Evolutionary perturbations of optimal life histories

Summary An optimal age-structured life history is perturbed by increasing the mortality factors specific to an agek. These can be density dependent (DD) or independent (DI), avoidable or unavoidable. The last two refer to whether their effect on any individual depends or not on how much energy it devotes to defence. Agespecific trade-offs between the allocation of energy to defence and fecundity exist: survival probabilities through each agex, P _x, are concave decreasing functions of the fecundity per unit size at that age,b _x. These are constraints for the optimal life history. The changes induced by perturbation are evaluated by equations that predict whether some extra energy is diverted towards survivorship at the expense of fecundity or vice versa. The model predicts that for DI environments the degree of avoidability of the mortality source perturbed, is a decisive factor for the strategy selected at agek, but not for any other age class. DD environments are more complex since all ages are simultaneously embedded in density effects. The perturbations not only act directly — as in the DI situation — but also indirectly through their effect on equilibrium density,N ^*. When any kind of mortality source becomes more intense at agek, N ^* always decreases and all ages react in consequence according to the effect of density on each age-specific trade-off. Either coincidental or opposing reactions can be expected from direct and indirect effects. The resultant strategy for any age would be a matter of magnitude comparisons. Some possible general patterns are discussed.     

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.     

Age and growth assessment of western North Atlantic spiny butterfly ray <Emphasis Type="Italic">Gymnura altavela</Emphasis> (L. 1758) using computed tomography of vertebral centra

Life history strategies of batoid fishes have evolved within dynamic marine ecosystems. Adaptations in reproductive and developmental biology are paramount to the survival of species, and therefore knowledge of growth rates to maturity is fundamental for identifying constraints on the conservation of populations. The butterfly rays (Myliobatiformes: Gymnuridae) are highly derived batoids with generally low reproductive potentials for which age and growth information remains unknown. In this study we applied high-resolution X-ray computed tomography (HRXCT) to vertebral centra from a stingray for the first time to estimate age, and used a multimodel approach to investigate growth of spiny butterfly ray, Gymnura altavela. Estimated ages of the oldest male and female were 11 and 18 yrs. at disk widths (WD) 1355 mm and 2150 mm, respectively. Disk width-at-age data were analyzed using three growth models (von Bertalanffy, logistic, Gompertz), and the most parsimonious and empirically supported model was the logistic function with sex treated as a fixed effect on asymptotic disk width (WD _∞ ) and k parameters. Model parameter estimates were (males) WD _∞  = 1285.46 ± 67.27 mm, k = 0.60 ± 0.10, and (females) WD _∞  = 2173.51 ± 129.78 mm, k = 0.27 ± 0.04. Results indicated sexually dimorphic growth patterns, with males growing faster and reaching asymptotic size at earlier ages than females. These age and growth results are the first reported for the genus, and suggest that G. altavela grows at a similar rate as some teleosts and batoids, and relatively fast among chondrichthyans.     

Age‐related variation and temporal patterns in the survival of a long‐lived scavenger

Although senescence has been described for various fitness components in a wide range of animal species, few studies have studied senescence in long‐lived species, and little is known about its interactions with varying environmental conditions. Using a 32 year capture–mark–recapture dataset on the griffon vulture Gyps fulvus, we examined the demographic patterns of actuarial senescence and the patterns of year‐to‐year variation in survival rates. We found a significant, surprisingly late, decrease of annual survival probabilities from the age of 28 years onward and divided individual lifetimes into to three categories (juvenile, mid‐age and senescent birds). In agreement with the environmental canalization hypothesis, our analyses uncovered 1) higher temporal variation of annual survival probabilities in both the juvenile and senescent age classes compared to the mid‐age class and 2) low sensitivity of the population growth rate to the survival of both the juvenile and senescent age classes. Our results further suggested that the temporal variation in the survival of senescent birds might be related to intra‐annual changes in air temperature amplitudes. Finally, using population dynamics modeling, we revealed contrasting effects of the inclusion of the senescent age class on predicted population growth, depending on how survival rates were modeled. Altogether, our results demonstrate the existence of a class of senescent birds that exhibit distinct demographic properties compared to juvenile and mid‐age classes.     

