Periodic Matrix Models Research Articles

TRANSIENT POPULATION DYNAMICS IN PERIODIC MATRIX MODELS: METHODOLOGY AND EFFECTS OF CYCLIC PERMUTATIONS

Many biological populations are subject to periodically changing environments such as years with or without fire, or rotation of crop types. The dynamics and management options for such populations are frequently investigated using periodic matrix models. However the analysis is usually limited to long-term results (asymptotic population growth rate and its sensitivity to perturbations of vital rates). In non-periodic matrix models it has been shown that long-term results may be misleading as populations are rarely in their stable structure. We therefore develop methods to analyze transient dynamics of periodic matrix models. In particular, we show how to calculate the effects of perturbations on population size within and at the end of environmental cycles. Using a model of a weed population subject to a crop rotation, we show that different cyclic permutations produce different patterns of sensitivity of population size and different population sizes. By examining how the starting environment interacts with the initial conditions, we explain how different patterns arise. Such understanding is critical to developing effective management and monitoring strategies for populations subject to periodically recurring environments.

Interactive Effects Of Prey Andp,p′-DDE On Burrowing Owl Population Dynamics

We used population models to explore the effects of the organochlorine contaminant p,p'-DDE and fluctuations in vole availability on the population dynamics of Burrowing Owls (Athene cunicularia). Previous work indicated an interaction between low biomass of voles in the diet and moderate levels of p,p'-DDE in Burrowing Owl eggs that led to reproductive impairment. We constructed periodic and stochastic matrix models that incorporated three vole population states observed in the field: average, peak, and crash years. We modeled varying frequencies of vole crash years and a range of impairment of owl demographic rates in vole crash years. Vole availability had a greater impact on owl population growth rate than did reproductive impairment if vole populations peaked and crashed frequently. However, this difference disappeared as the frequency of vole crash years declined to once per decade. Fecundity, the demographic rate most affected by p,p'-DDE, had less impact on population growth rate than adult or juvenile survival. A life table response experiment of time-invariant matrices for average, peak, and crash vole conditions showed that low population growth under vole crash conditions was due to low adult and juvenile survival rates, whereas the extremely high population growth under vole peak conditions was due to increased fecundity. Our results suggest that even simple models can provide useful insights into complex ecological interactions. This is particularly valuable when temporal or spatial scales preclude manipulative experimental work in the field or laboratory.

Comparative Population Dynamics of an Invading Species in its Native and Novel Ranges

Although the ecology of many exotic invaders has been intensively examined in the novel range, few studies have comparatively explored how population dynamics differ in native and novel parts of an invading plants’ range. The population dynamics of mile-a-minute weed, Polygonum perfoliatum L., was explored in both the native (Japan) and novel (northeastern USA) portions of its range and evaluated using periodic matrix models. Projected per capita population growth rate (λ) varied within and between native and novel range populations. Surprisingly, five of the six populations in the novel range were projected to fail to replace themselves (λ<1) while only two of the four native range populations were projected to decline, although these projections had wider confidence intervals than in the novel habitat. While changes in germination, survivorship, fecundity and seed banking would have equivalent effects on population growth in the invasive habitat, small increases in plant survivorship would greatly increase λ in native populations. The differences between native and novel population growth rates were driven by lower adult survival in the native range caused by annual flooding and higher fecundity. Simulation analyses indicated that a 50% reduction in plant survival would be required to control growing populations in the novel range. Further comparative studies of other invading species in both their native and novel ranges are needed to examine whether the high per capita population growth and strong regulatory effects of adult survival in the native habitat are generally predictive of invasive behavior in novel habitats.

