Traveling wave solutions in a two-group SIR epidemic model with constant recruitment

Journal of Mathematical Biology - Host heterogeneity can be modeled by using multi-group structures in the population. In this paper we investigate the existence and nonexistence of traveling waves...     

Traveling wave solutions for a one-dimensional crawling nematode sperm cell model

In this paper, we proved that the one-dimensional crawling nematode sperm cell model proposed by Mogilner and Verzi (2003) supports traveling wave solutions if there is no disassembly of unbundled filaments in the cell. Uniqueness of traveling wave is established under additional assumptions and numerical examples are also given in the paper. Mathematical methods used include dynamical system techniques, implicit function theorem and global bifurcation theory.Revised version: 16 September 2003     

Traveling wave solutions of a reaction diffusion model for competing pioneer and climax species

Presented is a reaction-diffusion model for the interaction of pioneer and climax species. For certain parameters the system exhibits bistability and traveling wave solutions. Specifically, we show that when the climax species diffuses at a slow rate there are traveling wave solutions which correspond to extinction waves of either the pioneer or climax species. A leading order analysis is used in the one-dimensional spatial case to estimate the wave speed sign that determines which species becomes extinct. Results of these analyses are then compared to numerical simulations of wave front propagation for the model on one and two-dimensional spatial domains. A simple mechanism for harvesting is also introduced.     

A stochastic population model for Lepidium papilliferum (Brassicaceae), a rare desert ephemeral with a persistent seed bank

Population viability analysis (PVA) is a valuable tool for rare plant conservation, but PVA for plants with persistent seed banks is difficult without reliable information on seed bank processes. We modeled the population dynamics of the Snake River Plains ephemeral Lepidium papilliferum using data from an 11-yr artificial seed bank experiment to estimate age-specific vital rates for viability loss and germination. We related variation in postgermination demographic parameters to annual variation in precipitation patterns and used these relationships to construct a stochastic population model using precipitation driver variables. This enabled us to incorporate realistic levels of environmental variability into the model. A model incorporating best estimates for parameter values resulted in a mean trajectory for seed bank size that remained essentially stable through time, although there was a measurable risk of extinction over a 100-yr period for the study population under this scenario. Doubling the annual seed viability loss rate resulted in near-certain extinction, as did increasing first-year germination to 100%, showing the importance of the persistent seed bank. Interestingly, increasing environmental variance substantially decreased the risk of extinction, presumably because this plant relies on extremely good years to restock the persistent seed bank, while extremely bad years have little impact. If every year were average in this desert environment, the species could not persist. Simulated effects of livestock trampling resulted in greatly increased extinction risk, even over time frames as short as 15 years.     

The effective size of annual plant populations: the interaction of a seed bank with fluctuating population size in maintaining genetic variation

Many annual plant populations undergo dramatic fluctuations in size. Such fluctuations can result in the loss of genetic variability. Here I formalize the potential for a seed bank to buffer against such genetic loss. The average time to seed germination (T) defines the generation time of "annuals" with a seed bank, and assuming random seed germination, I show that, under otherwise ideal conditions, a population's effective size (Ne) equals NT, where N is the number of adult plants. This result supports the general principle that lengthening the prereproductive period increases Ne. When adult numbers vary, Ne at any time depends on N and on the numbers contributing to the seed bank in previous seasons. Averaging these effects over time gives Ne approximately Nh + (T - 1)Na, where Nh and Na are the harmonic and arithmetic means of the adult population. Thus if T > 1, Ne is determined primarily by Na. Simulations showed that until fluctuations in N are large (>25x) this relationship is accurate. I extended the theory to incorporate a selfing rate (S) and reproductive variance (I) through seed production (k), outcrossed pollen (m), and variation in selfing rate: Ne = NT(1 -S/2)/(1 + I) = NT/[1 + FIS)(1 + I)]. Reproductive variance (I) equals [Ik(1 + S)2 + IM(1 - S)2 + 2(1 - S2)Ikm = S2IS(1 + Ik)]/4, , where Ij is the standardized variance (Vj/j2) of factor j and Ikm is the standardized covariance between k and m. These results are applicable to other organisms with a similar life history, such as freshwater crustaceans with diapausing eggs (e.g., tadpole shrimp, clam shrimp, and fairy shrimp) and other semelparous species with discrete breeding seasons and a variable maturation time (e.g., Pacific salmon).     