Frequency responses of age-structured populations: Pacific salmon as an example

Increasing evidence of the effects of changing climate on physical ocean conditions and long-term changes in fish populations adds to the need to understand the effects of stochastic forcing on marine populations. Cohort resonance is of particular interest because it involves selective sensitivity to specific time scales of environmental variability, including that of mean age of reproduction, and, more importantly, very low frequencies (i.e., trends). We present an age-structured model for two Pacific salmon species with environmental variability in survival rate and in individual growth rate, hence spawning age distribution. We use computed frequency response curves and analysis of the linearized dynamics to obtain two main results. First, the frequency response of the population is affected by the life history stage at which variability affects the population; varying growth rate tends to excite periodic resonance in age structure, while varying survival tends to excite low frequency fluctuation with more effect on total population size. Second, decreasing adult survival strengthens the cohort resonance effect at all frequencies, a finding that addresses the question of how fishing and climate change will interact.     

Static and dynamic expression of life history traits in the northern fulmar Fulmarus glacialis

Understanding the static and dynamic expression of life history traits is a prerequisite for the development of a causal theory of the evolution of aging and of life histories. We analyzed the statics and dynamics of reproduction and survival in a wild population of the northern fulmar, Fulmarus glacialis (Procellaridae). Survival rate is most influenced by year as compared to age and cohort. When temporal variation is ignored, survival rate increases slowly with age and then declines more rapidly at late ages. Survival rate contingent upon reproductive ‘stratum’ (producing an egg, hatching an egg, fledging a hatchling) also exhibits this pattern. Survival and reproduction have a positive static association in that survival rate increases as the apparent energy allocated to reproduction increases (as indexed by stratum). There is a broad distribution of realized lifetime reproductive success, which could be due to ‘fixed’ heterogeneity, with some individuals always having low survival and reproduction and others always having high survival and reproduction, or be due to ‘dynamic’ heterogeneity, with all individuals having the same expected reproductive and survival rates. Analysis of stochastic stratum dynamics indicates that individuals do not remain long in any given stratum and suggest that the variation among individuals with respect to lifetime reproductive success is due to dynamic heterogeneity. The probability of producing an egg increases with age for both sexes, whereas the probability of producing a fledgling initially declines with age and then increases. These results underscore the necessity of understanding the static and dynamic expression of demographic traits when making a causal claim about their evolution.     

Competing Risks and Time‐Dependent Covariates

Time‐dependent covariates are frequently encountered in regression analysis for event history data and competing risks. They are often essential predictors, which cannot be substituted by time‐fixed covariates. This study briefly recalls the different types of time‐dependent covariates, as classified by Kalbfleisch and Prentice [The Statistical Analysis of Failure Time Data, Wiley, New York, 2002] with the intent of clarifying their role and emphasizing the limitations in standard survival models and in the competing risks setting. If random (internal) time‐dependent covariates are to be included in the modeling process, then it is still possible to estimate cause‐specific hazards but prediction of the cumulative incidences and survival probabilities based on these is no longer feasible. This article aims at providing some possible strategies for dealing with these prediction problems. In a multi‐state framework, a first approach uses internal covariates to define additional (intermediate) transient states in the competing risks model. Another approach is to apply the landmark analysis as described by van Houwelingen [Scandinavian Journal of Statistics 2007, 34 , 70–85] in order to study cumulative incidences at different subintervals of the entire study period. The final strategy is to extend the competing risks model by considering all the possible combinations between internal covariate levels and cause‐specific events as final states. In all of those proposals, it is possible to estimate the changes/differences of the cumulative risks associated with simple internal covariates. An illustrative example based on bone marrow transplant data is presented in order to compare the different methods.     

Matrix models and sensitivity analysis of populations classified by age and stage: a vec-permutation matrix approach

Matrix population models in which individuals are classified by both age and stage can be constructed using the vec-permutation matrix. The resulting age-stage models can be used to derive the age-specific consequences of a stage-specific life history or to describe populations in which the vital rates respond to both age and stage. I derive a general formula for the sensitivity of any output (scalar, vector, or matrix-valued) of the model, to any vector of parameters, using matrix calculus. The matrices describing age-stage dynamics are almost always reducible; I present results giving conditions under which population growth is ergodic from any initial condition. As an example, I analyze a published stage-specific model of Scotch broom (Cytisus scoparius), an invasive perennial shrub. Sensitivity analysis of the population growth rate finds that the selection gradients on adult survival do not always decrease with age but may increase over a range of ages. This may have implications for the evolution of senescence in stage-classified populations. I also derive and analyze the joint distribution of age and stage at death and present a sensitivity analysis of this distribution and of the marginal distribution of age at death.     

