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Simplified conditions are given for the existence and positivity of wave speed for an integro-difference equation with a strong Allee effect and an unbounded habitat. The results are used to obtain the existence of a critical patch size for an equation with a bounded habitat. It is shown that if the wave speed is positive there exists a critical patch size such that for a habitat size above the critical patch size solutions can persist in space, and if the wave speed is negative solutions always approach zero. An analytical integral formula is developed to determine the critical patch size when the Laplace dispersal kernel is used, and this formula shows existence of multiple equilibrium solutions. Numerical simulations are provided to demonstrate connections among the wave speed, critical patch size, and Allee threshold.

We consider an integro-difference model to study the effect of a stationary barrier zone on invasion of a population with a strong Allee effect. It is assumed that inside the barrier zone a certain proportion of the population is killed. A Laplace dispersal kernel is used in the model. We provide a formula for the critical width [Formula: see text] of barrier zone. We show that when a barrier zone is set at the front of a population, if the width of barrier zone is bigger than [Formula: see text] then the barrier zone can stop the population invasion, and if the width of barrier zone is less than [Formula: see text] then the population crosses the barrier zone and eventually occupies the entire space. The results are proven by establishing the existence and attractivity of three types of equilibrium solutions. The mathematical proofs involve phase plane analysis and comparison.

We study an integro-difference competition model for the case that two species consecutively invade a habitat. We show that if a species spreads into a traveling wave of its rival, or if two species expand their spatial ranges in both directions, in a direction where open space is available, the species with larger invasion speed can always establish a wave moving into open space with its own speed. We demonstrate that when one species is stronger in competition, under appropriate conditions, the speeds at which the boundaries between two species move can be analytically determined. We find that in general there are multiple invasion speeds in the model. It is possible for a species to develop two separate waves propagating with different invasion speeds. It is also possible for each species to establish a single wave spreading with distinct speeds in both directions. The mathematical analysis relies on linear determinacy and new techniques developed.

We study a model that describes the spatial spread of a species along a habitat gradient on which the species' growth increases. Mathematical analysis is provided to determine the spreading dynamics of the model. We demonstrate that the species may succeed or fail in local invasion depending on the species' growth function and dispersal kernel. We delineate the conditions under which a spreading species may be stopped by poor quality habitat, and demonstrate how a species can escape a region of poor quality habitat by climbing a resource gradient to good quality habitat where it spreads at a constant spreading speed. We show that dispersal may take the species from a good quality region to a poor quality region where the species becomes extinct. We also provide formulas for spreading speeds for the model that are determined by the dispersal kernel and linearized growth rates in both directions.

The spread of an invasive species often results in a decline and subsequent disappearance of native competitors. Several models, primarily based on spatially explicit Lotka–Volterra competition dynamics, have been developed to understand this phenomenon. In general, the goal of these models is to relate fundamental life history traits, for example dispersal ability and competition strength, to the rate of spread of the invasive species, which is also the rate at which the invasive species displaces its native competitor. Stage-structure is often an important determinant of population dynamics, but it has received little attention within the context of Lotka–Volterra invasion models. For many species, behaviors like dispersal and competition depend on life-stage. To describe the processes of invasion in these species, it is important to understand how behaviors that vary as a function of life-stage can impact spread rate. In this paper, we develop a spatially explicit, stage-structured Lotka–Volterra competition model. By comparing spread speed predictions from this model to spread speed predictions from an analogous single-stage model, we are able to determine when stage-structure is important and how stage-dependent behavior can alter the characteristics of an invasion.

The process of invasion is fundamental to the study of the dynamics of ecological and epidemiological systems. Quantitatively, a crucial measure of species' invasiveness is given by the rate at which it spreads into new open environments. The so-called ``linear determinacy'' conjecture equates full nonlinear model spread rates with the spread rates computed from linearized systems with the linearization carried out around the leading edge of the invasion. A survey that accounts for recent developments in the identification of conditions under which linear determinacy gives the ``right" answer, particularly in the context of non-compact and non-cooperative systems, is the thrust of this contribution. Novel results that extend some of the research linked to some the contributions covered in this survey are also discussed.

