On the use of growth rate parameters for projecting population sizes: Application to aphids

This paper extends the application of the cumulative size based mechanistic model, which has previously been shown to describe diverse aphid population size data well. The mechanistic model is reviewed with a focus on the explanatory role of the birth and death rate formulation. An analysis of two data sets, one on the mustard aphid and the other on the pecan aphid, indicates that multiple linear regression equations based on the estimated birth and death rate parameters alone account for nearly all (R² > 0.95) of the variability in two key population attributes, namely the peak count and the cumulative density. This indicates that population size variables may be projected directly from the growth rate parameters using linear equations. Such linear relationships based on the birth and death rate parameters are shown to hold also for certain generalized mechanistic models for which the analytical solution is not available. The birth and death rate coefficients, therefore, constitute a new succinct set of variables that could be included in the predictive modeling of aphid populations, as well as other insect and animal populations with local collapse which follow similar growth dynamics.     

Population level effects of reduced fecundity in the fish species perch (Perca fluviatilis) and the implications for environmental monitoring

A 40% reduction in relative gonad size in perch (Perca fluviatilis) has been observed over that past two decades at the Swedish national reference site Kvädöfjärden. This biomarker response could be interpreted as a reduction in fecundity and increased risk of local extinction. However, abundance estimates from the same area has not provided any evidence of a reduction in population size. In the present study, a matrix population model was developed to investigate if a reduction in fecundity can be expected to have long term effects on population viability for perch and to evaluate the probability to detect such effects through abundance estimates. The model was parameterized from 17 years of population data from Kvädöfjärden as well as from other studies on perch. The model included density dependence and environmental stochasticity. The results indicated that a reduction in fecundity that is in level with the observed reduction in relative gonad size in Kvädöfjärden will cause a substantial risk for local extinction. The risk that the population will fall below 20% of the carrying capacity within 50 years is 44% when the fecundity is reduced by 40%. However, due to variability in abundance measurements it will take some time before a reduction in gonad size leads to statistically significant effects on the population. If the fecundity is reduced by 40% successively over a 10-year period, the probability to detect this through abundance estimates within 10 years is less than 50%. The results of the present study clearly show that relevant biomarkers have an important role in environmental monitoring as early warning signals, preferably in combination with measurements at higher levels of biological organization.     

Application of a spatial meta-population model with stochastic parameters to the management of the invasive grass Nassella trichotoma in North Canterbury, New Zealand

Optimising the management of invasive plants requires the identification of the population size outcomes for alternative management strategies. Mathematical models can be useful tools for making such management strategy comparisons. In this paper we develop a generic landscape meta-population model and apply it to the weedy grass, Nassella trichotoma, an invasive species occupying approximately 800 land parcels, predominantly pastoral farms, in the Hurunui district, North Canterbury, New Zealand. Empirical evidence reveals that this meta-population is currently stable (at a median density of 6 plants ha⁻¹) under a community strategy requiring manual removal (termed ‘grubbing’) of plants annually from all land parcels. Reduction in population size requires an alternative management strategy. Field data, collected over a 12 year period, were used to provide stochastic parameter values for land parcel size, carrying capacity, rates of local population growth and grubbing.The model reveals that at steady state, the most significant contribution to population growth on a land parcel comes from within the land parcel itself; the expected annual per capita growth on an individual land parcel is 4 orders of magnitude greater than the expected annual contribution from plants arising from other land parcels. This result implies that many of the farms currently supporting N. trichotoma may pose little or no threat to, nor are threatened themselves by, other farms infested by the weed. However, the steady state distribution (of the weed's population density) was sensitive to the spread rate, revealing a need for data on this process. It was also sensitive to how any increase in the grubbing rate is distributed; increasing it via a uniform distribution U(0, 1) where all rates between 0 and 100% year⁻¹ are equally probable did not affect the steady state, whereas increasing the rates via the uniform distribution U(0.25, 0.75) resulted in fewer farms with high population densities. In general the model provides a basis for exploring the effects of a wide range of alternative grubbing strategies on population growth in N. trichotoma.     

