Reproduction Threshold Research Articles

Mathematical study of in-host dynamics of Chlamydia trachomatis

The paper presents a deterministic model for the dynamics of Chlamydia trachomatis inside the body of an infected human host. The model is shown to have a globally asymptotically stable Chlamydia-free equilibrium (CFE) whenever a certain epidemiological threshold, known as the basic reproduction number, is less than unity. It has a unique Chlamydia-present equilibrium (CPE) whenever the threshold quantity exceeds unity. The unique CPE is globally asymptotically stable under certain conditions. The model is extended to incorporate the effect of humoral and cell-mediated immune responses. The extended model also has a globally asymptotically stable CFE whenever its associated reproduction threshold is less than unity, and the disease persists when the threshold exceeds unity. Numerical simulations of the extended model show that cell-mediated immune response is more effective than humoral immune response (and that humoral immune response only offers marginal impact in curtailing Chlamydia dynamics). Furthermore, based on the parameter values used in the numerical simulations, this study shows that a future Chlamydia vaccine that boosts cell-mediated immune response would be quite effective in reducing Chlamydia burden.

Seasonal migrations of black bears (Ursus americanus): causes and consequences

American black bears frequently abandon their home ranges in late summer and move to feeding areas to fatten themselves for hibernation. We examined seasonal movements of 206 radio-collared bears in north-central Minnesota during 1981–1990. We exploited the variability in this long-term data set to test tradeoffs for animals leaving their home range. Late summer movements were common for both sexes and all ages (39% of females, 44% of males), but were variable from year-to-year in prevalence, timing, and destination. Bears typically left their summer home ranges in August and returned ~6 weeks later in September or October. Most traveled southward, where acorns were more plentiful (median = 10 km for females, 26 km for males; maximum = 168 km). These facultative migrations were most common when rich resources were available outside home ranges. Bears were least apt to leave when foods were scarce in their home range, possibly sensing a risk of migrating during a widespread food failure. Among females, those whose body mass was close to a reproductive threshold were most prone to migrate. Migrating bears were less likely to be killed by hunters, suggesting that they were especially vigilant.

Does it pay to delay? Flesh flies show adaptive plasticity in reproductive timing

Life-history plasticity is widespread among organisms. However, an important question is whether it is adaptive. Most models for plasticity in life-history timing predict that animals, once they have reached the minimal nutritional threshold under poor conditions, will accelerate development or time to reproduction. Adaptive delays in reproduction are not common, especially in short-lived species. Examples of adaptive reproductive delays exist in mammalian populations experiencing strong interspecific (e.g., predation) and intraspecific (e.g., infanticide) competition. But are there other environmental factors that may trigger an adaptive delay in reproductive timing? We show that the short-lived flesh fly Sarcophaga crassipalpis will delay reproduction under nutrient-poor conditions, even though it has already met the minimal nutritional threshold for reproduction. We test whether this delay strategy is an adaptive response allowing the scavenger time to locate more resources by experimentally providing supplemental protein pulses (early, mid and late) throughout the reproductive delay period. Flies receiving additional protein produced more and larger eggs, demonstrating a benefit of the delay. In addition, by tracking the allocation of carbon from the pulses using stable isotopes, we show that flies receiving earlier pulses incorporated more carbon into eggs and somatic tissue than those given a later pulse. These results indicate that the reproductive delay in S. crassipalpis is consistent with adaptive post-threshold plasticity, a nutritionally linked reproductive strategy that has not been reported previously in an invertebrate species.

Dynamical analysis of a sex-structured Chlamydia trachomatis transmission model with time delay

The effect of using time delay to model the latency period of Chlamydia trachomatis infection is explored, by designing a deterministic two-sex model for Chlamydia transmission dynamics in a population. The resulting delay differential equation model is shown to undergo the phenomenon of backward bifurcation, where a stable disease-free equilibrium co-exists with one or more stable endemic equilibria when the associated reproduction threshold is less than unity. This phenomenon arises due to the re-infection of individuals who recovered from the disease. Using permanence theory, it is shown that Chlamydia will persist in the population whenever the associated reproduction threshold exceeds unity. It is further shown that long latency period could induce positive (decrease disease burden) or negative (increase disease burden) population-level impact depending on the sign of a certain epidemiological threshold quantity and some other conditions. Furthermore, this study shows that adding a time delay (to model the latency period) does not alter the main equilibrium dynamics (with respect to the effective control or persistence of the disease in the community) of the corresponding non-delayed Chlamydia transmission model considered in our earlier study Sharomi and Gumel (2009) [7].

