On spreading speeds and traveling waves for growth and migration models in a periodic habitat

 It is shown that the methods previously used by the author [Wei82] and by R. Lui [Lui89] to obtain asymptotic spreading results and sometimes the existence of traveling waves for a discrete-time recursion with a translation invariant order preserving operator can be extended to a recursion with a periodic order preserving operator. The operator can be taken to be the time-one map of a continuous time reaction-diffusion model, or it can be a more general model of time evolution in population genetics or population ecology in a periodic habitat. Methods of estimating the speeds of spreading in various directions will also be presented. Received: 12 July 2001 / Revised version: 19 July 2002 / Published online: 17 October 2002 Mathematics Subject Classification (2000): 92D40, 92D25, 35K55, 35K57, 35B40 Keywords or phrases: Periodic – Spreading speed – Traveling wave     

Stability of differential susceptibility and infectivity epidemic models

We introduce classes of differential susceptibility and infectivity epidemic models. These models address the problem of flows between the different susceptible, infectious and infected compartments and differential death rates as well. We prove the global stability of the disease free equilibrium when the basic reproduction ratio R₀ ￡ 1{\mathcal{R}_0 \leq 1} and the existence and uniqueness of an endemic equilibrium when ${\mathcal{R}_0 >1 }${\mathcal{R}_0 >1 } . We also prove the global asymptotic stability of the endemic equilibrium for a differential susceptibility and staged progression infectivity model, when ${\mathcal{R}_0 >1 }${\mathcal{R}_0 >1 } . Our results encompass and generalize those of Hyman and Li (J Math Biol 50:626–644, 2005; Math Biosci Eng 3:89–100, 2006).     

Existence of traveling waves for integral recursions with nonmonotone growth functions

A class of integral recursion models for the growth and spread of a synchronized single-species population is studied. It is well known that if there is no overcompensation in the fecundity function, the recursion has an asymptotic spreading speed c*, and that this speed can be characterized as the speed of the slowest non-constant traveling wave solution. A class of integral recursions with overcompensation which still have asymptotic spreading speeds can be found by using the ideas introduced by Thieme (J Reine Angew Math 306:94–121, 1979) for the study of space-time integral equation models for epidemics. The present work gives a large subclass of these models with overcompensation for which the spreading speed can still be characterized as the slowest speed of a non-constant traveling wave. To illustrate our results, we numerically simulate a series of traveling waves. The simulations indicate that, depending on the properties of the fecundity function, the tails of the waves may approach the carrying capacity monotonically, may approach the carrying capacity in an oscillatory manner, or may oscillate continually about the carrying capacity, with its values bounded above and below by computable positive numbers. B. Li’s research was partially supported by the National Science Foundation under Grant DMS-616445. M. A. Lewis research was supported by “The Canada Research Chairs program,” and a grant from the Natural Sciences and Engineering Research Council of Canada.     

Spreading speeds as slowest wave speeds for cooperative systems

It is well known that in many scalar models for the spread of a fitter phenotype or species into the territory of a less fit one, the asymptotic spreading speed can be characterized as the lowest speed of a suitable family of traveling waves of the model. Despite a general belief that multi-species (vector) models have the same property, we are unaware of any proof to support this belief. The present work establishes this result for a class of multi-species model of a kind studied by Lui [Biological growth and spread modeled by systems of recursions. I: Mathematical theory, Math. Biosci. 93 (1989) 269] and generalized by the authors [Weinberger et al., Analysis of the linear conjecture for spread in cooperative models, J. Math. Biol. 45 (2002) 183; Lewis et al., Spreading speeds and the linear conjecture for two-species competition models, J. Math. Biol. 45 (2002) 219]. Lui showed the existence of a single spreading speed c(*) for all species. For the systems in the two aforementioned studies by the authors, which include related continuous-time models such as reaction-diffusion systems, as well as some standard competition models, it sometimes happens that different species spread at different rates, so that there are a slowest speed c(*) and a fastest speed c(f)(*). It is shown here that, for a large class of such multi-species systems, the slowest spreading speed c(*) is always characterized as the slowest speed of a class of traveling wave solutions.     