Ungulate population dynamics and optimization models

Optimal harvesting strategies for an ungulate population are estimated using stochastic dynamic programming. Data on the Llano Basin white-tailed deer (Odocoileus virginianus) population were used to construct a 2-variable population dynamics model. The model provided the basis for estimating optimal harvesting strategies as a feedback function of the current values of the state variables (prefawning older deer and juveniles). Optimal harvest strategies were insensitive to assumptions about the probability distributions of the stochastic variable (rainfall). The response of the population components to harvesting and the returns obtained from applying optimal strategies were explored through simulation. Mean annual harvest is about 15% of the population. Simplified harvesting strategies based on age-ratios as well as a simplified version based on optimal strategies—but assuming persisting equilibrium juvenile deer density—were compared to optimal strategies through examining values of information. Simplified harvesting strategies lead to a lower harvest over a 50-year simulation period.     

A Voronoi diagram based population model for social species of wildlife

A population model is presented that accounts for spatial structure within habitat patches. It is designed for social species of wildlife that form social group home ranges that are much smaller than patch size. The model represents social group home ranges by Voronoi regions that tessellate a patch to form a Voronoi diagram. Neighbouring social groups are linked with habitat-confined shortest paths and form a dispersal network. The model simulates population dynamics and makes use of Voronoi diagrams and dispersal networks as a spatial component. It then produces density maps as outputs. These are maps that show predicted animal densities across the patches of a landscape. A construction procedure for the particular Voronoi diagram type used by the model is described. As a test case, the model is run for the squirrel glider (Petaurus norfolcensis), a small arboreal marsupial native to Australia. A time series of density maps are produced that show squirrel glider density changing across a landscape through time.     

Recruitment and attrition of associated plants under a shading crop canopy: Model selection and calibration

Associated plant and animal diversity provides ecosystem services within crop production systems. The importance of the maintenance or restoration of diversity is therefore increasingly acknowledged. Here we study the population dynamics of associated annual plants (‘weeds’) during the growth of a crop in a season and introduce a minimal model to characterize the recruitment and attrition of the associated plants under the influence of shading by the crop. A mechanistically based, logistic, light interception model was parameterized with light interception measurements in two single crops (barley and rye) and in mixtures of these cereals with peas. Population dynamics data were collected for the annuals Papaver rhoeas, Centaurea cyanus, Chrysanthemum segetum, and Misopates orontium. A minimal population dynamics model was identified for each annual plant species, using system identification techniques as model selection and calibration.     

Modeling the population dynamics of cotton bollworm Helicoverpa armigera (Hübner) (Lepidoptera: Noctuidae) over a wide area in northern China

The cotton bollworm Helicoverpa armigera (Lepidoptera: Noctuidae) is one of the most serious crop pests in northern China, calling for accurate prediction of pest outbreaks and strategies for pest control. A computer model is developed to simulate the population dynamics of H. armigera over a wide area in northern China. The area considered covers 12 provinces where serious outbreaks of H. armigera have been observed. In this model, pest development is driven by local ambient temperature, and adults migrate long distances between regions and select preferred hosts for oviposition within a region. Six types of host including cotton, wheat, corn, peanut, soybean and a single category composed of all other minor hosts are considered in this model. Survival rates of eggs and larvae are based on life-table data, and simulated as a function of host type, host phenology and temperature. The incidence of diapause depends on temperature and photoperiod experienced during the larval stage. Survival rate of non-diapause pupae is a nonlinear function of rainfall, and overwinter survival rate is a nonlinear function of temperature. Insecticide is applied when population density exceeds the economic threshold on a host crop within a region. Comparisons of model output with light-trap data indicate that our model reflects the pest population dynamics over a wide area, and could potentially be used for testing novel pest control strategies in northern China.     