Traveling wave solutions from microscopic to macroscopic chemotaxis models

In this paper, we study the existence and nonexistence of traveling wave solutions for the one-dimensional microscopic and macroscopic chemotaxis models. The microscopic model is based on the velocity jump process of Othmer et al. (SIAM J Appl Math 57:1044–1081, 1997). The macroscopic model, which can be shown to be the parabolic limit of the microscopic model, is the classical Keller–Segel model, (Keller and Segel in J Theor Biol 30:225–234; 377–380, 1971). In both models, the chemosensitivity function is given by the derivative of a potential function, Φ(v), which must be unbounded below at some point for the existence of traveling wave solutions. Thus, we consider two examples: F(v) = lnv{\Phi(v) = \ln v} and F(v) = ln[v/(1-v)]{\Phi(v) = \ln[v/(1-v)]}. The mathematical problem reduces to proving the existence or nonexistence of solutions to a nonlinear boundary value problem with variable coefficient on \mathbb R{\mathbb R}. The main purpose of this paper is to identify the relationships between the two models through their traveling waves, from which we can observe how information are lost, retained, or created during the transition from the microscopic model to the macroscopic model. Moreover, the underlying biological implications of our results are discussed.     

Traveling waves in a simple population model involving growth and death

The structure of solutions to a simple spatially dependent population model involving growth and death is investigated. Two forms of motility of the population are considered: (1) random motion only modeled by a Fickian law, and (2) a directed component of motion (chemotaxis), included in addition to the random motion. Under certain growth conditions a traveling wave of constant speed is approached. This speed can be increased by the addition of the chemotaxis with a corresponding increase in the asymptotic population. Development of initial conditions into a wave is illustrated numerically.     

Genetic comparisons between seed bank and Stipa krylovii plant populations

The soil seed bank represents the potential plant population since it is the source for population replacement. The genetic structure of a Stipa krylovii (Roshev.) plant population and its soil seed bank was investigated in the Xilinguole Steppe of Inner Mongolia using random amplified polymorphic DNA (RAPD) analyses. The population was sampled at two sites that were in close proximity to each other (0.5 km apart). Thirty plants and 18 seed bank samples were taken from each site to determine the genetic diversity between sites and between sources (plant or seed). The material was analyzed using 13 primers to produce 92 loci. Eighty-six were multi-loci, of which 23 loci (26.74%) of allele frequencies showed significant differences (P < or = 0.05). The genetic similarity between two seed bank sites was 0.9843 while the genetic similarity between two plant sites was 0.9619. Their similarities were all greater than that between the seed bank and plant populations. An analysis of their genetic structure showed that 87.86% of total variation was derived by two-loci. Genetic structures between plant and soil seed bank populations in S. krylovii were different due to the variance of mean gametic disequilibria and mean gene diversity. AMOVA results showed that the majority of variance (88.62%) occurred within sites, 12.75% was from between-groups. Further research is needed to investigate the selective function in maintaining the genetic diversity of Stipa krylovii plant populations.     

The probability of germination and establishment in discrete density-dependent plant populations with a seed bank: a correction formula

The effect of ignoring a seed bank in unstructured population models of annual plants is investigated under the assumption of ecological equilibrium. It is demonstrated that if delayed germination is an important life-history strategy and seed mortality in the seed bank is relatively low then it is important to take the effect of the seed bank into account. A formula that corrects the probability of germination and establishment in unstructured population models of annual plants for the effect of a seed bank is derived. This correction formula may be used in order to apply plant ecological data to a number of published unstructured plant ecological models.     

