response in per-capita growth rate to increasing density. Also, when the equation is employed in the Volterra's competition model, a familiar but incredible conclusion is derived which says that the outcome of competition is entirely independent of the reproductive potentialr of each species. It is shown that all such strange properties are mere artifacts arising peculiarly in thisr-K model from its misleading implicit supposition thatK could be independent ofr, and they can be readily removed by alternative use of a plainer, classical form of the model, dN/dt=(r−hN)N.     

Age-specific survivorship analysis ofHeliothis spp. Populations on cotton

Summary Population dynamics ofHeliothis virescens (F.) andHeliothis zea (Boddie) (Lepidoptera: Noctuidae) eggs and larvae were studied for two years in a small plot of cotton,Gossypium hirsutum (L.). Due to morphological and ecological similarities, the pooledHeliothis population was considered for most of the analyses. Two generations ofHeliothis eggs and larvae were completed during each year. Stage recruitment was estimated for the eggs and larval instars 2–6, and recruitment variances were estimated by a Monte Carlo method. A modified form of the Weibull distribution was developed and used as a model to characterize survivorship curves for each of the fourHeliothis generations. A Type I survivorship curve (mortality rate increasing with age) was inferred for both Generation 1 (early season) data sets, whereas a Type II survivorship curve (mortality rate constant and thus independent of age) was inferred for both Generation 2 (late season) data sets. The shapes of the survivorship curves for the individualH. virescens andH. zea populations were inferred to be the same as those for the pooled populations. Analysis of the contributions of various factors toHeliothis stage-specific mortality indicated that natural enemies (predators and parasites) and the availability of food for larvae were responsible for between-generation differences in survivorship patterns.     

Population growth trends in India: 1991 census

The decennial census counted the total population of India at 843.931 million as of the sunrise of March 1, 1991. The total is 160.6 million higher than that of a decade earlier in 1981. The actual census count exceeded by 45 million the official projections for 1991 based on the 1971 census. However, the official projections for the same year based on the 1981 census fell short by 7.6 million only. Most of the observed differences are explained by the slower decline in the fertility levels. The population growth ratepeaked during 1971–81, perhaps in 1972–73 (based on the Sample Registration Scheme data). The average annualexponential growth rate declined marginally to 2.11 per cent (4.5%) after having remained at a plateau for the previous two decades of 1961–71 and 1971–81. At this point in time, the fertility and mortality trends indicate that India will reach the replacement level fertility [Net Reproductive Rate of Unity] by the years 2010–2015. It can be said with a greater degree of certainty that the official target of reaching the replacement level fertility by the year 2000a.d. will not be reached. Based on the 1991 census results, it can be said that India will reach the billion mark by the turn of the century. The World Bank projects a population of 1,350 million by the year 2025a.d., and a stationary population of 1,862 million by the year 2150a.d., assuming that the replacement level fertility [Net Reproductive Rate = 1] in India is reached about the year 2015a.d.     

Dispersal of adult grasshoppers,Mecostethus magister, under the field condition

Summary Estimation of the number of adult grasshoppers,Mecostethus magister, was made by means of the mark-and-recapture method. The birth and death rates are possible to be estimated at the same time, but the immigration and the emigration rate are inevitably involved in these respectively. The immigration and emigration rates must be made clear to know the true birth and death rates. For this purpose the movement of the marked males in 1963 was analyzed. The grasshoppers dominantly moved in the directions of N, NW and W, and the difference in frequency among the movement directions was not so large. The distribution of the dispersal-distance relationship of each quadrate on each released day was fitted approximately to normal distribution. It could be concluded that almost all of the grasshoppers moved within the range of about 31–35m. The emigration rate from the quadrate (12×12m²) was about 0.73–0.77 and the difference in the rate among the released days was small. From these values the emigration rate from the station (84×60m²) was estimated as 0.21–0.23. Subtracting the emigration rate from the death-and-emigration rate, the true death rate was calculated. The death rate was very low until the number of males reached to the peak, then increased gradually. Supposing that immigration rate was equal to the emigration rate, the true birth rate was also estimated. But the presumption might not be pertinent, for the value of birth rates became negative.     