Integrating farming techniques in an ecological matrix model: Implementation on the primrose (Primula vulgaris)

Several studies have proven the importance of field margins in sustaining biodiversity and other work has been done on the effect of field management on field margin flora. However few models have been built to predict the effects of field management on the flora. Our project addresses this need for a model capable of predicting the effect of cropping techniques and their timing on the flora of field margins. Primula vulgaris is a biodiversity indicator, characteristic of undisturbed flora and found in field margins and woodlands: its population has been declining for several years. We created a temporal matrix model of P. vulgaris populations on field margins, taking into account the effects of field, field margin and roadside management based on literature and expert knowledge. We then analysed its sensitivity to demographic parameters by comparing lambda (growth rate) sensitivity and elasticity. We compared the management parameter effect using the relative growth rate of the population after 6 years of simulation. Sensitivity analysis to biological parameters showed the importance of adult survival and seed production and germination. Results show that P. vulgaris is particularly sensitive to broad-spectrum herbicides and that other management techniques like early mowing, scything and scrub-killer (diluted broad-spectrum herbicide or specific herbicide) are less aggressive. Our simulations show that management of cash crops in Brittany is too aggressive for P. vulgaris populations and that 4-5 years of grassland in the adjacent field are necessary to maintain populations.     

Refining logistic regression models for wildlife habitat suitability modeling—A case study with muntjak and goral in the Central Himalayas, India

High quality habitat suitability maps are indispensable for the management and planning of wildlife reserves. This is particularly important for megadiverse developing countries where shortages in skilled manpower and funding may preclude the use of mathematically complex modeling techniques and resource-intensive field surveys. In this study, we propose a simulation based k-fold partitioning and re-substitution approach to refine and update logistic regression models that are widely used for habitat suitability assessment and modeling. We test the modeling strategy using data from a rapid field survey conducted for habitat suitability assessment for muntjak (Muntiacus muntjak) and goral (Naemorrhaedus goral) in the central Himalayas, India. Results obtained from simulations match expectations in terms of model behavior and in terms of published habitat associations of the investigated species. Qualitative comparisons with predictions from the GARP, MaxEnt and Bioclimatic Envelopes modeling systems also show broad agreement with predictions obtained from the proposed technique. The proposed technique is suggested as a rapid-assessment precursor to detailed habitat studies such as patch occupancy modeling in situations where funds or trained manpower are not available.     

Data-directed modelling of Daphnia dynamics in a long-term micro-ecosystem experiment

The micro-ecosystem under consideration consists of three compartments forming a closed chain in which water circulates. Three trophic levels are represented in different compartments: autotrophs (algae, mainly Chlorella vulgaris), herbivores (Daphnia magna) and microbial decomposers. From a 20 years experiment with this system, data has been selected for this study. The dynamics of algae and Daphnia magna in only one of the compartments were modeled by different systems of differential and difference equations. We describe the successive steps in the process of model development, and the fitting of parameters using a Nelder-Mead simplex calibration method. Identification problems were overcome by taking values for physiological parameters in agreement with the literature. It turned out that a logistic type of model gives the best result for the structured Daphnia population because of the set up of the experiment: algae grow and reproduce in the upstream compartment. For this reason well-known plant–herbivore models did not comply with the data. The results of the parameter estimation procedure are discussed. The estimated grazing rate by Daphnia was smaller than expected. Possibly the Daphnia fed also on detritus and decomposing algae which were not measured.     

Population as an oscillating system

In this paper we present a new approach describing population dynamics based on the view of a population as an oscillating system. To develop a mathematical model of an oscillating population, we applied a third-order differential equation. Our model describes population dynamics within a parametric-temporal continuum, formed by the relative values of population growth and decrease over time. In this paper we also illustrate how our oscillative model effectively compliments the existing suite of models in population dynamics.     