Dynamically-consistent non-standard finite difference method for an epidemic model

This paper considers the problem of constructing finite-difference methods that are qualitatively consistent with the original continuous-time model they approximate. To achieve this goal, a deterministic continuous-time model for the transmission dynamics of two strains of an arbitrary disease, in the presence of an imperfect vaccine, is considered. The model is rigorously analysed, first of all, to gain insights into its dynamical features. The analysis reveal that it undergoes a vaccine-induced backward bifurcation when the associated reproduction threshold is less than unity. For the case where the vaccine is 100% effective, the disease-free equilibrium of the model is shown to be globally-asymptotically stable if the reproduction number is less than unity. The model also exhibits the phenomenon of competitive exclusion, where the strain with the higher reproduction number dominates (and drives out) the other. Two finite-difference methods are presented for numerically solving the model. The central objective is to determine which of the two methods gives solutions that are dynamically consistent with those of the continuous-time model. The first method, an implicitly-derived explicit finite-difference method, is considered for its computational simplicity, being a Gauss–Seidel-like algorithm. However, this method is shown to suffer numerous scheme-dependent numerical instabilities and spurious behaviour (such as convergence to the wrong steady-state solutions and failing to preserve many of the main essential dynamical features of the model), particularly when relatively large step-sizes are used in the simulations. On the other hand, the second numerical method, constructed based on Mickens’ non-standard finite-difference discretization framework, is shown to be free of any numerical instabilities and contrived behaviour regardless of the size of the step-size used in the numerical simulations. In other words, unlike the first method, the non-standard method is shown to be dynamically consistent with the original continuous-time model, and, therefore, it is more suited for use to study the asymptotic dynamics of the disease transmission model being considered.

The effect of incidence functions on the dynamics of a quarantine/isolation model with time delay

The problem of the asymptotic dynamics of a quarantine/isolation model with time delay is considered, subject to two incidence functions, namely standard incidence and the Holling type II (saturated) incidence function. Rigorous qualitative analysis of the model shows that it exhibits essentially the same (equilibrium) dynamics regardless of which of the two incidence functions is used. In particular, for each of the two incidence functions, the model has a globally asymptotically stable disease-free equilibrium whenever the associated reproduction threshold quantity is less than unity. Further, it has a unique endemic equilibrium when the threshold quantity exceeds unity. For the case with the Holling type II incidence function, it is shown that the unique endemic equilibrium of the model is globally asymptotically stable for a special case. The permanence of the disease is also established for the model with the Holling type II incidence function. Furthermore, it is shown that adding time delay to and/or replacing the standard incidence function with the Holling type II incidence function in the corresponding autonomous quarantine/isolation model with standard incidence (considered in Safi and Gumel (2010) [10]) does not alter the qualitative dynamics of the autonomous system (with respect to the elimination or persistence of the disease). Finally, numerical simulations of the model with standard incidence show that the disease burden decreases with increasing time delay (incubation period). Furthermore, models with time delay seem to be more suitable for modeling the 2003 SARS outbreaks than those without time delay.

Global asymptotic properties of an SEIRS model with multiple infectious stages

The paper presents a rigorous mathematical analysis of a deterministic model, which uses a standard incidence function, for the transmission dynamics of a communicable disease with an arbitrary number of distinct infectious stages. It is shown, using a linear Lyapunov function, that the model has a globally-asymptotically stable disease-free equilibrium whenever the associated reproduction threshold is less than unity. Further, the model has a unique endemic equilibrium when the threshold exceeds unity. The equilibrium is shown to be locally-asymptotically stable, for a special case, using a Krasnoselskii sub-linearity trick. Finally, a non-linear Lyapunov function is used to show the global asymptotic stability of the endemic equilibrium (for the special case). Numerical simulation results, using parameter values relevant to the transmission dynamics of influenza, are presented to illustrate some of the main theoretical results.

