The minimum effort required to eradicate infections in models with backward bifurcation

We study an epidemiological model which assumes that the susceptibility after a primary infection is r times the susceptibility before a primary infection. For r = 0 (r = 1) this is the SIR (SIS) model. For r > 1 + (μ/α) this model shows backward bifurcations, where μ is the death rate and α is the recovery rate. We show for the first time that for such models we can give an expression for the minimum effort required to eradicate the infection if we concentrate on control measures affecting the transmission rate constant β. This eradication effort is explicitly expressed in terms of α,r, and μ As in models without backward bifurcation it can be interpreted as a reproduction number, but not necessarily as the basic reproduction number. We define the relevant reproduction numbers for this purpose. The eradication effort can be estimated from the endemic steady state. The classical basic reproduction number R ₀ is smaller than the eradication effort for r > 1 + (μ/α) and equal to the effort for other values of r. The method we present is relevant to the whole class of compartmental models with backward bifurcation.Dedicated to Karl Peter Hadeler on the occasion of his 70th birthday.     

Diploidy and the selective advantage for sexual reproduction in unicellular organisms

This article develops mathematical models describing the evolutionary dynamics of both asexually and sexually reproducing populations of diploid unicellular organisms. The asexual and sexual life cycles are based on the asexual and sexual life cycles in Saccharomyces cerevisiae, Baker’s yeast, which normally reproduces by asexual budding, but switches to sexual reproduction when stressed. The mathematical models consider three reproduction pathways: (1) Asexual reproduction, (2) self-fertilization, and (3) sexual reproduction. We also consider two forms of genome organization. In the first case, we assume that the genome consists of two multi-gene chromosomes, whereas in the second case, we consider the opposite extreme and assume that each gene defines a separate chromosome, which we call the multi-chromosome genome. These two cases are considered to explore the role that recombination has on the mutation-selection balance and the selective advantage of the various reproduction strategies. We assume that the purpose of diploidy is to provide redundancy, so that damage to a gene may be repaired using the other, presumably undamaged copy (a process known as homologous recombination repair). As a result, we assume that the fitness of the organism only depends on the number of homologous gene pairs that contain at least one functional copy of a given gene. If the organism has at least one functional copy of every gene in the genome, we assume a fitness of 1. In general, if the organism has l homologous pairs that lack a functional copy of the given gene, then the fitness of the organism is κ _l . The κ _l are assumed to be monotonically decreasing, so that κ₀ = 1 > κ₁ > κ₂ > ⋯ > κ_∞ = 0. For nearly all of the reproduction strategies we consider, we find, in the limit of large N, that the mean fitness at mutation-selection balance is max{2 e^-m-1, 0} ,\hbox{max}\{2 e^{-\mu}-1, 0\} , where N is the number of genes in the haploid set of the genome, ε is the probability that a given DNA template strand of a given gene produces a mutated daughter during replication, and μ = Nε. The only exception is the sexual reproduction pathway for the multi-chromosomed genome. Assuming a multiplicative fitness landscape where κ _l  = α ^l for α ∈ (0, 1), this strategy is found to have a mean fitness that exceeds the mean fitness of all the other strategies. Furthermore, while other reproduction strategies experience a total loss of viability due to the steady accumulation of deleterious mutations once μ exceeds ln2 ,\ln 2 , no such transition occurs in the sexual pathway. Indeed, in the limit as α → 1 for the multiplicative landscape, we can show that the mean fitness for the sexual pathway with the multi-chromosomed genome converges to e ^−2μ, which is always positive. We explicitly allow for mitotic recombination in this study, which, in contrast to previous studies using different models, does not have any advantage over other asexual reproduction strategies. The results of this article provide a basis for understanding the selective advantage of the specific meiotic pathway that is employed by sexually reproducing organisms. The results of this article also suggest an explanation for why unicellular organisms such as Saccharomyces cerevisiae (Baker’s yeast) switch to a sexual mode of reproduction when stressed. While the results of this article are based on modeling mutation-propagation in unicellular organisms, they nevertheless suggest that, in more complex organisms with significantly larger genomes, sex is necessary to prevent the loss of viability of a population due to genetic drift. Finally, and perhaps most importantly, the results of this article demonstrate a selective advantage for sexual reproduction with fewer and much less restrictive assumptions than those of previous studies.     