Modulation of predator-prey interactions by the Allee effect

An Allee effect arising from density-dependent mating success can have significant impacts at the ecosystem level when considered in the context of predator-prey interactions. These are captured by a mathematical model for the exchange of biomass between a structured predator population (continuous weight distribution) and a resource. Because the predator’s mating success affects the amount of resources required for the production of offsprings and their future growth into mature organisms, it influences the flux of biomass between trophic levels. Under simple assumptions, the equations can be reduced to an equivalent unstructured predator-prey model in which the Allee effect modulates the predation rate: the mating probability multiplies the rate of predator growth as well as the rate of resource depletion. Implications of the Allee effect for the bifurcation structure and equilibrium densities are examined. The model is compared to a modified version in which the Allee effect instead modulates the assimilation efficiency, hence the mating probability does not appear in the dynamical equation for the resource density. Both models exhibit qualitatively similar dynamics. However, compared to the model in which the Allee effect modulates predation, the model in which the Allee effect modulates assimilation efficiency predicts (i) unrealistically inefficient resource assimilation when predator density is low, (ii) a higher risk of catastrophic extinction resulting from a change in the parameter controlling the strength of the Allee effect, and (iii) no possibility of an increase in population size when the density dependence is enhanced.     

Lyapunov stability for an age-structured population model

This is a short notice on the McKendrick equation that I actually learned from Yu.M. Svirezhev in the 1990s. This McKendrick equation modelling the evolution in time of an age-structured population has received attention recently from mathematicians. The initial and boundary conditions for the McKendrick equation imposed by the population model are not the standard side conditions one sees in PDE theory for an evolution equation. In the simplest case, the problem reduces to a well-known model in demography, the Lotka integral equation.     

A hydrologically driven model of swamp water mosquito population dynamics

We develop a swamp water mosquito population model that is forced solely by environmental variability. Measured temperature and land surface wetness conditions are used to simulate Anopheles walkeri population dynamics in a northern New Jersey habitat. Land surface wetness conditions, which represent oviposition habitat availability, are derived from simulations using a dynamic hydrology model. Using only these two density-independent effects, population model simulations of biting Anoph. walkeri correlate significantly with light trap collections. These results suggest that prediction of mosquito populations and the diseases they transmit could be better constrained by inclusion of environmental variability.     

A survival-analysis-based simulation model for Russian wheat aphid population dynamics

A simulation model for Russian wheat aphid (RWA), Diuraphis noxia (Mordvilko), populations is built by integrating survival-analysis-based development and survivor functions and the same-shape reproduction distribution model in the framework of Leslie [Leslie, P.H., 1945. On the use of matrices in certain population mathematics. Biometrika 33, 183–212] matrix structure. Survival analysis is utilized to model both the development and survival of RWA populations, and the Cox (1972) proportional hazards model is fitted with the data sets from our laboratory observation of 1800 RWA individuals under 25 factorial combinations of five temperature regimes and five barley plant-growth stages. Rather than using simple age-specific survivor rates as in the traditional Leslie matrix, the survivor functions based on survival analysis describe age-specific, temperature and plant stage-dependent RWA survival probabilities. Similarly, a probability model from survival analysis to estimate the probability that an individual will reach mature adult stage is utilized to describe the development process; this makes the transition from nymphal stage to mature adult stage dependent on RWA age as well as temperature and plant-growth stage.Inspired by the same-shape distribution and rate-summation approach for modeling insect development, a similar approach for modeling insect reproduction under variable temperature is developed. This new same-shape reproduction distribution model incorporates individual variation in reproduction capability, as well as the effects of RWA age, temperature and plant-growth stage. Consequently, the same-shape reproduction distribution model replaces the simple age-specific fecundities in Leslie matrix model. To the best of our knowledge, this work is the first to introduce survival analysis to simulation modeling in entomology and ecology and also the first to integrate our newly developed same-shape reproduction distribution model into application.     