Qualitative dynamics of a vaccination model for HSV-2

A new mathematical model for the transmission dynamics of herpes simplex virus type 2 (HSV-2), which takes into account disease transmission by infected individuals in the quiescent state and an imperfect HSV-2 vaccine, is designed and qualitatively analysed. In the absence of vaccination, it is shown that the model has a globally asymptotically stable (GAS) disease-free equilibrium (DFE) point whenever an epidemiological threshold, known as the ‘basic reproduction number’, is less than unity. Further, this model has a unique endemic equilibrium whenever the reproduction number exceeds unity. Using a non-linear Lyapunov function, it is shown that the unique endemic equilibrium is GAS (for a special case) when the associated reproduction threshold is greater than unity. On the other hand, the model with vaccination undergoes a vaccine-induced backward bifurcation, where the stable DFE coexists with a stable endemic equilibrium when the reproduction threshold is less than unity. Threshold analysis of the vaccination model reveals that the use of an imperfect HSV-2 vaccine could have positive or negative population-level impact (in reducing disease burden). Simulations of the vaccination model show that an HSV-2 vaccine could lead to effective disease control or elimination if the vaccine efficacy and the fraction of susceptible individuals vaccinated at steady state are high enough (at least 80% each).

Diuraphis noxiaReproduction and Development With a Comparison of Intrinsic Rates of Increase to Other Important Small Grain Aphids: A Meta-Analysis

The Russian wheat aphid, Diuraphis noxia (Kurdjumov), is a significant pest of small grains in the United States and worldwide. There is an increasing need for quality population dynamic models to aid in development of integrated pest management strategies. Unfortunately, there exists high variability in published data regarding basic life history traits that frequently direct model parameterization. Metadata were analyzed to develop relationships between temperature and reproductive and developmental traits of D. noxia. Specifically, functions were developed between temperature and the following traits: lifespan, fecundity, fecundity rate, pre-nymphipositional period, reproductive period, and intrinsic rate of increase. Lower and upper temperature reproductive thresholds were calculated as 0.6 and 36.9 degrees C, respectively. The lower temperature developmental threshold was calculated as -0.69 degrees C. Modeled longevity reached its maximum at approximately 80 d. Meta-analysis indicates maximum fecundity at approximately 18.5 degrees C, with a maximum fecundity rate of approximately 2.1 nymphs per day over the nymphipositional period. The calculated maximum total fecundity was approximately 55 nymphs per female. The maximum reproductive period was calculated to be 29.9 d. Compared with other aphid species, as temperature increased, the intrinsic rate of increase of D. noxia increased more slowly relative to Schizaphis graminum (Rondani) and Rhopalosiphum padi L., but at a similar rate to Sitobian avenae (F.).

Mathematical Study of the Role of Gametocytes and an Imperfect Vaccine on Malaria Transmission Dynamics

A mathematical model is developed to assess the role of gametocytes (the infectious sexual stage of the malaria parasite) in malaria transmission dynamics in a community. The model is rigorously analysed to gain insights into its dynamical features. It is shown that, in the absence of disease-induced mortality, the model has a globally-asymptotically stable disease-free equilibrium whenever a certain epidemiological threshold, known as the basic reproduction number (denoted by R(0)), is less than unity. Further, it has a unique endemic equilibrium if R(0) > 1. The model is extended to incorporate an imperfect vaccine with some assumed therapeutic characteristics. Theoretical analyses of the model with vaccination show that an imperfect malaria vaccine could have negative or positive impact (in reducing disease burden) depending on whether or not a certain threshold (denoted by nabla) is less than unity. Numerical simulations of the vaccination model show that such an imperfect anti-malaria vaccine (with a modest efficacy and coverage rate) can lead to effective disease control if the reproduction threshold (denoted by R(vac)) of the disease is reasonably small. On the other hand, the disease cannot be effectively controlled using such a vaccine if R(vac) is high. Finally, it is shown that the average number of days spent in the class of infectious individuals with higher level of gametocyte is critically important to the malaria burden in the community.