The theory of invasions and invasion speeds has traditionally been studied in macroscopic systems. Surprisingly, microbial invasions have received less attention. Although microbes share many of the features associated with competition between larger-bodied organisms, they also exhibit distinctive behaviors that require new mathematical treatments to fully understand invasions in microbial systems. Most notable is the possibility for long-distance interactions, including competition between populations mediated by diffusible toxins and cooperation among individuals of a single population using quorum sensing. In this paper, we model bacterial invasion using a system of coupled partial differential equations based on Fisher's equation. Our model considers a competitive system with diffusible toxins that, in some cases, are expressed in response to quorum sensing. First, we derive analytical approximations for invasion speeds in the limits of fast and slow toxin diffusion. We then test the validity of our analytical approximations and explore intermediate rates of toxin diffusion using numerical simulations. Interestingly, we find that toxins should diffuse quickly when used offensively, but that there are two optimal strategies when toxins are used as a defense mechanism. Specifically, toxins should diffuse quickly when their killing efficacy is high, but should diffuse slowly when their killing efficacy is low. Our approach permits an explicit investigation of the properties and characteristics of diffusible compounds used in non-local competition, and is relevant for microbial systems and select macroscopic taxa, such as plants and corals, that can interact through biochemicals.

We propose an integro-difference equation model to predict the spatial spread of a plant population with a seed bank. The formulation of the model consists of a nonmonotone convolution integral operator describing the recruitment and seed dispersal and a linear contraction operator addressing the effect of the seed bank. The recursion operator of the model is noncompact, which poses a challenge to establishing the existence of traveling wave solutions. We show that the model has a spreading speed, and prove that the spreading speed can be characterized as the slowest speed of a class of traveling wave solutions by using an asymptotic fixed point theorem. Our numerical simulations show that the seed bank has the stabilizing effect on the spatial patterns of traveling wave solutions.

We study a chemostat model that describes competition between two species for two essential resources based on storage. The model incorporates internal resource storage variables that serve the direct connection between species growth and external resource availability. Mathematical analysis for the global dynamics of the model is carried out by using the monotone dynamical system theory. It is shown that the limiting system of the model basically exhibits the familiar Lotka-Volterra alternatives: competitive exclusion, coexistence, and bi-stability, and most of these results can be carried over to the original model.

We provide rigorous analysis for a discrete-time model composed of the Ricker function and Beverton–Holt function. This model was proposed by Lewis and Li [Bull. Math. Biol. 74 (2012) 2383–2402] in the study of a population in which reproduction occurs at a discrete instant of time whereas death and competition take place continuously during the season. We show analytically that there exists a period-doubling bifurcation curve in the model. The bifurcation curve divides the parameter space into the region of stability and the region of instability. We demonstrate through numerical bifurcation diagrams that the regions of periodic cycles are intermixed with the regions of chaos. We also study the global stability of the model.

We show the existence of a periodic solution in which four species coexist in competition for three essential resources in the standard model of resource competition. By assuming that species i is limited by resource i for each i near the positive equilibrium, and that the matrix of contents of resources in species is a combination of cyclic matrix and a symmetric matrix, we obtain an asymptotically stable periodic solution of three species on three resources via Hopf bifurcation. A simple bifurcation argument is then employed which allows us to add a fourth species. In principle, the argument can be continued to obtain a periodic solution adding one new species at a time so long as asymptotic stability can be assured at each step. Numerical simulations are provided to illustrate our analytical results. The results of this paper suggest that competition can generate coexistence of species in the form of periodic cycles, and that the number of coexisting species can exceed the number of resources in a constant and homogeneous environment.

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].

In this study, we expand on the susceptible-infected-susceptible (SIS) heterosexual mixing setting by including the movement of individuals of both genders in a spatial domain in order to more comprehensively address the transmission dynamics of competing strains of sexually-transmitted pathogens. In prior models, these transmission dynamics have only been studied in the context of nonexplicitly mobile heterosexually active populations at the demographic steady state, or, explicitly in the simplest context of SIS frameworks whose limiting systems are order preserving. We introduce reaction-diffusion equations to study the dynamics of sexually-transmitted diseases (STDs) in spatially mobile heterosexually active populations. To accomplish this, we study a single-strain STD model, and discuss in what forms and at what speed the disease spreads to noninfected regions as it expands its spatial range. The dynamics of two competing distinct strains of the same pathogen on this population are then considered. The focus is on the investigation of the spatial transition dynamics between the two endemic equilibria supported by the nonspatial corresponding model. We establish conditions for the successful invasion of a population living in endemic conditions by introducing a strain with higher fitness. It is shown that there exists a unique spreading speed (where the spreading speed is characterized as the slowest speed of a class of traveling waves connecting two endemic equilibria) at which the infectious population carrying the invading stronger strain spreads into the space where an equilibrium distribution has been established by the population with the weaker strain. Finally, we give sufficient conditions under which an explicit formula for the spreading speed can be found.