Modeling the brown bear population in Slovenia: A tool in the conservation management of a threatened species

In this paper, we address three aspects of the brown bear population in Slovenia: its size (and its evolution over time), its spatial expansion out of the core area, and its potential habitat based on natural habitat suitability. Data collected through measurement/observation of the bear population and from the literature are used. A model is developed for each aspect. The results are estimates of population size, a picture of the spatial expansion of the population and maps of its optimal and maximal potential habitat (based on natural suitability). Overall, the brown bear population has been increasing since the establishment of a core protective area and has been expanding outside this area. The habitat suitability maps show that there is room for further expansion. Based on habitat suitability and bear population density, as well as human activity and current damage reports, we recommend that the Alps should be temporarily kept free of the bears, until the necessary mitigation measures regarding human–bear conflicts are carried out. On the other hand it is of crucial importance to adapt human activities and improve bear management in the optimal habitat, with which the goals of successful conservation of the species might be achieved.     

A finite element method to solve a population dynamics stage-structured model of intertidal barnacles

Many marine organisms are fixed or highly sedentary as adults but the adult population may be strongly dependent on the oceanic transport of planktonic larvae. In order to assess interactions between oceanographic and biological processes that determine the population dynamics of marine organisms with a sessile adult phase restricted to the coastline and a planktonic larval phase, we present a stage-structured finite element model for the barnacle Balanus glandula that inhabits the rocky intertidal zone of central California, USA.     

Development and validation of an individual based Daphnia magna population model: The influence of crowding on population dynamics

An individual-based model was developed to predict the population dynamics of Daphnia magna at laboratory conditions from individual life-history traits observed in experiments with different feeding conditions. Within the model, each daphnid passes its individual life cycle including feeding on algae, aging, growing, developing and – when maturity is reached – reproducing. The modelled life cycle is driven by the amount of ingested algae and the density of the Daphnia population. At low algae densities the population dynamics is mainly driven by food supply, when the densities of algae are high, the limiting factor is “crowding” (a density-dependent mechanism due to chemical substances released by the organisms or physical contact, but independent of food competition).     

Dissecting the drivers of population cycles: Interactions between parasites and mountain hare demography

There is a growing awareness that cyclic population dynamics in vertebrate species are driven by a complex set of interactions rather than a single causal factor. While theory suggests that direct host-parasite interactions may destabilise population dynamics, the interaction between host and parasite may also influence population dynamics through indirect effects that result in delayed responses to either density or to life-history traits. Using empirical data on mountain hares (Lepus timidus) infected with a nematode parasite (Trichostrongylus retortaeformis), we developed an individual-based model (IBM) that incorporated direct effects and delayed life-history effects (DLHEs) of a macroparasite, alternative transmission mechanisms and seasonality in host population dynamics. The full model describes mean characteristics of observed mountain hare time series and parasite abundance, but by systematically removing model structure we dissect out dynamic influences of DLHEs. The DLHEs were weakly destabilising, increasing the propensity for cyclic dynamics and suggesting DLHEs could be important processes in host-parasite systems. Further, by modifying model structure we identify a strong influence of parasite transmission mechanism on host population stability, and discuss the implications for parasite aggregation mechanisms, host movement and natural geographical variation in host population dynamics. The effect of T. retortaeformis on mountain hares likely forms part of a complex set of interactions that lead to population cycles.     

A note on density dependence in population models

One of the most studied phenomena in ecology is density dependent regulation. The model most frequently used to study this behaviour is the theta-logistic model. However, disagreement has developed within the ecology community pertaining to the interpretation of this model’s parameters, and thus as to appropriate values for the parameters to assume. In particular, the parameter θ$\theta$ has been allowed to take negative values, resulting in the ‘growth rate parameter’ estimated to be negative for species which are extant and exhibit no signs of becoming extinct in the short-term. Here we explain this phenomenon by formulating the theta-logistic model in the manner in which the original logistic model was formulated by Verhulst (1838), in doing so providing a simple interpretation of model parameters and thus restrictions on values the parameters may assume. We conclude that θ$\theta$ should (almost always) be restricted to values greater than −1$-1$. This has implications for studies assessing the form of density dependence from data. Additionally, another model appearing in the literature is presented which provides a more flexible model of density dependence at the expense of only one additional parameter.