Life‐history plasticity after attaining a dietary threshold for reproduction is associated with protein storage in flesh flies

Body condition affects the timing and magnitude of life history transitions. Therefore, identifying proximate mechanisms involved in assessing condition is critical to understanding how these mechanisms affect the expression of life history plasticity. Nutrient storage is an important body condition parameter, likely playing roles in both attaining minimum body-condition thresholds for life history transitions and expression of life history traits.We manipulated protein availability for females of the flesh fly Sarcophaga crassipalpis to determine whether reproductive timing and output would remain plastic or become fixed. Liver was provided for 0, 2, 4, or 6 days of adult pre-reproductive development. Significantly, liver was removed after the feeding threshold had been attained and females had committed to producing a clutch.We also identified the major storage proteins and monitored their abundances, because protein stores may serve as an index of body condition and therefore may play an important role in life history transitions and plasticity.Flesh flies showed clear post-threshold plasticity in reproductive timing. Females fed protein for 2 days took ~30% longer to provision their clutch than those fed for 4 or 6 days. Observations of oogenesis showed the 2-day group expressed a different developmental program including slower egg provisioning.Protein availability also affected reproductive output. Females fed protein for 2 days produced ~20% fewer eggs than females fed 4 or 6 days. Six-day treated females provisioned larger eggs than 4-day treated females, followed by 2-day treated females with the smallest eggs.Two storage proteins were identified, LSP-1 and LSP-2. LSP-2 accumulation differed across feeding treatments. The 2- and 4-day treatment groups accumulated LSP-2 stores but depleted them during provisioning of the first clutch, whereas the 6-day group accumulated the greatest quantity of LSP-2 and had substantial LSP-2 stores remaining at the end of the clutch. This pattern of accumulation and depletion suggests that LSP-2 could play roles in both provisioning the current clutch and future clutches, making it a good candidate molecule for affecting reproductive timing and allotment. LSP-1 was not associated with post-threshold plasticity; it was carried over from larval feeding into adulthood and depleted uniformly across all feeding groups.

Mathematical analysis of the role of repeated exposure on malaria transmission dynamics

This paper presents a deterministic model for assessing the role of repeated exposure on the transmission dynamics of malaria in a human population. Rigorous qualitative analysis of the model, which incorporates three immunity stages, reveals the presence of the phenomenon of backward bifurcation, where a stable disease-free equilibrium co-exists with a stable endemic equilibrium when the associated reproduction threshold is less than unity. This phenomenon persists regardless of whether the standard or mass action incidence is used to model the transmission dynamics. It is further shown that the region for backward bifurcation increases with decreasing average life span of mosquitoes. Numerical simulations suggest that this region increases with increasing rate of re-infection of first-time infected individuals. In the absence of repeated exposure (re-infection) and loss of infection-acquired immunity, it is shown, using a non-linear Lyapunov function, that the resulting model with mass action incidence has a globally-asymptotically stable endemic equilibrium when the reproduction threshold exceeds unity.

Sensory Systems: Auditory Action Streams?

The visual system is suggested to have two main processing streams, dorsal and ventral, the former being an 'action stream' concerned with motor responses, as opposed to perception. Two recent studies suggest the existence of a comparable mechanism in the auditory system.

Does global warming impact on migration patterns and recruitment of Allis shad (Alosa alosa L.) young of the year in the Loire River, France?

The hydrological and thermal changes in the Loire River were investigated to test the influence of climatic changes on a short freshwater stage anadromous fish species, Allis shad, for the 1995- 2004 period. The mean water temperatures during the adult migration and juvenile growth phases showed significant increase, and mean water flow during these two phases decreased significantly. The period below the threshold of 18� C shortened, and the period between 18� C and the maximum lengthened, while the temperature amounts (number of degree-days) received by aquatic organisms between 18 and 24� C showed an increase. The pattern of young-of-the-year downstream migration was modified. The first day when the juvenile catches reached 5% occurred 17 days earlier at the end of the 1995-2004 period than at the beginning. The first day when the juvenile catches reached 50% was related to the 18� C threshold (reproductive threshold) and the tempera- ture amounts accumulated between the 18 and 20� C thresholds. The year-on-year levels of young-of-the- year abundance showed wide variations, which were not explained by environmental parameters, probably because of the long distance between the study site and the spawning grounds.