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.

This work presents an example of a cooperative system of truncated linear recursions in which the interaction between species causes one of the species to have an anomalous spreading speed. By this we mean that this species spreads at a speed which is strictly greater than its spreading speed in isolation from the other species and the speeds at which all the other species actually spread. An ecological implication of this example is discussed in Sect. 5. Our example shows that the formula for the fastest spreading speed given in Lemma 2.3 of our paper (Weinberger et al. in J Math Biol 45:183–218, 2002) is incorrect. However, we find an extra hypothesis under which the formula for the faster spreading speed given in (Weinberger et al. in J Math Biol 45:183–218, 2002) is valid. We also show that the hypotheses of all but one of the theorems of (Weinberger et al. in J Math Biol 45:183–218, 2002) whose proofs rely on Lemma 2.3 imply this extra hypothesis, so that all but one of the theorems of (Weinberger et al. in J Math Biol 45:183–218, 2002) and all the examples given there are valid as they stand.

We consider the dynamics of the standard model of 3 species competing for 3 essential (non-substitutable) resources in a chemostat using Liebig's law of the minimum functional response. A subset of these systems which possess cyclic symmetry such that its three single-population equilibria are part of a heteroclinic cycle bounding the two-dimensional carrying simplex is examined. We show that a subcritical Hopf bifurcation from the coexistence equilibrium together with a repelling heteroclinic cycle leads to the existence of at least two limit cycles enclosing the coexistence equilibrium on the carrying simplex- the ``inside'' one is an unstable separatrix and the ``outside'' one is at least semi-stable relative to the carrying simplex. Numerical simulations suggest that there are exactly two limit cycles and that almost every positive solution approaches either the stable limit cycle or the stable coexistence equilibrium, depending on initial conditions. Bifurcation diagrams confirm this picture and show additional features. In an alternative scenario, we show that the subcritical Hopf together with an attracting heteroclinic cycle leads to an unstable periodic orbit separatrix.

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.

The discrete-time recursion system $\u_{n+1}=Q[\u_n]$ with $\u_n(x)$ a vector of population distributions of species and $Q$ an operator which models the growth, interaction, and migration of the species is considered. Previously known results are extended so that one can treat the local invasion of an equilibrium of cooperating species by a new species or mutant. It is found that, in general, the resulting change in the equilibrium density of each species spreads at its own asymptotic speed, with the speed of the invader the slowest of the speeds. Conditions on $Q$ are given which insure that all species spread at the same asymptotic speed, and that this speed agrees with the more easily calculated speed of a linearized problem for the invader alone. If this is true we say that the recursion has a single speed and is linearly determinate. The conditions are such that they can be verified for a class of reaction-diffusion models.

A chemostat model of three species of microorganisms competing for three essential, growth-limiting nutrients is considered. J. Husiman and F.J. Weissing [Nature 402 (1999) 407] show numerically that this model can generate periodic oscillations. The present contribution is concerned with rigorous analysis regarding the existence of periodic oscillations in this model. Our analysis is based on the following observation made by Huisman and Weissing: there is a cyclic replacement of species, if each species becomes limited by the resource for which it is the intermediate competitor. Using a permanence theory, an index theory, and a Poincare–Bendixson theory for three-dimensional competitive systems, we analytically succeed to give sufficient conditions for the existence of periodic orbits in the limit sets in this model. The results in this paper suggest that with a wide range of parameter values, sustained periodic oscillations of species abundances for the model are possible, without involving external disturbances. Our results also suggest that competition is not necessarily destructive, i.e., in the case of existence of sustained periodic oscillations, if one of three competitors is absent, one of the other two rivals cannot survive.