Periodic Matrix Population Models: Growth Rate,Basic Reproduction Number,and Entropy

This article considers three different aspects of periodic matrix population models. First, a formula for the sensitivity analysis of the growth rate λ is obtained that is simpler than the one obtained by Caswell and Trevisan. Secondly, the formula for the basic reproduction number ℛ₀ in a constant environment is generalized to the case of a periodic environment. Some inequalities between λ and ℛ₀ proved by Cushing and Zhou are also generalized to the periodic case. Finally, we add some remarks on Demetrius’ notion of evolutionary entropy H and its relationship to the growth rate λ in the periodic case.     

Individual- and population-level effects of antibiotics on the rotifers, <Emphasis Type="Italic">Brachionus calyciflorus</Emphasis> and <Emphasis Type="Italic">B. plicatilis</Emphasis>

The toxicity of three common antibiotics (streptomycin sulfate, tetracycline hydrochloride, and tylosin tartrate) to the freshwater rotifer Brachionus calyciflorus and brackish-water rotifer B. plicatilis was investigated using full-lifespan exposure durations. Effects of each antibiotic on lifespan, lifetime reproduction, and Malthusian parameter were assessed at seven nominal concentrations (ranging from 5.6 mg l⁻¹ to 2,000 mg l⁻¹) and a negative control. Lowest Observed Effect Concentrations (LOECs) were determined for reproduction and lifespan, while 1%, 10%, 25%, and 50% Inhibitory Concentrations (IC₁, IC₁₀, IC₂₅, IC₅₀) and 95% confidence intervals were estimated for all three endpoints. LOECs ranged from 5.6 mg l⁻¹ to 90 mg l⁻¹, with all LOECs less than 90 mg l⁻¹ occurring in B. calyciflorus. The lowest IC1 concentrations were 3.91 mg l⁻¹ for the effect of tetracycline on lifetime reproduction in B. calyciflorus and 4.06 mg l⁻¹ for the effect of tylosin on lifetime reproduction in B. plicatilis. Overall, lifetime reproduction was the most sensitive endpoint and the Malthusian parameter was the least sensitive. IC₁ values for lifetime reproduction were roughly one to two orders of magnitude lower than the corresponding IC₅₀ values. Guest editors: S. S. S. Sarma, R. D. Gulati, R. L. Wallace, S. Nandini, H. J. Dumont and R. Rico-Martínez Advances in Rotifer Research     

Modeling the Population Level Effects of an HIV-1 Vaccine in an Era of Highly Active Antiretroviral Therapy

First generation HIV vaccines may have limited ability to prevent infection. Instead, they may delay the onset of AIDS or reduce the infectiousness of vaccinated individuals who become infected. To assess the population level effects of such a vaccine, we formulate a deterministic model for the spread of HIV in a homosexual population in which the use of highly active antiretroviral therapy (HAART) to treat HIV infection is incorporated. The basic reproduction number R ₀ is obtained under this model. We then expand the model to include the potential effects of a prophylactic HIV vaccine. The reproduction number R _f is derived for a population in which a fraction f of susceptible individuals is vaccinated and continues to benefit from vaccination. We define f ^* as the minimum vaccination fraction for which R _f ≤1 and describe situations in which it equals the critical vaccination fraction necessary to eliminate disease. When R ₀ is large or an HIV vaccine is only partially effective, the critical vaccination fraction may exceed one. HIV vaccination, however, may still reduce the prevalence of disease if the reduction in infectiousness is at least as great as the reduction in the rate of disease progression. In particular, a vaccine that reduces infectiousness during acute infection may have an important public health impact especially if coupled with counseling to reduce risky behavior.     