Spreading speed and linear determinacy for two-species competition models

 One crucial measure of a species' invasiveness is the rate at which it spreads into a competitor's environment. A heuristic spread rate formula for a spatially explicit, two-species competition model relies on `linear determinacy' which equates spread rate in the full nonlinear model with spread rate in the system linearized about the leading edge of the invasion. However, linear determinacy is not always valid for two-species competition; it has been shown numerically that the formula only works for certain values of model parameters when the model is diffusive Lotka-Volterra competition [2]. This paper derives a set of sufficient conditions for linear determinacy in spatially explicit two-species competition models. These conditions can be interpreted as requiring sufficiently large dispersal of the invader relative to dispersal of the out-competed resident and sufficiently weak interactions between the resident and the invader. When these conditions are not satisfied, spread rate may exceed linearly determined predictions. The mathematical methods rely on the application of results established in a companion paper [11]. Received: 7 August 2000 / Revised version: 5 January 2002 / Published online: 17 July 2002     

Using mean first passage times to quantify equilibrium resilience in perturbed intraguild predation systems

Regime shift inducibility depends on equilibrium resilience, which depends on species interactions. When species interactions include intraguild predation (IGP), integrated pest management may induce regime shifts because enhancing the abundance of intraguild predators simultaneously increases competition with, and predation on, invasive prey. To explore the dynamical consequences of such manipulations, we use a bistable, deterministic IGP model with stochastic removals that perturb invader density from the high-density equilibrium. We quantify the combined effects of IGP and such perturbations in terms of mean first passage times (MFPTs) to target invader densities such as thresholds between regimes. Analytical MFPTs compare favorably with those generated by Monte Carlo numerical solutions of the stochastically perturbed IGP model. MFPTs can therefore usefully quantify equilibrium resilience in terms of perturbation schedules. Electronic supplementary material The online version of this article (doi:) contains supplementary material, which is available to authorized users.     

The existence of globally stable equilibria of ecosystems of the generalized Volterra type

In this paper, global asymptotic stability of ecosystems of the generalized Volterra type $$dx_i /dt = x_i \left( {b_{i - } \mathop \sum \limits_{j = 1}^n a_{ij} x_j } \right),{\text{ }}i = 1,...,n,$$ is investigated. We obtain the conditions for the existence of a nonnegative and stable equilibrium point of the system by applying a result of linear complementarity theory. The results of this paper show that there exists a class of systems that do not have multiple domains of attractions. This class is defined in terms of the species interactions alone, and does not involve carrying capacities or species net birth rates.     

Asymptotic Enumeration of RNA Structures with Pseudoknots

In this paper, we present the asymptotic enumeration of RNA structures with pseudoknots. We develop a general framework for the computation of exponential growth rate and the asymptotic expansion for the numbers of k-noncrossing RNA structures. Our results are based on the generating function for the number of k-noncrossing RNA pseudoknot structures, , derived in Bull. Math. Biol. (2008), where k−1 denotes the maximal size of sets of mutually intersecting bonds. We prove a functional equation for the generating function and obtain for k=2 and k=3, the analytic continuation and singular expansions, respectively. It is implicit in our results that for arbitrary k singular expansions exist and via transfer theorems of analytic combinatorics, we obtain asymptotic expression for the coefficients. We explicitly derive the asymptotic expressions for 2- and 3-noncrossing RNA structures. Our main result is the derivation of the formula .     