Effects of spatial scale and foraging efficiency on the predictions made by spatially-explicit models of fish growth rate potential

Synopsis Spatially-explicit modeling of fish growth rate potential is a relatively new approach that uses physical and biological properties of aquatic habitats to map spatial patterns of fish growth rate potential. Recent applications of spatially-explicit models have used an arbitrary spatial scale and have assumed a fixed foraging efficiency. We evaluated the effects of spatial scale, predator foraging efficiency (combined probabilities of prey recognition, attack, capture, and ingestion), and predator spatial distribution on estimates of mean growth rate potential of chinook salmon,Oncorhynchus tshawytscha. We used actual data on prey densities and water temperatures taken from Lake Ontario during the summer, as well as, simulated data assuming binomial distribution of prey. Results show that a predator can compensate for low foraging efficiency by inhabiting the most profitable environments (regions of high growth rate potential). Differences exist in predictions of growth rate potential across spatial scales of observation and a single scale may not be adequate for interpreting model results across seasons. Continued refinements of this modeling approach must focus on the assumptions of stationary distributions of predator and prey populations and predator foraging tactics.     

Living on the Edge: Viability of Moose in Northeastern Minnesota

ABSTRACT North temperate species on the southern edge of their distribution are especially at risk to climate-induced changes. One such species is the moose (Alces alces), whose continental United States distribution is restricted to northern states or northern portions of the Rocky Mountain cordillera. We used a series of matrix models to evaluate the demographic implications of estimated survival and reproduction schedules for a moose population in northeastern Minnesota, USA, between 2002 and 2008. We used data from a telemetry study to calculate adult survival rates and estimated calf survival and fertility of adult females by using results of helicopter surveys. Estimated age- and year-specific survival rates showed a sinusoidal temporal pattern during our study and were lower for younger and old-aged animals. Estimates of annual adult survival (when assumed to be constant for ages >1.7 yr old) ranged from 0.74 to 0.85. Annual calf survival averaged 0.40, and the annual ratio of calves born to radiocollared females averaged 0.78. Point estimates for the finite rate of increase (λ) from yearly matrices ranged from 0.67 to 0.98 during our 6-year study, indicative of a long-term declining population. Assuming each matrix to be equally likely to occur in the future, we estimated a long-term stochastic growth rate of 0.85. Even if heat stress is not responsible for current levels of survival, continuation of this growth rate will ultimately result in a northward shift of the southern edge of moose distribution. Population growth rate, and its uncertainty, was most sensitive to changes in estimated adult survival rates. The relative importance of adult survival to population viability has important implications for harvest of large herbivores and the collection of information on wildlife fertility.     

Non‐additive effects of density‐dependent and density‐independent factors on brown trout vital rates

Interactions between density‐dependent and density‐independent processes can lead to variation in both growth and survival rates. Detecting such effects, however, will often require sampling on an individual level and at the appropriate spatial and temporal scale. This study documents substantial variation in survival and growth of stream‐dwelling brown trout Salmo trutta from a small Norwegian stream. The data is based on seasonal capture–recaptures of individually marked trout on fixed stations during eight years. The fish were small‐sized, rarely reaching sizes larger than 20 cm and ages older than seven years. Density varied between 0.2–0.8 fish m^?2. Variation in survival and recapture probabilities was analysed using program MARK. Apparent survival (the probability of being alive and present within the study area) generally decreased with increasing trout density and increasing drought level (measured as lowest observed water flow) during both winter and summer. Further, there was a significant interaction effect between density and water flow, indicating that density‐dependent effects on survival predominated when environmental conditions were benign (no drought), while density‐independent processes were most important under harsh environmental conditions (drought). Observed length‐at‐age during autumn indicated a more or less linear growth trajectory throughout life, and no effect of density, water flow or temperature was found. However, using the individual‐based capture–recapture data to estimated specific growth rate, significant positive effects of water flow and temperature and a negative effect of density were identified. Thus, the capture–recapture data suggest a strong potential for population regulation at the rather low densities found in this stream, and regulation may occur both through effects on survival and growth.     