High survival during hibernation affects onset and timing of reproduction

The timing of reproduction is one of the most crucial life history traits, with enormous consequences for the fitness of an individual. We investigated the effects of season and timing of birth on local survival probability in a small mammalian hibernator, the common dormouse (Muscardinus avellanarius). Local monthly survival probability was lowest in the early active season (May–August, ϕ_adult = 0.75–0.88, ϕ_juvenile = 0.61–0.68), increased during the late active season (August–October), and highest during hibernation (October–May, ϕ_adult = 0.96–0.98, ϕ_juvenile = 0.81–0.94). Consequently, dormice had an extremely high winter survival probability. We observed two peaks in the timing of reproduction (June and August/September, respectively), with the majority of juveniles born late in the active season. Although early investment in reproduction seems the better life history tactic [survival probability until onset of reproduction: ϕ_{born early} = 0.46, 95% confidence interval (CI) 0.28–0.64; ϕ_{born late} = 0.19, 95% CI = 0.09–0.28], only females with a good body condition (significantly higher body mass) invest in reproduction early in the year. We suggest the high over-winter survival in dormice allows for a unique life history pattern (i.e., combining slow and fast life history tactics), which leads to a bimodal seasonal birth pattern: (1) give birth as early as possible to allow even the young to breed before hibernating, and/or (2) give birth as late as possible (leaving just enough time for these young to fatten) and enter directly into a period associated with the highest survival rates (hibernation) until maturity.     

Multiregional Periodic Matrix for Modeling the Population Dynamics of Sardine (<Emphasis Type="Italic">Sardina pilchardus</Emphasis>) Along the Moroccan Atlantic Coast: Management Elements for Fisheries

In this paper, we present a deterministic time discrete mathematical model based on multiregional periodic matrices to describe the dynamics of Sardina pilchardus in the Central Atlantic area of the Moroccan coast. This model deals with two stages (immature and mature) and three spatial zones where sardines are supposed to migrate from one zone to another. The population dynamics is described by an autonomous recurrence equation N(t + 1) = A.N(t), where A is a positive matrix whose entries are estimated using data collected during biannual acoustic surveys carried out from 2001 to 2003 onboard the Norwegian research vessel “Dr Fridtjof Nansen”. The dominant eigenvalue λ of A that gives the long-term growth rate of fish population is smaller than one. This agrees with the stock decrease observed in the data collected. We show that λ is highly sensitive to the recruitment rate and much less sensitive to the reproduction rate. These results can clearly be used to define an efficient scenario in order to fight for instance against a stock decrease.     

Malaria model with periodic mosquito birth and death rates

In this paper, we introduce a model of malaria, a disease that involves a complex life cycle of parasites, requiring both human and mosquito hosts. The novelty of the model is the introduction of periodic coefficients into the system of one-dimensional equations, which account for the seasonal variations (wet and dry seasons) in the mosquito birth and death rates. We define a basic reproduction number R ₀ that depends on the periodic coefficients and prove that if R ₀<1 then the disease becomes extinct, whereas if R ₀>1 then the disease is endemic and may even be periodic.     

Genealogy with seasonality,the basic reproduction number,and the influenza pandemic

The basic reproduction number R ₀ has been used in population biology, especially in epidemiology, for several decades. But a suitable definition in the case of models with periodic coefficients was given only in recent years. The definition involves the spectral radius of an integral operator. As in the study of structured epidemic models in a constant environment, there is a need to emphasize the biological meaning of this spectral radius. In this paper we show that R ₀ for periodic models is still an asymptotic per generation growth rate. We also emphasize the difference between this theoretical R ₀ for periodic models and the “reproduction number” obtained by fitting an exponential to the beginning of an epidemic curve. This difference has been overlooked in recent studies of the H1N1 influenza pandemic.     

Quantifying BSE control by calculating the basic reproduction ratio R 0 for the infection among cattle

The safety of using meat and bone meal (MBM) in mammal feed was studied in view of BSE, by quantifying the risk of BSE transmission through different infection routes. This risk is embodied in the basic reproduction ratio R ₀ of the infection, i.e. the average number of new infections induced by one initial infection. Only when R ₀ is below 1, will the disease die out with certainty and the population will become free from BSE. Unfortunately this is a slow process due to the slow progression of the disease. We calculate R ₀ explicitly from basic ingredients taking several different transmission routes into account. Several of the basic ingredients are functions of age or of infection-age. We also calculate the exponential growth rate r in terms of the same basic ingredients. Next we quantify the ingredients from available data and compute the effects on R ₀ of various scenarios for controlling BSE, with examples for the UK and the Netherlands. This revised version was published online in November 2003. A correction was made to formula 10 of this paper, indicies were previously printed in an incorrect order and an extraneous element has been removed.     