Prediction of plant cover from seed bank analysis in a semi‐arid plant community on gypsum

Abstract. Question: Does the seed bank filter annual plant composition and determine cover at the species level? Location: 510 m a.s.l., central Spain. Methods: Seven transects and 136 quadrats were established in a semi‐arid gypsum system. Seed bank samples were collected in each quadrat in September. The community was sampled the following April. For each quadrat we measured slope, microslope, landform, elevation, perennial cover and crust cover. Seed bank was estimated using the direct emergence method in glasshouse. Relationship among seed bank and annual community was assessed by Mantel correlations. Above‐ground cover for the five most abundant species was modelled with GLMs. Results: Seed bank density was the best predictor for annual community cover; perennial cover and landform were also included in the model. Species composition between September seed bank and April annual community cover was also highly related according to the Mantel test. This relationship was constant, even when the effect due to other abiotic (landform, microslope) or biotic (perennial cover, crust cover) parameters were partialled out. Microslope, elevation and seed bank density were the best parameters to predict spring cover of the five most abundant species. Conclusions: Above‐ground and below‐ground community compartments are strongly related in terms of abundance and species composition. This relationship is filtered by several environmental factors (e.g. perennial cover, landform, microslope) that exert a strong control at community and individual levels. Our results support the hypothesis that annual community performance is affected by seed bank pattern.     

Similarity between seed bank and vegetation in Mediterranean grassland: a predictive model

Abstract. The similarity in species composition between seed bank and vegetation was analysed in Mediterranean grasslands in relation to altitude, topography and grazing. Soil samples were collected in permanent plots in autumn at the end of the summer drought period and in spring, before the new seed fall and after the natural winter seed stratification. The seed bank composition was determined by greenhouse germination over a nine-month period. Presence/absence of species in the standing vegetation throughout the complete annual cycle, and the percentage area of bare ground in October, were recorded in the same plots. The species composition of the standing vegetation is clearly determined by altitude, topography and grazing, while the floristic composition of the seed banks is only related to altitude and topography in the case of autumn seed bank and with any of the three factors in the spring seed bank. Relative abundances of grasses, legumes and forbs also show different patterns in vegetation and seed bank data. Sørensen similarity between the autumn seed bank and the vegetation declines as altitude rises, but there are no significant differences for topography and grazing. This similarity decreases in the case of the spring seed bank and does not show any significant relationship with any of the factors. The perennial/ annual ratio and the proportion of bare soil in October are proposed as explanatory variables in a predictive model of similarity between the seed composition of the seed bank and vegetation.     

Restoration of ditch bank plant species richness: The potential of the soil seed bank

Abstract. Small‐scale landscape elements, such as ditch banks, play an important role in preserving plant species richness in agricultural landscapes. In this study, we investigate whether the seed bank might be useful for restoring the above‐ground plant species richness. We studied the vegetation and seed bank composition at six species‐rich and six species‐poor ditch banks, where agri‐environment schemes are running to maintain and enhance ditch bank plant diversity. We show that the number of species in the seed bank was low, regardless of the number of species in the established vegetation. Moreover, the seed bank was always dissimilar to the established vegetation. Target species for nature conservation were occasionally present in the seed bank at both species‐poor and species‐rich sites, but rarely so if the species was absent from the established vegetation. We conclude that the potential use of the seed bank for restoration of ditch banks is minimal. At present, plant species richness seems to be largely controlled by germination opportunities; high biomass and competition appear to hamper germination at species‐poor sites. We recommend continued nutrient reduction at such sites. Soil disturbance measures and deliberate sowing should also be considered.     

Existence of traveling wave solutions in a diffusive predator-prey model

 We establish the existence of traveling front solutions and small amplitude traveling wave train solutions for a reaction-diffusion system based on a predator-prey model with Holling type-II functional response. The traveling front solutions are equivalent to heteroclinic orbits in R ⁴ and the small amplitude traveling wave train solutions are equivalent to small amplitude periodic orbits in R ⁴ . The methods used to prove the results are the shooting argument and the Hopf bifurcation theorem. Received: 25 May 2001 / Revised version: 5 August 2002 / Published online: 19 November 2002 RID="*" ID="*" Research was supported by the National Natural Science Foundations (NNSF) of China. RID="*" ID="*" Research was partially supported by the Natural Sciences and Engineering Research Council (NSERC) of Canada. On leave from the Department of Mathematics and Statistics, Dalhousie University, Halifax, Nova Scotia B3H 3J5, Canada. Mathematics Subject Classification (2000): 34C35, 35K57 Key words or phrases: Traveling wave solution – Wazewski set – Shooting argument – Hopf bifurcation Acknowledgements. We would like to thank the two referees for their careful reading and helpful comments.     