Relationship between growth rates and xylem hydraulic characteristics in young, mature and old-growth ponderosa pine trees

The first objective of the present study was to quantify the effects of tree age and stem position on specific conductivity (k_s), vulnerability to embolism and water storage capacity (capacitance) in trunks of young, mature and old‐growth ponderosa pine. The second objective was to determine relationships between hydraulic characteristics and radial and height growth rates to increase the understanding of possible tradeoffs. Within sapwood at all heights and in all ages of trees, outer sapwood had 25–60% higher k_s than inner sapwood. The water potential at which embolism started (air entry point) was 1.3 MPa lower in inner sapwood than outer sapwood within the mature trees, but there was no difference in the other trees. There was no significant difference in capacitances between the tops of the old growth trees, the mature trees and the young trees. Taking all data together, the capacitances increased sharply with an increase in k_s and an increase in vulnerability to embolism. The hydraulic characteristics of the three age classes were correlated with the height growth rate but not with the diameter growth rate. Within these age classes, high k_s was associated with the slowest yearly increase in sapwood area and with a low percentage of latewood, whereas high vulnerability to embolism and high capacitance were more closely associated with high height growth rates.     

Calorie restriction in the rotifer Brachionus plicatilis enhances hypoxia tolerance in association with the increased mRNA levels of glycolytic enzymes

The rotifer Brachionus plicatilis shows a typical sigmoid growth curve, where calorie restriction (CR) and hypoxia are thought to be introduced at high population density in the stationary phase. CR may induce a shift from aerobic to anaerobic metabolism in this stationary phase, possibly contributing to an increased hypoxia tolerance. This study was undertaken to investigate the effect of CR on hypoxia tolerance at the molecular level. When rotifers were cultured under CR (fed every second day) or fed ad libitum (AL), and subsequently exposed to hypoxia, those in the CR group had a higher survival rate than their AL counterparts. We then cloned cDNAs encoding three glycolytic enzymes, glyceraldehyde 3-phosphate dehydrogenase (GAPDH), enolase (ENO), and phosphoglycerate mutase (PGM) and compared their accumulated mRNA levels between CR and AL rotifers at ages of 1–8 days by quantitative real-time PCR. The CR group showed significantly higher mRNA levels of GAPDH and ENO than their AL counterparts. Furthermore, rotifers in the stationary phase showed higher mRNA levels of these enzymes than those in the exponential growth phase. These results suggest that CR induces anaerobic metabolism, which possibly contributes to population stability under hypoxia in the stationary phase.     

Life‐history traits of flatfish in the Northeast Atlantic and Mediterranean Sea

Flatfishes are poorly represented in published literature regarding life‐history strategies, as opposed to some other taxonomic groups of teleost fish. The present work constitutes an integrated approach to life‐history traits of the Order Pleuronectiformes occurring in the Northeast Atlantic and Mediterranean Sea. Data was exhaustively collected for several species of the families Scophthalmidae, Pleuronectidae, Bothidae and Soleidae, namely life‐history parameters from the von Bertalanffy growth model (L_∞ and k), absolute fecundity and size at maturity (L_m), as well as ecological and environmental data, with regard to the species and their area of occurrence. An inter‐specific ordination analysis based on life‐history parameters and environmental factors revealed distinct patterns of life‐history strategies, not necessarily grouped by phylogenetic affinities. Species with a more northern range of distribution were distinguishable from those with southern affinities by showing larger sizes, lower growth rates, earlier spawning, shorter spawning periods and higher fecundity. Possible environmental causes for these traits are discussed. Intra‐specific analyses were also performed and were generally in agreement with other authors, yet some disagreement was found for the inter‐specific analysis.     