Recruitment variability and stochastic population growth of the soft-shell clam, Mya arenaria

Variability in recruitment is a striking feature of the life histories of most marine invertebrates and many fishes. The distribution of recruitment is typically lognormal rather than normal. It has been conjectured that such variation plays a major role in population growth of these species, but no complete stochastic demographic analysis has ever been reported. Here, we developed a set of deterministic and stochastic models for the clam, Mya arenaria. We constructed a size-classified, periodic matrix model to describe seasonal population dynamics, based on a mark-recapture study in Barnstable Harbor, MA, USA. In the deterministic model, an equilibrium recruitment rate of ERR = 19.4 recruits per adult was required to maintain constant population size. Periodic elasticity analysis showed that, at the ERR, fertility and larval survival accounted for 99% of the population growth rate λ . In a stochastic model with lognormally distributed recruitment, the stochastic growth rate log ⁡ λ s increased with increases in either E ( log ⁡ R ) or SD ( log ⁡ R ) . The positive effect of SD ( log ⁡ R ) is due to the highly skewed nature of the lognormal distribution. Variation in population size increases dramatically with increases in SD ( log ⁡ R ) , which makes it difficult to predict future population size. If variability in recruitment is high, quasi-extinction is nearly certain even when log ⁡ λ s is well above zero. Stochastic elasticity analysis shows that the contribution of adult growth and survival to λ s increases dramatically when recruitment is variable. These results suggest that, for marine invertebrates with a lognormally distributed recruitment pattern, the amount of uncertainty in recruitment should be positively associated with adult life span.

STOCHASTIC FLOOD AND PRECIPITATION REGIMES AND THE POPULATION DYNAMICS OF A THREATENED FLOODPLAIN PLANT

Boltonia decurrens is an endangered plant restricted to the Illinois River Valley. Its complex life cycle has evolved in response to the dynamics of the historic flood regime, which has changed dramatically in the last century due to the construction of navigation dams and agricultural levees. To explore the effects of these changes, we developed deterministic and stochastic matrix population models of the demography of Boltonia. We used periodic matrix models to incorporate intra‐annual seasonal variation. We estimated parameters as a function of the timing of spring flood recession (early or late) and of growing season precipitation (high or low). Late floods and/or low precipitation reduce population growth (λ). Early floods and high precipitation lead to explosive population growth. Elasticity analysis shows that changes in floods and precipitation alter the life history pathways responsible for population growth, from annual to biennial and eventually clonal pathways. We constructed and analyzed a stochastic model in which flood timing and precipitation vary independently, and we computed the stochastic growth rate (log λs) and the variance growth rate (σ2) as functions of the frequency of late floods and low precipitation. Using historical data on floods and rainfall over the last 100 years, we found that log λs has declined as a result of hydrological changes accompanying the regulation of the river. Stochastic elasticity analysis showed that over that time the contribution of annual life history pathways to log λs has declined as the contributions of biennial and clonal pathways have increased. Over the same time period, σ2 has increased, in agreement with observations of large fluctuations in local B. decurrens populations. Undoubtedly, many plant and animal species evolved in concert with dynamic habitats and are now threatened by anthropogenic changes in those dynamics. The data and analyses used in this study can be applied to management and development strategies to preserve other dynamic systems.

Cropping system effects on giant foxtail (Setaria faberi) demography: II. Retrospective perturbation analysis

Cropping system characteristics affect weed management by altering key demographic rates of weeds, including recruitment, seedling survival, fecundity, and seed survival. To facilitate the design and improvement of cropping systems that limit weed population growth, analytical methods are needed to identify weed management “choke points” (weed life stages that vary in response to management and whose variation strongly affects weed population growth rate). The objectives of this study were to (1) determine whether wheat–red clover green manure can limit giant foxtail population growth rate (λ) in a wheat–corn–soybean crop sequence and (2) identify choke points in the giant foxtail life cycle with respect to the green manure treatment. Demographic data were used to construct a periodic matrix model of giant foxtail population growth in a wheat–corn–soybean crop sequence, with either a wheat sole crop (W) or a wheat–red clover intercrop (R) in the wheat phase. Identification of choke points was accomplished by adapting the life-table response experiment (LTRE) design for retrospective perturbation analysis of the periodic matrix model. The difference inλ(Δλ) between the two treatments was decomposed into contributions from each parameter in each rotation phase of the periodic model. Each LTRE contribution was equal to the product of the sensitivity ofλto changes in a given parameter by the treatment difference in that parameter. Those parameters making large contributions to Δλrepresented weed management choke points. Giant foxtail population growth rate in the simulation was more than twice as great in the W treatment (λ= 2.54) than in the R treatment (λ= 1.16). Retrospective perturbation analysis indicated that the proportion of seeds surviving predation in the wheat phase made the largest LTRE contribution (0.55) to Δλ, followed by seedling recruitment in the soybean (0.41) and corn (0.20) phases. By identifying weed management choke points within a given system, retrospective perturbation analysis can target research and management efforts for greater reductions in weed population growth.