A reaction–diffusion malaria model with incubation period in the vector population

Malaria is one of the most important parasitic infections in humans and more than two billion people are at risk every year. To understand how the spatial heterogeneity and extrinsic incubation period (EIP) of the parasite within the mosquito affect the dynamics of malaria epidemiology, we propose a nonlocal and time-delayed reaction–diffusion model. We then define the basic reproduction ratio R₀{\mathcal{R}_0} and show that R₀{\mathcal{R}_0} serves as a threshold parameter that predicts whether malaria will spread. Furthermore, a sufficient condition is obtained to guarantee that the disease will stabilize at a positive steady state eventually in the case where all the parameters are spatially independent. Numerically, we show that the use of the spatially averaged system may highly underestimate the malaria risk. The spatially heterogeneous framework in this paper can be used to design the spatial allocation of control resources.     

Year class coexistence or competitive exclusion for strict biennials?

 We consider a discrete time model of semelparous biennial population dynamics. Interactions between individuals are modelled with the aid of an ``environmental' variable I. The impact on and the sensitivity to the environmental condition is age specific. The main result is that competitive exclusion between the year classes is possible as is their coexistence. For moderate values of the basic reproduction ratio R ₀ there is a strict dichotomy: depending on the other parameters we either find competitive exclusion or coexistence. We characterize rather precisely the patterns of age specific impact and sensitivity that lead to either of these outcomes. Received: 13 July 2001 / Revised version: 26 June 2002 / Published online: 19 November 2002 Key words or phrases: Competitive exclusion – Semelparous species – Periodical insects     

Thresholds for macroparasite infections

We analyse here the equilibria of an infinite system of partial differential equations modelling the dynamics of a population infected by macroparasites. We find that it is possible to define a reproduction number R₀ that satisfies the intuitive definition, and yields a sharp threshold in the behaviour of the system: if R₀ < 1, the parasite-free equilibrium (PFE) is asymptotically stable and there are no endemic equilibria; if R₀ > 1, the PFE is unstable and there exists a unique endemic equilibrium. The results mainly confirm what had been obtained in simplified models, except for the fact that no backward bifurcation occurs in this model. The stability of equilibria is established by extending an abstract linearization principle and by analysing the spectra of appropriate operators.Revised version: 14 November 2003Supported in part by CNR under Grant n. 00.0142.ST74 Metodi e modelli matematici nello studio dei fenomeni biologici     

Progression age enhanced backward bifurcation in an epidemic model with super-infection

 We consider a model for a disease with a progressing and a quiescent exposed class and variable susceptibility to super-infection. The model exhibits backward bifurcations under certain conditions, which allow for both stable and unstable endemic states when the basic reproduction number is smaller than one. Received: 11 October 2001 / Revised version: 17 September 2002 / Published online: 17 January 2003 Present address: Department of Biological Statistics and Computational Biology, 434 Warren Hall, Cornell University, Ithaca, NY 14853-7801 This author was visiting Arizona State University when most of the research was done. Research partially supported by NSF grant DMS-0137687. This author's research was partially supported by NSF grant DMS-9706787. Key words or phrases: Backward bifurcation – Multiple endemic equilibria – Alternating stability – Break-point density – Super-infection – Dose-dependent latent period – Progressive and quiescent latent stages – Progression age structure – Threshold type disease activation – Operator semigroups – Hille-Yosida operators – Dynamical systems – Persistence – Global compact attractor     

Approximation of the Basic Reproduction Number <Emphasis Type="Italic">R</Emphasis><Subscript>0</Subscript> for Vector-Borne Diseases with a Periodic Vector Population

The main purpose of this paper is to give an approximate formula involving two terms for the basic reproduction number R ₀ of a vector-borne disease when the vector population has small seasonal fluctuations of the form p(t) = p ₀ (1+ε cos (ωt − φ)) with ε ≪ 1. The first term is similar to the case of a constant vector population p but with p replaced by the average vector population p ₀. The maximum correction due to the second term is (ε²/8)% and always tends to decrease R ₀. The basic reproduction number R ₀ is defined through the spectral radius of a linear integral operator. Four numerical methods for the computation of R ₀ are compared using as example a model for the 2005/2006 chikungunya epidemic in La Réunion. The approximate formula and the numerical methods can be used for many other epidemic models with seasonality. MSC 92D30 ⋅ 45C05 ⋅ 47A55     

Subfossil chydorid (Cladocera, Chydoridae) ephippia as paleoenvironmental proxies: evidence from boreal and subarctic lakes in Finland