Postnatal growth of skull linear measurements of Cape Hare Lepus capensis in northern China: an analysis in an adaptive context

304 skulls of Cape hare (Lepus capensis) were collected from two climatically distinct localities in northern China. With eye lens weight as a continuous age variable, postnatal growth patterns of 25 cranial linear measurements in relation to sex, growth season and region were analysed to understand the morphological basis of life history adaptation. In almost all the skull measurements, no significant differences were found between either sex or growth seasons. Principal component analysis revealed that facial elements accounted for the greatest proportion of skull morphological variation. Von Bertalanffy function was applied to describe growth trajectories of the skull elements. Based on this model, the growth rates of skull elements and the age at which they reached a certain proportion (95%) of asymptotes were compared. The results showed that skull growth exhibited an allometric pattern, with neural components attaining their final size more rapidly (at about 2–3 months old in tympanic bulla and 4–6 months old in others) than did the facial, which continued to grow well into postnatal life (at 6–10 months old). The earlier establishment of neurocranial morphology was associated with a fully developed central nervous system, which may play a key role in improving the survival of animals during the early phase of life. There was a regional difference in developmental rate of the hare skull. For all the skull parameters, northern hares had a more rapid rate of cranial growth compared to the southern, i.e. skull elements of juveniles from northern population were relatively larger at comparable ages and achieved adult size 0.5–4.0 months earlier than those from the south. In adult hares, however, no significant regional differences in any of the skull parameters were present. Adaptive explanations for the regional difference in ontogenetic pattern of skull morphology include age‐specific thermoregulation constraint, season‐related food availability and age‐dependent predation pressure. Based on the findings of this study, it is suggested that the postnatal growing period represents a crucial time of life, and that improvement of survivorship when young by growth adaptation forms an important aspect of the hare's life history strategies. © 2003 The Linnean Society of London, Biological Journal of the Linnean Society, 2003, 78 , 343–353.     

Partial life cycle analysis: a model for pre-breeding census data

Matrix population models have become popular tools in research areas as diverse as population dynamics, life history theory, wildlife management, and conservation biology. Two classes of matrix models are commonly used for demographic analysis of age‐structured populations: age‐structured (Leslie) matrix models, which require age‐specific demographic data, and partial life cycle models, which can be parameterized with partial demographic data. Partial life cycle models are easier to parameterize because data needed to estimate parameters for these models are collected much more easily than those needed to estimate age‐specific demographic parameters. Partial life cycle models also allow evaluation of the sensitivity of population growth rate to changes in ages at first and last reproduction, which cannot be done with age‐structured models. Timing of censuses relative to the birth‐pulse is an important consideration in discrete‐time population models but most existing partial life cycle models do not address this issue, nor do they allow fractional values of variables such as ages at first and last reproduction. Here, we fully develop a partial life cycle model appropriate for situations in which demographic data are collected immediately before the birth‐pulse (pre‐breeding census). Our pre‐breeding census partial life cycle model can be fully parameterized with five variables (age at maturity, age at last reproduction, juvenile survival rate, adult survival rate, and fertility), and it has some important applications even when age‐specific demographic data are available (e.g., perturbation analysis involving ages at first and last reproduction). We have extended the model to allow non‐integer values of ages at first and last reproduction, derived formulae for sensitivity analyses, and presented methods for estimating parameters for our pre‐breeding census partial life cycle model. We applied the age‐structured Leslie matrix model and our pre‐breeding census partial life cycle model to demographic data for several species of mammals. Our results suggest that dynamical properties of the age‐structured model are generally retained in our partial life cycle model, and that our pre‐breeding census partial life cycle model is an excellent proxy for the age‐structured Leslie matrix model.     

Investigating life‐history traits of Steller sea lions with multistate hidden Markov mark–recapture models: Age at weaning and body size effects