Impact assessment revisited: improving the theoretical basis for management of invasive alien species

The theoretical underpinnings of the assessment of invasive alien species impacts need to be improved. At present most approaches are unreliable to quantify impact at regional scales and do not allow for comparison of different invasive species. There are four basic problems that need to be addressed: (1) Some impacted ecosystem traits are spatially not additive; (2) invader effects may increase non-linearly with abundance or there may be effect thresholds impairing estimates of linear impact models; (3) the abundance and impact of alien species will often co-vary with environmental variation; and (4) the total invaded range is an inappropriate measure for quantifying regional impact because the habitat area available for invasion can vary markedly among invasive species. Mathematical models and empirical data using an invasive alien plant species (Heracleum mantegazzianum) indicate that ignoring these issues leads to impact estimates almost an order of magnitude from the real values. Thus, we propose a habitat-sensitive formula for regional impact assessment that is unaffected by non-linearity. Furthermore, we make some statistical suggestions on how to assess invader effects properly and we discuss the quantification of the invaded range. These improvements are crucial for impact assessment with the overall aim of prioritizing management of invasive species.     

Species diversity reduces invasion success in pathogen‐regulated communities

The loss of natural enemies is thought to explain why certain invasive species are so spectacularly successful in their introduced range. However, if losing natural enemies leads to unregulated population growth, this implies that native species are themselves normally subject to natural enemy regulation. One possible widespread mechanism of natural enemy regulation is negative soil feedbacks, in which resident species growing on home soils are disadvantaged because of a build‐up of species‐specific soil pathogens. Here we construct simple models in which pathogens cause resident species to suffer reduced competitive ability on home soils and consider the consequences of such pathogen regulation for potential invading species. We show that the probability of successful invasion and its timescale depend strongly on the competitive ability of the invader on resident soils, but are unaffected by whether or not the invader also suffers reduced competitive ability on home soils (i.e. pathogen regulation). This is because, at the start of an invasion, the invader is rare and hence mostly encounters resident soils. However, the lack of pathogen regulation does allow the invader to achieve an unusually high population density. We also show that increasing resident species diversity in a pathogen‐regulated community increases invasion resistance by reducing the frequency of home‐site encounters. Diverse communities are more resistant to invasion than monocultures of the component species: they preclude a greater range of potential invaders, slow the timescale of invasion and reduce invader population size. Thus, widespread pathogen regulation of resident species is a potential explanation for the empirical observation that diverse communities are more invasion resistant.     

Competitive displacement or biotic resistance? Disentangling relationships between community diversity and invasion success of tall herbs and shrubs

Questions: Are negative invasion–diversity relationships due to biotic resistance of the invaded plant community or to post‐invasion displacement of less competitive species? Do invasion–diversity relationships change with habitat type or resident traits? Location/species: Lowlands and uplands of western and southern Germany, Heracleum mantegazzianum; mountain range in central Germany, Lupinus polyphyllus; and coastal dunes of northwest Germany, Rosa rugosa. Methods: We tested the significance and estimated regression slopes of invasion–diversity relationships using generalized linear (mixed effects) models relating invader cover and habitat type to species richness in different plant groups, stratified based on size, life cycle and community association. Results: We found negative, positive and neutral relationships between invader cover and species richness. There were negative linear correlations of invader cover with small plant species throughout, but no negative linear correlation with tall species. Invasion–diversity relationships tended to be more negative in early‐successional habitats, such as dunes or abandoned grasslands, than in late‐successional habitats. Conclusions: Invasion diversity–relationships are complex; they vary among habitat types and among different groups of resident species. Negative invasion–diversity relationships are due to asymmetric competitive displacement of inferior species and not due to biotic resistance. Small species are displaced in early‐successional habitats, while there is little effect on persistence of tall species.     