Chydorids (Cladocera, Chydoridae) have two reproductive strategies: asexual reproduction that prevails during favorable environmental conditions and sexual reproduction that is induced by environmental stimuli associated with seasonal or aperiodic environmental stresses. These modes of reproduction can be recognized in the subfossil sedimentary records as parthenogenetic shells of females (asexual reproduction) and by ephippia (sexual reproduction). We studied the interrelations between subfossil chydorid ephippia and environmental variables by analyzing surface sediment samples obtained from 76 Finnish lakes across a latitudinal gradient (60–70°N). The results showed that the total chydorid ephippia (TCE) increases along the climate gradient from ~2 to 3% in the south to ~25% in the north and suggested a significant dependence (r ~ −0.8, P < 0.001) with several climate factors, especially that of mean July air temperature. We used this relationship to create a model for reconstructing past mean July air temperatures. A linear regression of the log₁₀ transformed TCE as a single independent variable explained 76% (SE ± 0.76°C) of the variance of the observed mean July air temperatures. Accordingly, we propose that this novel tool may be highly suitable for reconstructing paleotemperatures in cold-temperate environments.     

Capillary operators—II

This work continues with an examination of capillary exchange models as operators, namely the operatorsO _k andK _αk relating extravascular and intravascular concentration to input for the Krogh cylinder model of a single capillary, a model basic to many organ models. Fundamental algebraic and analytic properties are presented: the operators belong to a commutative Banach algebra; an addition theorem holdsK _αk +K _βk =K _α+β,k ; the operatorK _αk has an inverse;K _αk ^-1 , (as an operator on LebesgueL _p space or on the locally integrable functions); partial derivatives are given forK _αk [f](t) andO _k [f](t) (sensitivity functions); and inequalities are established for the derivatives. Dominance relations between model curves are inferred. Error bound formulas are presented forK andO as bounds on ‖K _αk f-K _βl f‖ _p and ‖O _k f-O _l f‖ _p for allL _p . Consequent limitations on relative errors are shown. The implications for operators on a finite time interval are deduced. This work supported in part by PHS Grant Nos HL-19153 (SCOR and Pulmonary Vascular Disease) and HL-19370 at Vanderbilt University Medical School.     

Variation in life-cycle between three rare and endangered floodplain violets in two regions: implications for population viability and conservation

We studied the demography of Viola elatior, V. pumila, and V. stagnina, three rare and endangered Central European floodplain species, to (i) analyse variation in life-cycles among congeners and between regions (Dyje-Morava floodplains, Czech Republic; Upper Rhine, Germany), (ii) to define sensitive stages in the life-cycles, and (iii) to identify possible threats for population viability and species conservation. Matrix models were based on the fate of marked individuals from a total of 27 populations over two years. We analysed population growth rate (λ), stage distribution, net reproductive rate (R ₀), generation time, age at first reproduction, and elasticity and calculated a life table response experiment (LTRE). Most populations were declining and λ did not differ between species or regions during the observed interval. Despite higher probabilities for survival and flowering in the Dyje populations, R ₀ was higher in the Rhine populations. Also other demographic traits showed consistent differences between regions and/or species. Complex life-cycles and large variation in λ precluded unequivocal identification of sensitive stages or vital rates for conservation. Variation between regions may be a consequence of differences in habitat quality. Our results suggest that deterministic processes such as reduced management, succession, habitat destruction, and lack of disturbance through reduced or eliminated flooding present the strongest threat for the viability and persistence of populations of the three floodplain violets as compared with stochastic processes. However, the persistent seed bank of the species may buffer populations against environmental variation and represents a reservoir for recovery after resumption of suitable land-use management.     

The root–soil system of Norway spruce subjected to turning moment: resistance as a function of rotation