Relationship between surplus energy and body weight in fish populations

Summary This paper theoretically analyses the relationship between surplus energy, which is available for either somatic growth or reproduction, and body weight. From the data of metabolism and growth of the biwamasu,Oncorhynchus rhodurus, obtained by Miura et al., a Bernoulli's differential equation is induced to represent the relationship between body weight and the sum of surplus energy and active metabolic rate. Solving this equation gives the amount of surplus energy,f(W_x), as follows:f(W_x) = (αW _x ^1−γ +β^1−γ)^1/(1−γ)−W_x, in which α, β and γ are constants andW _x is body weight at agex. The function is applied to ten fish populations and consequently it is found to be useful for a wider age range and a wider variety of fishes than the conventional function.     

Analysis of changes of insect-plant ratio-improvement of key-factor analysis for evaluating bottom-up effects in insect population dynamics

A revised key-factor analysis was presented for analyzing the temporal changes in the ratio of insect absolute number to plant resource. Ten data sets for 5 insect species were then analyzed. In this key-factor analysis, the key factor is defined as the factor contributing highly to between-year variation inR _r , the log rate of the inter-year change of the insect-plant ratio. The yearly change of plant resource was handled as a separate factor, expressed byr _pl , log ratio of plant resource in yearn to plant resource in yearn+1. The following was revealed: 1) In 7 of the 10 data sets examined,r _pl influenced variations ofR _r ; in particular in 3 casesr _pl was the main key factor. 2) Generation-to-generation fluctuations of absolute insect densities showed density dependence in 4 cases, while those of insect-plant ratios, in 8 cases. 3) The Royama model or a linear model, explained well the relationship between log insect-plant ratio (X _r ) andR _r and the relationship betweenX _r and log yearly change rate of absolute insect density (R _abs ). However, in the 7 cases in whichr _pl was a critical factor for variations ofR _r , with, increase ofX _r ,R _r showed a steeper, decrease around the equilibrium point (the point for whichR _r is 0) thanR _abs . This occurred becauser _pl tended to be negatively correlated withX _r . Consequently, in two casesX _r fluctuated cyclicly or chaotically although without the changes in plant resource, fluctuations ofX _r would be damped oscillations approaching equilibrium.     

A stochastic computer model for simulating population growth

Summary A model is described for investigating the interactions of age-specific birth and death rates, age distribution and density-governing factors determining the growth form of single-species populations. It employs Monte Carlo techniques to simulate the births and deaths of individuals while density-governing factors are represented by simple algebraic equations relating survival and fecundity to population density. In all respects the model’s behavior agrees with the results of more conventional mathematical approaches, including the logistic model andLotka’s Law, which predicts a relationship betwen age-specific rates, rate of increase and age distribution. Situations involving exponential growth, three different age-independent density functions affecting survival, three affecting fecundity and their nine combinations were tested. The one function meeting the assumptions of the logistic model produced a logistic growth curve embodying the correct values orr _m andK. The others generated sigmoid curves to which arbitrary logistic curves could be fitted with varying success. Because of populational time lags, two of the functions affecting fecundity produced overshoots and damped oscillations during the initial approach to the steady state. The general behavior of age-dependent density functions is briefly explored and a complex example is described that produces population fluctuations by an egg cannibalism mechanism similar to that found in the flour beetleTribolium. The model is free of inherent time lags found in other discrete time models yet these may be easily introduced. Because it manipulates separate individuals, the model may be combined readily with the Monte Carlo simulation models of population genetics to study eco-genetic phenomena.     