Sensitivity analysis in periodic matrix models: A postscript to Caswell and Trevisan

Periodic matrix population models are a useful approach t@ modelling cyclic variations in demographic rates. Caswell and Trevisan [1] introduced the perturbation analysis (sensitivities and elasticities) @f the per-cycle population growth rate for such models. Although powerful, their method can be time-consuming when the dimension of the matrices is large @r when cycles are composed of many phases. We present a more efficient method, based @n a very simple matrix product. We compared the two methods for matrices of different sizes. We observed a reduction in calculation time on the order of 24% with the new method for a set @f 26 within-year Leslie matrices of size 287 × 287. The time saving may become particularly significant when sensitivities are used in Monte Carlo or bootstrap simulations.

WEED POPULATIONS AND CROP ROTATIONS: EXPLORING DYNAMICS OF A STRUCTURED PERIODIC SYSTEM

The periodic growing of a certain set of crops in a prescribed order, called a crop rotation, is considered to be an important tool for managing weed populations. Nevertheless, the effects of crop rotations on weed population dynamics are not well understood. Explanations for rotation effects on weed populations usually invoke the diversity of environments caused by different crops that a weed population encounters. Using a periodic matrix model, we show that the number of different crops is not the sole factor, and that the sequence of a given set of crops can play an important role. In the model the weed population is structured by seed depth in the soil, and plowing moves seeds between layers. For illustration of concepts, we use parameter values thought to be characteristic for Polygonum persicaria growing in carrots (crop A) and spring wheat (crop B) in the Netherlands. We systematically examine the population growth rates for P. persicaria and their sensitivity to changes for all rotations of 2–6 years based on crops A and B. We include eight scenarios that differ in the effects of plowing and seed survival over winter. Differences between rotations can be striking. For example the weed population growth rate in the baseline rotation AABB (assuming 100% winter survival) is nearly 25% lower than in rotation ABAB. The elasticity (a measure to quantify the effect of proportional changes in model parameters on population growth) to seedling survival is nearly 75% higher in the B years of rotation ABAB than in the B years of rotation AABB. Changing parameter values changes the relation between population dynamics and rotation organization, but not the conclusion that there are consequences for population dynamics and management due to choice of a rotation. While our example is an agronomic one, the question “Does sequence matter?” and the methods applied should be of interest to researchers and managers concerned with the periodic management of other ecosystems.

AN EXTENDED FLOWERING AND FRUITING SEASON HAS FEW DEMOGRAPHIC EFFECTS IN A MEDITERRANEAN PERENNIAL HERB

The Mediterranean perennial herb Lobularia maritima shows an exceptionally extended flowering and fruiting that lasts for 10 months, from early September to late June. We hypothesized that such an extended phenology may be a flexible mechanism that enhances population persistence in variable Mediterranean environments, as fecundity in one part of the season could compensate for reproductive failure in another part of the season. We predicted that (1) fecundities throughout the year would significantly contribute to mean population growth rate λ, (2) negative covariances among fecundities in different periods of the year would reduce variance in population growth rate, and (3) extended flowering and fruiting would enhance long-term stochastic population growth rates. We used a periodic matrix model to describe the demography of a L. maritima population over five years, in which each year was divided into six periods of two months each. Population growth rates varied from a low of 0.25 to a high of 4.81 ove...