The interactive effects of calcium concentration and temperature on the survival and reproduction of Daphnia pulex at high and low food concentrations

We reared Daphnia pulex in a fully defined medium at seven calcium (Ca) concentrations (0.1, 0.5, 1, 1.5, 2, 5, and 10 mg Ca L−1), two mixed algal densities (high food: 30 mg chlorophyll a L−1 and low food: 3 mg Chl a L−1), and four temperatures (20ºC, 24ºC, 28ºC, and 32ºC) in a fully factorial design. The minimum Ca concentration required for the survival and reproduction of D. pulex was between 0.1 and 0.5 mg Ca L−1. Daphniids reared at 0.1 mg Ca L−1 lived no longer than 10 d and did not reproduce. Although offspring were produced at 0.5 mg Ca L−1 and above, reproduction was reduced below 1.5 mg Ca L−1 due to delays in maturity and reductions in both the brood size and number of broods produced within the 15 d of the experiment. As a result, the intrinsic rate of natural increase, r, decreased dramatically between 1.5 and 1 mg Ca L−1, and was undefined at 0.1 mg Ca L−1. Higher temperatures and reduced algal biomass enhanced the susceptibility of D. pulex to low Ca concentrations by raising the reproductive threshold to 1.5 mg Ca L−1. Thermal stress at 32ºC was so great that daphniids lived no longer than 5 d and did not reproduce. Hence, r was undefined at 32ºC. In order to sustain D. pulex populations and potentially other Ca‐sensitive daphniids, fresh waters must maintain a Ca concentration of at least 0.5 mg Ca L−1, although concentrations as high as 1.5 mg Ca L−1 may be required for daphniids to withstand the concurrent reductions in algal biomass and rising water temperatures that are now commonplace on the south‐central Canadian Shield.

Interactions between poaching and management policy affect marine reserves as conservation tools

To explore the effects of poaching within marine reserve boundaries under three different management policies this analysis uses a simple age-structured reserve model based on yield maximization or reproductive thresholds of Black rockfish ( Sebastes melanops). Departures from the traditional assumptions of full compliance to reserve boundaries alter the conclusions of prior modeling work that demonstrate yield equivalence to no-reserve effort control management and augmented reproductive benefits when small reserves are implemented. By degrading the recruitment subsidization effect to nonreserve areas from protected reserve populations, poaching resulted in negative externalities for compliant fishermen in open areas in terms of yield and degraded the reproductive output and age-structure of the system. All three policies required effort reduction in open areas as a response to poaching in reserves. The strength of the impacts from poaching varied with policy choice and harvest intensity in the reserve, where at the highest level of poaching modeled here (15% annual exploitation rate of the vulnerable reserve population) biological and fishery benefits of implementing reserves were totally negated. Under the assumptions of this model, a policy managing for a reproductive threshold that excludes the reserve population is the precautionary choice if poaching is likely. The results of this exercise emphasize the importance of garnering compliance to reserve boundaries from resource-users for spatial closures to be successful ocean management tools.

Mathematical analysis of the transmission dynamics of HIV/TB coinfection in the presence of treatment