A Measure of Plankton Community Similarity Utilizing Variable Weighting of Total Density and Taxonomic Proportionality

Four commonly used formulae for measuring percentage similarity (PS) of biological communities were tested for their usefulness in relating to two plankton community properties, species proportional differences and total density differences. The formula best combining species proportionality and total density in the expression of PS is new: where min (xi,yi) is the lesser percentage (doubly standardized) of a species in two samples X and Y and where 2 q, 2xi and 2yi are the total quantities of all species in samples 8,X and Y, where \documentclass{article}\pagestyle{empty}\begin{document}$ \sum\limits_i {z_i } ,\,\sum\limits_i {x_i } \,and\sum\limits_i {y_i } $\end{document} are the total quantities of all species in samples Z, X and Y, respectively. Sample 2 contains the highest density of all species in the set; \documentclass{article}\pagestyle{empty}\begin{document}$ \sum\limits_i {z_i \, > \,(\sum\limits_i {x_i ,\,} \sum\limits_i {y_i } )} $\end{document}. The new expression of PS is simple to use and has the additional advantage of offering the analyst an unlimited choice of weighting factors or importance values for proportionality of species content and total density. The method has been applied to data from Gravenhurst Bay (Ontario) and effectively demonstrates the consequences of phosphorus loading reductions for phytoplankton communities.     

Vessel distensibility and flow distribution in vascular trees

 In a class of model vascular trees having distensible blood vessels, we prove that flow partitioning throughout the tree remains constant, independent of the nonzero driving flow (or nonzero inlet to terminal outlet pressure difference). Underlying assumptions are: (1) every vessel in the tree exhibits the same distensibility relationship given by $D/D_0 = f(P)$ where $D$ is the diameter which results from distending pressure $P$ and $D_0$ is the diameter of the individual vessel at zero pressure (each vessel may have its own individual $D_0$). The choice of $f(P)$ includes distensibilities often used in vessel biomechanics modeling, e.g., $f(P) = 1 + \alpha P$ or $f(P) = b + (1-b) \exp(-c P)$, as well as $f(P)$ which exhibit autoregulatory behavior. (2) Every terminal vessel in the tree is subjected to the same terminal outlet pressure. (3) Bernoulli effects are ignored. (4) Flow is nonpulsatile. (5) Blood viscosity within any individual vessel is constant. The results imply that for a vascular tree consistent with assumptions 2–5, the flow distribution calculations based on a rigid geometry, e.g., $D=D_0$, also gives the flow distribution when assuming the common distensibility relationships. Received: 30 October 2001 / Published online: 14 March 2002     

Global analysis on delay epidemiological dynamic models with nonlinear incidence

In this paper, we derive and study the classical SIR, SIS, SEIR and SEI models of epidemiological dynamics with time delays and a general incidence rate. By constructing Lyapunov functionals, the global asymptotic stability of the disease-free equilibrium and the endemic equilibrium is shown. This analysis extends and develops further our previous results and can be applied to the other biological dynamics, including such as single species population delay models and chemostat models with delay response.     

Competition between similar invasive species: modeling invasional interference across a landscape

As the number of biological invasions increases, interactions between different invasive species will become increasingly important. Several studies have examined facilitative invader–invader interactions, potentially leading to invasional meltdown. However, if invader interactions are negative, invasional interference may lead to lower invader abundance and spread. To explore this possibility, we develop models of two competing invaders. A landscape simulation model examines the patterns created by two such species invading into the same region. We then apply the model to a case study of Carduus nutans L. and C. acanthoides L., two economically important invasive weeds that exhibit a spatially segregated distribution in central Pennsylvania, USA. The results of these spatially-explicit models are generally consistent with the results of classic Lotka–Volterra competition models, with widespread coexistence predicted if interspecific effects are weaker than intraspecific effects for both species. However, spatial segregation of the two species (with lower net densities and no further spread) may arise, particularly when interspecific competition is stronger than intraspecific competition. A moving area of overlap may result when one species is a superior competitor. In the Carduus system, our model suggests that invasional interference will lead to lower levels of each species when together, but a similar net level of thistle invasion due to the similarity of intra- and interspecific competition. Thus, invasional interference may have important implications for the distribution and management of invasive species.     