The duration of offspring care is critical to female fitness and population resilience by allowing flexibility in life‐history strategies in a variable environment. Yet, for many mammals capable of extended periods of maternal care, estimates of the duration of offspring dependency are not available and the relative importance of flexibility of this trait on fitness and population viability has rarely been examined. We used data from 4,447 Steller sea lions Eumetopias jubatus from the Gulf of Alaska and multistate hidden Markov mark–recapture models to estimate age‐specific weaning probabilities. Maternal care beyond age 1 was common: Weaning was later for animals from Southeast Alaska (SEAK) and Prince William Sound (PWS, weaning probabilities: 0.536–0.648/0.784–0.873 by age 1/2) compared with animals born to the west (0.714–0.855/0.798–0.938). SEAK/PWS animals were also smaller than those born farther west, suggesting a possible link. Females weaned slightly earlier (+0.080 at age 1 and 2) compared with males in SEAK only. Poor survival for weaned versus unweaned yearlings occurred in southern SEAK (female survival probabilities: 0.609 vs. 0.792) and the central Gulf (0.667 vs. 0.901), suggesting poor conditions for juveniles in these areas. First‐year survival increased with neonatal body mass (NBM) linearly in the Gulf and nonlinearly in SEAK. The probability of weaning at age 1 increased linearly with NBM for SEAK animals only. Rookeries where juveniles weaned at earlier ages had lower adult female survival, but age at weaning was unrelated to population trends. Our results suggest the time to weaning may be optimized for different habitats based on long‐term average conditions (e.g., prey dynamics), that may also shape body size, with limited short‐term plasticity. An apparent trade‐off of adult survival in favor of juvenile survival and large offspring size in the endangered Gulf of Alaska population requires further study.     

Variation in northern bobwhite demography along two temporal scales

Quantification and understanding of demographic variation across intra- and inter-annual temporal scales can benefit from the development of theoretical models of evolution and applied conservation of species. We used long-term survey data for northern bobwhites (Colinus virginianus) collected at the northern and southern extent of its geographic range to develop matrix population models which would allow investigation of intra- and inter-annual patterns in bobwhite population dynamics. We first evaluated intra-annual patterns in the importance of a seasonal demographic rate to asymptotic population growth rate with prospective perturbation analysis (elasticity analysis). We then conducted retrospective analysis (life table response experiments) of inter-annual patterns in the contribution of observed changes in demography to the observed change in population growth rate. Survival in the earliest age class during the nonbreeding season had the greatest potential influence in both the northern and southern populations. Examination of inter-annual variation in demography indicated that variation in nonbreeding season survival in the earliest age class contributed the most to observed changes in population growth rate in the northern population. In contrast, changes in fertility in the earliest age class in the southern population had the greatest influence on changes in population growth rate. Prospective elasticity analyses highlight the similarities in bobwhite demography throughout different parts of its geographic range, while retrospective life table response experiments revealed important patterns in the temporal differences of bobwhite life history at the northern and southern extent of its geographic range.     

Elasticity Analysis of Green Sturgeon Life History

I provide an analysis of a simplified life history model for green sturgeon, Acipenser medirostris, based on published and recent estimates of reproduction and growth rates and survival rates from life history theory. The deterministic life cycle models serve as a tool for qualitative analysis of the impacts of perturbations on green sturgeon, including harvest regulations based on minimum and maximum size limits (“slot limits”). Elasticity analysis of models with two alternative age–length relationships give similar results, with a high sensitivity of population growth rate to changes in the survival rate of subadult and adult fish. A dramatic increase in the survival of young of the year sturgeon or annual egg production is required to compensate for relatively low levels of fishing mortality. Peak reproductive values occur from ages 25 to 40. An increase or decrease in the maximum and minimum size limits can have a profound effect on the elasticity of population growth to changes in the annual survival rate of age classes specified by the slot, due to changes in the number of age classes of subadults and adults that are available for harvest. This analysis provides managers with a simple tool to assess the relative impacts of alternative harvest regulations. In general, green sturgeon follow life history patterns similar to other sturgeon, but species-specific demographic information is needed to produce more complex assessment and viability analysis models.     

Dynamics of an age-structured population with Allee effects and harvesting

We propose a discrete-time, age-structured population model to study the impact of Allee effects and harvesting. It is assumed that survival probabilities from one age class to the next are constants and fertility rate is a function of weighted total population size. Global extinction is certain if the maximal growth rate of the population is less than one. The model can have multiple attractors and the asymptotic dynamics of the population depend on its initial distribution if the maximal growth rate is larger than one. An Allee threshold depending on the components of the unstable interior equilibrium is derived when only the last age class can reproduce. The population becomes extinct if its initial population distribution is below the threshold. Harvesting on any particular age class can decrease the magnitude of the possible stable interior equilibrium and increase the magnitude of the unstable interior equilibrium simultaneously.