Control of insecticide resistance in a field population of houseflies,Musca domestica, by releasing susceptible flies

Summary Susceptible houseflies,Musca domestica, were released at a waste disposal site to control insecticide resistance in a field housefly population. In the first experiment, a total of 163,000 pupae of the susceptible Takatsuki strain were released in October–November 1977. LD₅₀ values to fenitrothion and diazinon decreased to about one-sixth in April 1978, five months after the releases, of those before the releases. For the second experiment, a susceptible colony was derived by cross and backcross between a white-eyed substrain of the Takatsuki and a field colony. This susceptible colony consisted of whiteeyed flies with low activity and normal-eyed flies bearing no or one white eye gene. The results of large cage experiments suggested that the normal-eyed males of the susceptible colony had half the mating competitiveness of wild males. Approximately 31,000–46,000 susceptible pupae were used in each of five releases from October to November 1980. The population number of each sex, estimated by a mark-release-recapture method, increased from 12,000 in late September to 35,000–43,000 in middle November and then decreased to 5,000–8,000 in early December. The frequency of field-collected males bearing one white eye gene and those bearing one male determining factor, which were characteristics of the susceptible colony released, increased gradually during the period of releases. The susceptibility of the field population to fenitrothion and diazinon was examined five times in the period from September to December 1980. With time, the dosage-mortality regression gradually shifted towards that of the susceptible colony after starting the releases. LD₅₀ values to fenitrothion and diazinon decreased to about one-sixth and one-fifth, respectively, in June 1981, six months after the second series of susceptible fly releases, of those before the releases. Ratios of the wild flies to the released fiies were estimated to be between 4.7∶1 and 9.8∶1 in males and between 3.0∶1 and 3.9∶1 in females by taking the quality of the released colony and the population parameters of the field houseflies into consideration. Under several assumptions, the manner of resistant phenotype reduction was discussed, based on the dosage-mortality regressions and the ratios of released flies. These results showed that the releases of susceptible flies were successful in suppression of insecticide resistance in the field housefly population.     

Survivorship and fertility schedules of two Sumatran tortoise beetles,Aspidomorpha miliaris andA. sanctaecrucis (Coleoptera: Chrysomelidae) under laboratory conditions

Summary Two species of tortoise beetles,Aspidomorpha miliaris (AM) andA. sanctaecrucis (AS) feeding on a shrub-like morning glory,Ipomoea carnea, were reared under laboratory conditions to study their survivorship and fertility schedules. AM and AS required 34–39 days and 30–37, respectively, for the development of the immature stages. The mean longevity of the males was 88.4 days in AM and 63.8 in AS, and that of females was 87.9 days in AM and 83.3 in AS. The mean length of the pre-reproductive period (27.2 days in AM and 33.8 in AS) was much longer than that of the post-reproductive period (10.9 days in AM and 14.3 in AS). Females laid eggs at a nearly constant rate throughout their reproductive period. The reproductive valueV _x /V ₀ of the two species remained high for most of their adult life, as a result of prolonged survivorship and fertility periods. The total number of eggs produced per female was 442.9 (AM) and 80.1 (AS). The intrinsic rate of natural increaser was 0.070 (AM) and 0.044 (AS) per capita per day. The prolonged reproductive schedules, coupled with strong dispersal power, of these species no doubt have an adaptive value for living in highly disturbed tropical environments, where rainfall is ample but unpredictable and food resources are available throughout the year in a wide area, but distributed in widely flung patches. Contributions to the knowledge of population dynamics of tortoise beetles in Sumatra 3. Contribution No. 33 of Sumatra Nature Study (Entomology). Partly supported by Grants from Japan Society for Promotion of Science for JSPS-DGHE Scientific Cooperation (1980, 1982) and Grants-in-Aid for Overseas Scientific Survey from Ministry of Education, Science and Culture of Japan (Nos. 56041027 and 58041030).     

Postmetamorphic growth, survival, and age at first reproduction of the salamander,hynobius nebulosus tokyoensis Tago in relation to a consideration on the optimal timing of first reproduction