Aboveground plant community and seed bank composition along an invasion gradient

Invasive plants can reduce plant diversity and abundance in native grassland communities; however, the effect on the native seed bank is less clear. The objective of this study was to assess the effects of invasion by the exotic grass old world bluestem (OWB; Bothriochloa spp.) on native aboveground plant species composition and seed bank diversity and abundance (i.e., cover, density). In this central Great Plains grassland, OWB invasion had differential effects on native diversity and abundance of both aboveground and seed bank plant communities. Native plant species diversity and cover showed a steep decline as OWB cover increased. No change in native seed density or richness was observed in response to OWB invasion, however, OWB seed density increased with increasing invasion, thus increasing total seed density. Our results indicate that as OWB invasion increases, the native plant community decreases in diversity and abundance. Although, no effect on native seed bank diversity and density was observed in this study, as native seeds are lost through a loss of native species in the plant communities, native seed bank diversity and density is expected to decline.     

Modeling the population dynamics of annual plants with seed bank and density dependent effects

A model is proposed for the population dynamics of an annual plant (Sesbania vesicaria) with a seed bank (i.e. in which a proportion of seeds remain dormant for at least one year). A simple linear matrix model is deduced from the life cycle graph. The dominant eigenvalue of the projection matrix is estimated from demographic parameters derived from field studies. The estimated values for population growth rate () indicates that the study population should be experiencing a rapid exponential increase, but this was not the case in our population.The addition of density dependent effects on seedling survivorship and adult fecundity, effects for which field studies provide evidence, considerably improves our model. Depending on the demographic parameters, the model leads to stable equilibrium, oscillations, or chaos. Study of the behaviour of this model in the parameter space shows that the existence of a seed bank allows higher among-year variation of adult fecundity, without leaving the region of demographic stability. Field data obtained over 3 years confirm this prediction.     

Legacy effects of afforestation on prairie plant and seed bank communities in a northern Canadian prairie

Afforestation resulting from fire suppression, modified grazing, plantation establishment and climate change poses a threat to northern prairie ecosystems. Trees alter the composition and function of plant and soil fauna communities and can compromise the restoration of afforested prairies. To evaluate the hypothesis that legacies of afforestation persist in restored prairie communities and decrease the potential for restoration, we examined the composition, structure, and diversity of plant and seed bank communities along a 20 year chronosequence of plantation tree removal from a northern fescue prairie in Riding Mountain National Park, Manitoba, Canada. Tree removal increased the abundance of weedy species in the plant and seed bank communities of restored prairies and plant diversity peaked and declined over the 20 year period of passive restoration. As a result, time since tree removal and the encroachment of invasive species were key in explaining the composition of restored prairie communities. Low correlation between the species composition of plant and seed bank communities, including the complete absence of Festuca hallii in restored treatments, demonstrated that legacies of afforestation compromised the potential of seed banks to facilitate prairie restoration. We conclude that tree removal alone is insufficient for the restoration of northern fescue prairies and that, in the absence of active management, the persistence of low-diversity plant and seed bank communities constitutes an important legacy of afforestation and an important barrier to future restoration.     

Traveling waves in a chemotactic model

A model for chemotaxis in a bacteria-substrate mixture introduced by Keller and Segel, which is described by nonlinear partial differential equations, is studied analytically. The existence of traveling waves is shown for the system in which the substrate diffusion is taken into account and the chemotactic coefficient is greater than the motility one, and the instability of traveling waves is discussed.     

On periodic cohort solutions of a size-structured population model

 We consider a size-structured population model with discontinuous reproduction and feedback through the environmental variable ‘substrate’. The model admits solutions with finitely many cohorts and in that case the problem is described by a system of ODEs involving a bifurcation parameter β. Existence of nontrivial periodic n-cohort solutions is investigated. Moreover, we discuss the question whether n cohorts (n≧2) with small size differences will tend to a periodic one-cohort solution as t→∞. Received 16 March 1995; received in revised form 7 January 1997