Effects of fine‐scale disturbances on the demography and population dynamics of the clonal moss Hylocomium splendens

Summary The boreal forest is subject to disturbance but the effects on population dynamics are poorly understood. We investigated how a prominent species of the forest floor, the clonal moss Hylocomium splendens, responded to different types of fine‐scale disturbance. Canopy gap formation was simulated in H. splendens populations by removing 50% of the bryophyte layer (Removal treatment) and herbivory by clipping all of its growing points (Clipped treatment). Demographic parameters and population dynamics were compared with an unmanipulated reference population (Ref). Both types of disturbance caused a significant decrease in mean segment size (particularly following clipping) and an increase in branching rates. Branching rates in the Removal population were enhanced at all architectural opportunities for enhanced branching, but only the Clipped segments themselves were affected during the 2 years following disturbance. The population effects of disturbance were analysed by a periodic matrix model, with 3‐year as the period, followed by elasticity analysis. The Ref population was approximately stable (growth rate, λ = 1.012), with the survival of mature segments making an overwhelming contribution to its growth rate. The higher λ (1.740) in the Removal population was associated with higher elasticities of branching. Clipping reduced λ to 0.841 by blocking segment survival, although regeneration of new growing points on the remnant main shoot axis after 1 year, and its subsequent maturation in the next year, then made a major contribution. Life table response experiment analyses revealed that almost all demographic processes including branching and the survival of mature segments contributed to the increased λ in Removal treatment. The relatively small decrease in λ following clipping was mainly due to the absence of mature segment survival, being almost balanced by the pulse of regeneration of growing points the year after disturbance. We conclude that H. splendens populations show a good ability to recover from common types of disturbances in their environment. The general mechanisms behind the responses are discussed.

Demography of the invasive woody perennial Prosopis glandulosa (honey mesquite)

Summary 1 To assess the effect of annual environmental variability on the long‐term population dynamics of the invasive woody perennial Prosopis glandulosa var. torreyana (honey mesquite), we employed elasticity analyses of annual, mean and periodic matrix models. Growth, survival and reproduction were recorded for 1306 individuals in a 1‐ha plot over a 4‐year period. The volume of each individual was estimated, and transition matrices with nine size classes were constructed. Standard matrix analysis was performed, and the relative importance of individual life‐cycle components to changes in the finite rate of population increase (λ) was determined. 2 Periodic matrix analysis projected a 29% annual increase in population size (λ= 1.29), while annual projection λ‐values varied between 0.99 and 1.44. For both methods, elasticity of seedling recruitment was always very low in all 4 years, and the highest elasticities were generally associated with permanence in the same size class. 3 Periodic and annual projections predicted similar elasticity patterns, with the relative contribution to a change in λ of different demographic processes changing between ‘good’ and ‘bad’ years. However, annual matrices identified two bad years, with fecundity and growth elasticity decreasing relative to survival, whereas periodic analysis identified only one such year, and elasticity changes were seen only in fecundity and survival.

Dynamics of a sheep population in a Sahelian area (Ndiagne district in Senegal): a periodic matrix model

The demography of livestock populations in Sahelian areas is highly sensitive to periodic stresses generated by climatic fluctuations. A deterministic age-transition matrix model which explicitly accounts for within-year demographic variability was developed. The annual cycle was divided into a number of 24 two-week phases and the annual projection matrix was the product of the 24 projection matrices for each phase. This model was applied to an extensively managed sheep population in North Senegal. Results such as growth rates, age structures and population size projections were compared to field data. We showed that such models can be useful in exploring demographic patterns and thus be powerful descriptive tools for livestock dynamic analysis in periodic environmental variations.

Sensitivity Analysis of Periodic Matrix Models

Periodic matrix models are used to describe the effects of cyclic environmental variation, both seasonal and interannual, on population dynamics. If the environmental cycle is of length m, with matrices B(1), B(2),...., B(m) describing population growth during the m phases of the cycle, then population growth over the whole cycle is given by the product matrix A = B(m)B(m—1)...B(1). The sensitivity analysis of such models is complicated because the entries in A are complicated combinations of the entries in the matrices B(i), and thus do not correspond to easily interpreted life history parameters. In this paper we show how to calculate the sensitivity and elasticity of population growth rate to changes in the entries in the individual matrices B(i) making up a periodic matrix product. These calculations reveal seasonal patterns in sensitivity that are impossible to detect with sensitivity analysis based on the matrix A. We also show that the vital rates interact in important ways: the sensitivity to changes in a rate at one point in the cycle may depend strongly on changes in other rates at other points in the cycle.