Evolutionarily stable strategy and invader strategy in matrix games

In this paper, we consider the concepts of evolutionarily stable strategy (ESS), neighborhood invader strategy (NIS) and global invader strategy (GIS) in single species with frequency-dependent interactions. We find some general relationships among the three concepts in matrix games. The main conclusion is that ESS and NIS are equivalent to each other and are both equivalent to local superiority; a strategy with global superiority must be a GIS; a GIS may not be equivalent to its global superiority in games with more than two players; and in any two-player matrix game a GIS is just equivalent to its global superiority. In two-player games, globally asymptotic stability in the replicator dynamics has also been shown. Equivalent conditions for the three concepts stated by payoff comparisons are given and are applied to examples involved.     

A Tale of Four “Carp”: Invasion Potential and Ecological Niche Modeling

Background

Invasive species are a serious problem in ecosystems, but are difficult to eradicate once established. Predictive methods can be key in determining which areas are of concern regarding invasion by such species to prevent establishment [1]. We assessed the geographic potential of four Eurasian cyprinid fishes (common carp, tench, grass carp, black carp) as invaders in North America via ecological niche modeling (ENM). These “carp” represent four stages of invasion of the continent (a long-established invader with a wide distribution, a long-established invader with a limited distribution, a spreading invader whose distribution is expanding, and a newly introduced potential invader that is not yet established), and as such illustrate the progressive reduction of distributional disequilibrium over the history of species'' invasions.

Methodology/Principal Findings

We used ENM to estimate the potential distributional area for each species in North America using models based on native range distribution data. Environmental data layers for native and introduced ranges were imported from state, national, and international climate and environmental databases. Models were evaluated using independent validation data on native and invaded areas. We calculated omission error for the independent validation data for each species: all native range tests were highly successful (all omission values <7%); invaded-range predictions were predictive for common and grass carp (omission values 8.8 and 19.8%, respectively). Model omission was high for introduced tench populations (54.7%), but the model correctly identified some areas where the species has been successful; distributional predictions for black carp show that large portions of eastern North America are at risk.

Conclusions/Significance

ENMs predicted potential ranges of carp species accurately even in regions where the species have not been present until recently. ENM can forecast species'' potential geographic ranges with reasonable precision and within the short screening time required by proposed U.S. invasive species legislation.     

Rates of decay of linkage disequilibrium under two-locus models of selection

The effect of selection and linkage on the decay of linkage disequilibrium, D, is investigated for a hierarchy of two-locus models. The method of analysis rests upon a qualitative classification of the dynamic of D under selection relative to the neutral dynamic. To eliminate the confounding effects of gene frequency change, the behavior of D is first studied with gene frequencies fixed at their invariant values. Second, the results are extended to certain special situations where gene frequencies are changing simultaneously.A wide variety of selection regimes can cause an acceleration of the rate of decay of D relative to the neutral rate. Specifically, the asymptotic rate of decay is always faster than the neutral rate in the neighborhood of a stable equilibrium point, when viabilities are additive or only one locus is selected. This is not necessarily the case for models in which there is nonzero additive epistasis. With multiplicative viabilities, decay is always accelerated near a stable boundary equilibrium, but decay is only faster near the stable central equilibrium (with = 0) if linkage is sufficiently loose. In the symmetric viability model, decay may even be retarded near a stable boundary equilibrium. Decay is only accelerated near a stable corner equilibrium when the double homozygote is more fit than the double heterozygotes. Decay near a stable edge equilibrium may be retarded if there is loose linkage. With symmetric viabilities there is usually an acceleration of the decay process for gene frequencies near 1/2 when the central equilibrium (with = 0) is stable. This is always the case when the sign of the epistasis is negative or zero.Conversely, the decay ofD is retarded in the neighborhood of a stable equilibrium in the multiplicative and symmetric viability models if any of the conditions above are violated. Near an unstable equilibrium of any of the models considered,D may either increase or decay at a rate slower than, equal to, or faster than the neutral rate. These analytic results are supplemented by numerical studies of the symmetric viability model.