Summary The postmetamorphic growth and survival of the salamanderHynobius nebulosus tokyoentis Tago were surveyed in the study site located in Habu village of Hinodemachi, a suburb of Tokyo City, during 1975–1981. A laboratory experiment on the growth rate of juveniles was conducted in parallel with the field survey. The result indicated that this salamander grew at the rate of 8,mm in s.v.l. per year during the juvenile stage, but its growth rate decreased markedly as low as 1.8 mm for males and 1.1 mm for females, once it had attained sexual maturity. According to the “capture-recapture” procedure the annual survival rate after metamorphosis was found to be quite high; that is, approximately 0.7. By using the growth rate of juveniles and the difference between the sizes at metamorphosis and sexual maturity, the age at first reproduction was estimated to be 4 year for males and 5 year for females. From the data obtained in this study, the intrinsic rates of increase (r) were calculated for various values of age at first reproduction under different survival schedules, and the relationship between the age at first reproduction and fitness as measured byr was examined. The result indicated that an optimal age maximizing fitness always existed under respective survival schedules, and the observed age at first reproduction of this salamandei was found to coincide well with the predicted optimal age.     

Adult population parameters and life tables of an epilachnine beetle (Coleoptera: Coccinellidae) feeding on bitter cucumber in Sumatra

Summary The population dynamics of an epilachnine beetle, which is closely related toEpilachna sparsa Dieke (henceforth called “sp. C”) and feeds on bitter cucumberMomordica charantia, was studied by mark-recapture of adults and the construction of life tables. The study was repeated three times, i.e., March–May, July–September and October–December in 1982, in Padang, Sumatra, Indonesia. After the establishment of the host plants, adults of “sp. C” soon colonized, and each study period ended in the death of the plants due to defoliation by the larvae and adults. The estimated mean length of residence of adults ranged from 6–11 days, but this was probably much shorter than the actual longevity, because the adults were so active that they flew away, or dropped off the plants, when they were approached or slightly disturbed. Life tables indicated that egg mortality ranged from 17.8–53.9%, and a parasitic waspTetrastichus sp. B made up 41.1–64.2% of egg mortality. Two wasps,Tetrastichus sp. C andPediobius foveolatus killed 1.2–19.4% (7.6–100%)^* of 4th instars and only the latter species attacked the pupae, killing 24.6–59.1% (45.1–72.4%). Parasitism and starvation by overcrowding contributed most to the total mortality from egg to adult emergence, which ranged from 89.4–99.5%. “Sp. C” had a higher diversity and level of parasitism than the Japanese species,E. vigintioctopunctata. The high dispersal power of “sp. C”, coupled with the prolongedl _x−m_x schedules shown under laboratory conditions, was advantageous for exploiting the food plant which was available throughout the year, but was rather patchily distributed in space.     

Life tables for worker honeybees

Summary Life tables for worker honeybees covering all life span, and those for adults, were prepared for three seasonal cohorts,June bees, July bees andwintering bees. Survivorship curves forJune andJuly bees show a convex type being exceptional for insects, with relatively high mortality at egg and feeding larval stages and at later adult stage after most bees became potential foragers. Adult longevity greatly lengthens inWinteriing bees and survivorship curve drops approximately with the same rate. A remarkable similarity of survivorship curves for men and honeybees was demonstrated, apparently due to highly developed social care in both. Some comments were given on mortality factors. The importance of life tables for population researches was shown by applying our result to the population growth curve made byBodenheimer, based upon the data byNolan. At the asymptote of the uncorrected curve, the ratio of total population estimated by uncorrected curve to that by corrected curve reaches about 3∶2. Contribution No. 821 from the Zoological Institute, Faculty of Science, Hokkaido University, Sapporo, Japan. Contributions from JIBP-PT No. 45. This study was in part supportod by a grant in aid from the Ministry of Education for the special project research, “Studies on the dynamic status of biosphere.” Population and bioeconomic studies on the honeybee colonies. II. We express our sincere thanks to Dr. YosiakiIt?, National Institute of Agricultural Sciences, Tokyo, for his kind stimulation and advices to the present work.     

The species-area relation II. A second model for continuous sampling

Summary A second mathematical model describing the species-area relation was proposed for continuous expanding of sample area. This model is expressed asS=λ ln(1+x/E) whereS is the number of species occurring in an areax, and λ andE are the constants termedspecific diversity andelemental area respectively. As a result of testing the validity of the model for several sets of data, it was shown that the above equation would provide an adequate fit to a group of species belonging to a single synusia which exists in an open habitat. The ecological implications of parameters involved were discussed and the characteristic area presented previously (Kobayashi, 1974) was defined in terms ofE. The relation between results obtained by discrete sampling and continuous sampling was examined and the possibility of converting one to another was suggested. Contribution from the Laboratory of Applied Zoology, Yamagata University, No. 79.     