This paper addresses the synergistic interaction between HIV and mycobacterium tuberculosis using a deterministic model, which incorporates many of the essential biological and epidemiological features of the two dis- eases. In the absence of TB infection, the model (HIV-only model) is shown to have a globally asymptotically stable, disease-free equilibrium whenever the associated reproduction number is less than unity and has a unique endemic equilibrium whenever this number exceeds unity. On the other hand, the model with TB alone (TB-only model) undergoes the phenomenon of back- ward bifurcation, where the stable disease-free equilibrium co-exists with a stable endemic equilibrium when the associated reproduction threshold is less than unity. The analysis of the respective reproduction thresholds shows that the use of a targeted HIV treatment (using anti-retroviral drugs) strategy can lead to effective control of HIV provided it reduces the relative infectiousness of individuals treated (in comparison to untreated HIV-infected individuals) below a certain threshold. The full model, with both HIV and TB, is simulated to evaluate the impact of the various treatment strategies. It is shown that the HIV-only treatment strategy saves more cases of the mixed infection than the TB-only strategy. Further, for low treatment rates, the mixed-only strategy saves the least number of cases (of HIV, TB, and the mixed infection) in comparison to the other strategies. Thus, this study shows that if resources are limited, then targeting such resources to treating one of the diseases is more beneficial in reducing new cases of the mixed infection than targeting the mixed infection only diseases. Finally, the universal strategy saves more cases of the mixed infection than any of the other strategies.

Mercury and drought along the lower Carson River, Nevada: II. Snowy egret and black-crowned night-heron reproduction on Lahontan Reservoir, 1997–2006

Mercury concentrations in the floodplain of the Carson River Basin in northwestern Nevada are some of the highest ever reported in a natural system. Thus, a portion of the basin including Lahontan Reservoir was placed on the U.S. Environmental Protection Agency's Natural Priorities List for research and cleanup. Preliminary studies indicated that reproduction in piscivorous birds may be at risk. Therefore, a 10-year study (1997--2006) was conducted to evaluate reproduction of snowy egrets (Egretta thula) and black-crowned night-herons (Nycticorax nycticorax) nesting on Gull Island in Lahontan Reservoir. Special attention was given to the annual flow of the Carson River, the resultant fluctuation of this irrigation reservoir, and the annual exposure of snowy egrets and night-herons to methylmercury (MeHg). The dynamic character of the river due to flooding and drought (drought effect) influenced snowy egret and night-heron reproduction more so than did MeHg contamination of eggs. During an extended drought (2000--2004) in the middle of the study, snowy egret nests containing eggs with concentrations of MeHg (measured as total mercury [THg] approximately 100% MeHg) > or =0.80 microg THg/g, ww, all failed, but in 1997 and 2006 (wet years with general flooding), substantial numbers of young were produced (but fewer than at nests where eggs contained <0.80 microg/g). Thus, a variable reproductive threshold of tolerance to MeHg may be associated with habitat quality (food type and abundance). Clearly, drought was the most important factor affecting snowy egret annual productivity. In contrast to snowy egrets, night-herons generally had fewer nests meeting the 0.80 microg THg/g criterion, and those above the criterion were less sensitive to mercury than were snowy egrets. Furthermore, night-herons appeared more tolerant of drought conditions than snowy egrets because they nested earlier, selected more protected nesting sites, and had a more generalist diet that provided additional food options including terrestrial organisms, which also reduced exposure to MeHg. A putative biological effect threshold of 2.0 microg THg/g in whole blood for young of both species was evaluated, which was frequently exceeded, but with no evidence, while still in the colony, of an association with direct mortality. An evaluation of physiological associations with blood residues and post-fledging survival will be presented in future reports in this series.

Temperature-Dependent Regulation of Reproduction in the Diving Beetle Dytiscus sharpi (Coleoptera: Dytiscidae)

The effects of temperature on the mating behavior, gonad development, germ cell maturation, and egg spawning of the predaceous diving beetle Dytiscus sharpi (Coleoptera; Dytiscidae), were investigated. By field observations, we found that mating behavior started in October and occurred more frequently from November to December. Under our laboratory breeding conditions, we observed almost the same seasonal variation in mating behavior. We found that temperatures lower than 20 degrees C were required to trigger mating behavior. We also found the same temperature threshold triggered gonadogenesis as well as spermatogenesis. Furthermore, for females, exposure to lower temperatures (<8 degrees C) during the winter was required for egg maturation and spawning in spring; that is, there was a second threshold for successful female reproduction. We conclude that the termination of summer reproductive diapause of D. sharpi is regulated in a temperature-dependent manner, thus effecting the adaptation of D. sharpi to southern warm habitats.