The reactions of trees to wind, rockfall, and snow and debris flow depend largely on how strong and deformable their anchorage in the soil is. Here, the resistive turning moment M of the root–soil system as a function of the rotation ϕ at the stem base plays the major role. M(ϕ) describes the behavior of the root–soil system when subject to rotational moment, with the maximum M(ϕ) indicating the anchorage strength M _a of the tree. We assessed M(ϕ) of 66 Norway spruce (Picea abies L. Karst) by pulling them over with a winch. These 45- to 170-year-old trees grew at sites of low and high elevation, with a diameter at breast height DBH = 14–69 cm and a height H = 9–42 m. M(ϕ) displayed a strong nonlinear behavior. M _a was reached at a lower ϕ for large trees than for small trees. Thus overhanging tree weight contributed less to M _a for the large trees. Overturning also occurred at a lower ϕ for the large trees. These observations show that the rotational ductility of the root–soil system is higher for small trees. M _a could be described by four monovariate linear regression equations of tree weight, stem weight, stem volume and DBH ² ·H (0.80 < R ² < 0.95), and ϕ at M _a, ϕ _a, by a power law of DBH²·H (R ² = 0.85). We found significantly higher M _a for the low-elevation spruces than for the high-elevation spruces, which were more shallowly anchored, but no significant difference in ϕ _a. The 66 curves of M(ϕ), normalized (_n) by M _a in M-direction and by ϕ _a in ϕ-direction, yielded one characteristic average curve: . Using and the predictions of M _a and ϕ _a, it is shown that M(ϕ) and the curves associated with M(ϕ) can be predicted with a relative standard error ≤25%. The parameterization of M(ϕ) by tree size and weight is novel and provides useful information for predicting with finite-element computer models how trees will react to natural hazards.     

Inheritance of resistance to the two-spotted spider mite from L. pimpinellifolium (Jusl.) Mill. accession ‘TO-937’

The two-spotted spider mite (Tetranychus urticae Koch) is an important pest of tomato (Lycopersicon esculentum Mill.) crops in temperate regions as this spider mite has a very large capacity for population increase and causes severe tomato yield losses. There is no described tomato cultivar fully resistant to this pest, although resistant accessions have been reported within the green-fruited tomato wild species L. pennellii (Corr.) D’Arcy and L. hirsutum Humb. & Bonpl. We observed a L. pimpinellifolium (Jusl.) Mill. accession, ‘TO-937’, which seemed to be completely resistant to mite attacks and we crossed it with the susceptible L. esculentum cultivar. ‘Moneymaker’ to obtain a family of generations consisting of the two parents, the F₁, the F₂, the BC₁ to L. esculentum, and the BC₁ to L. pimpinellifolium. This family was evaluated for mite resistance in a polyethylene greenhouse using an experimental design in 60 small complete blocks distributed along 12 double rows. Each block consisted of five F₂ plants in one row and one plant of each of the two parents, the F₁, the BC₁ to L. esculentum, and the BC₁ to L. pimpinellifolium in the adjacent row. Plants at the 10–15 leaf stage were artificially infested by putting on them two pieces of French bean leaf heavily infested with T. urticae. After two months, evaluations of infestation were made by visual observation of mite nets and leaf damage. Plants that were free of signs of mite reproduction on the top half were considered as resistant, plants with silky nets only on their basal leaves, intermediate, and plants with mite reproduction on both basal and top canopies were scored as susceptible. Dominance for resistance appeared because all the ‘To-937’, BC₁ to L. pimpinellifolium, and F₁ plants were resistant. Not all ‘Moneymaker’ plants behaved as susceptible because 35% of plants were intermediate. In the BC₁ to L. pimpinellifolium and the F₂, most plants were scored as resistant, only 7 % BC₁ and 3 % F₂ plants were intermediate, and a single F₂ plant (0.3 %) was susceptible. With these figures, resistance seemed to be controlled by either four or two genes according to whether segregation in the BC₁ or in the F₂, respectively, were considered. These results could in part be explained because of appearance of negative interplot interference due to the high frequency of resistant genotypes within most of the generations. Therefore, the family was evaluated again but using a different experimental design. In the new experiment, 16 ‘TO-937’, 17 ‘Moneymaker’, 17 F₁, 37 BC₁ to L. pimpinellifolium, 38 BC₁ to L. esculentum, and 125 F₂ plants were included. Each of these test plants was grown besides a susceptible ‘Moneymaker’ auxilliary plant that served to keep mite population high and homogeneous in the greenhouse. Negative interplot interference was avoided with this design and all the ‘TO-937’, F₁, and BC₁ to L. pimpinellifolium plants were resistant, all ‘Moneymaker’ test plants were susceptible, and 52 % BC₁ to L. esculentum and 25 % F₂ plants were susceptible, which fitted very well with the expected for resistance governed by a single dominant gene. The simple inheritance mode found will favour sucessful introgression of mite resistance into commercial tomatoes from the very close relative L. pimpinellifolium.