Time-specific life tables for the pea aphid,Acyrthosiphon pisum (Harris ), on alfalfa

Summary Time-specific life tables were constructed for three pea aphid,Acyrthosiphon pisum (Harris) (Homoptera: Aphididae), populations using a modification ofHughes' analytical procedure. All populations were studied on second-growth alfalfa (mid-June to mid-July) in south central Wisconsin; data for two populations were collected during 1980, and data for the third population were collected during 1982. The intrinsic rate of increase (r _m) estimated on a physiological time (day-degree) scale under field conditions but in the absence of natural enemies, provided a reliable estimate of potential population growth rate and was used in preference toHughes' approach of estimating potential population growth rates directly from stage structure data. Emigration by adult alatae and fungal disease were the major sources ofA. pisum mortality in each of the three populations studied. These factors were most important because of their impact on reducing birth rates within the local population. Parasitism was never greater than 9 percent. Mortality attributable to predation ranged from 0.0 to about 30.0%; however, even at the highest predator densitiesA. pisum populations increased exponentially.     

A mathematical model of the food-consumer system I. A case without food replenishment

Summary A simple mathematical model was proposed to describe the dynamics of a food-consumer system. The model was based on the Logistic Theory and consisted of Eqs. (4), (5) and (6). The model was divided into the following three cases for further analyss; i) without food supply except at the initial time, ii) with continuous food supply at a constant rate, and iii) with food supply at varying rates. Only the first model was dealth with in this paper. The assumptions of the model 1 are that a definite amount of food is given only once at the initial time and only the feeding by animals is responsible for the decrease of food, and that the rate of decrease is proportional to the amount of animals. It is also assumed that the growth of animal population is represented by the logistic curve, and that the upper limit of the population is proportional to the amount of food at that time. For simplicity the parameters of basic differential equations are assumed to be constant throughout the time course. Analytical solutions of this non-linear model were given by Eqs. (8), (9), (10) and (11). The properties of time course of the food amount and consumer population were discussed from the mathematical and biological points of view. The method of the estimation of the three constants λ,b, andc from the experimental data was also suggested. Since we had no available data for animal populations, we applied the model, regarding reserve substance asx and new plant body asy, to the data of the initial growth of Azuki bean plant in the dark. This model is very simple, but it may be useful for analyzing the behavior of food-consumer system. And it may give some clue to the analysis of the more complex systems.     

Loss of genetic variability in a fragmented continuously distributed population

An individual-based simulation model was used to examine the effect of population subdivision, dispersal distance of offspring, and migration rates between subpopulations on genetic variability(H ₁ H _S andH _T ) in a continuously distributed population. Some difficulties with mathematical models of a continuously distributed population have been pointed out. The individual-based model can avoid these difficulties and can be used to examine genetic variability in a population within which individuals are distributed continuously and in which the dispersal of individuals is disturbed by geographical or artificial barriers. The present simulation showed that the pattern of decrease inH ₁ had three stages. During the first stage,H ₁ decreased at the rates predicted by Wright’s neighborhood size. During the second stage,H ₁ decreased more rapidly when the migration rate decreased, while during the third stage, it decreased less rapidly when the migration rate decreased. Increasing the number of subdivisions increased the rate of decrease after the 200th generation. The pattern of decrease inH _T was classified into 2 stages. During the first stage, the rates of decrease corresponded with those of a randomly mating population. During the second stage, a decrease in the migration rates of the subpopulations slowed the rate of decrease inH _T . A uniform spatial distribution and a reduced total dispersal distance of offspring causedH ₁ H _S , andH _T to decrease more rapidly. Habitat fragmentation in a continuously distributed population usually was detrimental to the genetic variability in the early generations. Other implications of the results for conservation are discussed.