On the hyperbolic competition model: a comment to Weigelt et al. (2007)

It is a common assumption that the plant biomass or plant yield displays an asymptotic response to increasing density, and it is true that empirical evidence for over-compensation (the phenomenon where plant biomass or yield reaches an optimum at an intermediary plant density) is scarce (e.g. Rees & Crawley 1991; Freckleton & Watkinson 2002). Furthermore, there are theoretical reasons why one may expect that plants have an asymptotic constant biomass or fecundity per area, i.e. the modular construction of plants, the process of self-thinning at high densities, size-asymmetric competition against the small individuals, and a relatively small size threshold for reproduction are all effects that will reduce the effect of over-compensation (Freckleton & Watkinson 2002; Rees & Crawley 1989). However, there are reported possible examples of over-compensation in plants (e.g. Silvertown 1991; Damgaard & Borksted 2004) and whether over-compensation plays a role in plant population dynamics is still open to question (Damgaard 2004a). In a recent paper by Weigelt et al. (2007), the mechanisms of competition in multi-species plant communities are investigated under the implicit assumption of an asymptotic response to increasing density. The effect of the densities of four different plant species on individual biomass was investigated in a hyperbolic competition model originally proposed by Watkinson (1980) and Vandermeer (1984) for the single species case (here shown in the simpler case of the effect of the densities of two species on the biomass of species one, and somewhat re-parameterized in order to use the notion of competition coefficients) where x1 is the density of species i, is the biomass of isolated plants of species 1, α is a measure of the effect of density, while φ is a shape parameter that measures the rate at which competition decays as a function of the distance between plants (Vandermeer 1984). If φ > 1, then over-compensation is occurring; and if φ = 1, as implicitly assumed by Weigelt et al. (2007), then plant biomass is assumed to display an asymptotic response to increasing densities of both plant species. In the paper by Weigelt et al. (2007) the above model (1) (with φ set to 1) was tested against more complicated models with extra terms that are functions of new parameters and density and size of the neighbouring plants. If the inclusion of the extra terms increased the fit of the model significantly, then this was interpreted as a deviation from ‘additivity’ (Weigelt et al. 2007). I find the replacing of the nonlinear shape parameter φ used in model (1) with the β terms used in model (2) problematic, because (i) most of the earlier studies that motivate the use of the empirical hyperbolic competition model, including some of the studies referenced by Weigelt et al. (2007), include one or more nonlinear shape parameter (for a more thorough discussion see, for example, Damgaard 2004b), (ii) the possibly important and thoroughly studied ecological concept of over-compensation, which also provides information of the underlying mechanisms of plant–plant interactions (Vandermeer 1984) and may lead to chaotic population dynamics (e.g. Damgaard 2004a), is ignored, and (iii) it makes it more difficult to retain the use of the very fruitful concept of constant competition coefficients introduced by Lotka and Volterra, which consequently makes comparisons between studies more difficult. To illustrate the effect of choosing the relevant hyperbolic competition model, the non-additivity model (2) was fitted to data that was simulated using model (1) with φ set slightly above one (= 1.1), and the estimated β-term in the fitted model (2) was negative and significantly different from zero. Consequently, if the simulated data had been analysed with model (2), the phenomenon of over-compensation would have been incorrectly interpreted as complex non-additive interactions. On the other hand, I do not consider the hyperbolic competition model, as once perceived in earlier studies, to be sacred in any way. There may be significant effects of species interactions (e.g. Law & Watkinson 1987; Damgaard 1998), the spatial arrangement of plants (e.g. Bolker & Pacala 1999; Law et al. 1997; Damgaard 2004c), and attained biomass as suggested by Weigelt et al. (2007) that need to be studied in further developments of the basic competition models, especially as the ecological relevance and realism of the competition experiments are increased by measuring plant cover, biomass, or by classifying different size classes (e.g. Rees & Paynter 1997) of perennial species in natural plant communities instead of counting the number of individuals, or trying to discover the mechanisms underlying the observed competitive interactions (e.g. Weigelt et al. 2007). While we do not expect over-compensation to occur for all plant species in all plant communities, and even less often when biomass is measured rather than fecundity, in my opinion it is not prudent a priori to accept the hypothesis of no over-compensation.