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 speeds and traveling waves in competitive recursion systems

This paper is concerned with the spreading speeds and traveling wave solutions of discrete time recursion systems, which describe the spatial propagation mode of two competitive invaders. We first establish the existence of traveling wave solutions when the wave speed is larger than a given threshold. Furthermore, we prove that the threshold is the spreading speed of one species while the spreading speed of the other species is distinctly slower compared to the case when the interspecific competition disappears. Our results also show that the interspecific competition does affect the spread of both species so that the eventual population densities at the coexistence domain are lower than the case when the competition vanishes.     

The minimal speed of traveling fronts for a diffusive Lotka-Volterra competition model

This paper concerns the minimal speed of traveling wave fronts for a two-species diffusion-competition model of the Lotka-Volterra type. An earlier paper used this model to discuss the speed of invasion of the gray squirrel by estimating the model parameters from field data, and predicted its speed by the use of a heuristic analytical argument. We discuss the conditions which assure the validity of their argument and show numerically the existence of the realistic range of parameter values for which their heuristic argument does not hold. Especially for the case of the strong interaction of two competing species compared with the intraspecific competition, we show that all parameters appearing in the system affect the minimal speed of invasion. Dedicated to the Memory of Akira Okubo     

Conditions for the existence of stationary densities for some two-dimensional diffusion processes with applications in population biology

Conditions are derived that we conjecture are necessary and sufficient for the existence of stationary densities for a class of two-dimensional diffusion processes. The derivation of the conditions rests on the assumption that a two-dimensional stationary density (which can be viewed as a stable “internal equilibrium”) exists if and only if all “boundary equilibria” are unstable in the sense that small perturbations lead to moving away from the boundaries with high probability. For the models considered, the boundary equilibria are one-dimensional stationary densities and equilibrium points. To demonstrate the usefulness of the conditions, three random environment models are analyzed: a three-allele selection model, a two-species competition model, and a two-locus selection model. Several of the results obtained have been verified by alternate methods.     

Autocatalysis in Reaction Networks

The persistence conjecture is a long-standing open problem in chemical reaction network theory. It concerns the behavior of solutions to coupled ODE systems that arise from applying mass-action kinetics to a network of chemical reactions. The idea is that if all reactions are reversible in a weak sense, then no species can go extinct. A notion that has been found useful in thinking about persistence is that of “critical siphon.” We explore the combinatorics of critical siphons, with a view toward the persistence conjecture. We introduce the notions of “drainable” and “self-replicable” (or autocatalytic) siphons. We show that: Every minimal critical siphon is either drainable or self-replicable; reaction networks without drainable siphons are persistent; and nonautocatalytic weakly reversible networks are persistent. Our results clarify that the difficulties in proving the persistence conjecture are essentially due to competition between drainable and self-replicable siphons.     

Advantages and disadvantages of interference‐competitive ability and resource‐use efficiency when invading established communities

Invaders into established communities must overcome low resource availability. To establish, invaders must either appropriate resources from existing individuals through interference competition or efficiently use the small amount of resource that remains. Although both strategies may be important, they are rarely considered together and, in particular, resource‐use efficiency is often ignored in systems dominated by interference competition. To identify the traits that confer invasion success, we experimentally invaded resource patches in established communities with multiple species from two functional groups that differ in interference competitive ability and resource‐use efficiency. In contrast to previous assessments, we show that resource‐use efficiency can facilitate invasion in systems dominated by interference competition. Furthermore, large resource requirements can be a liability when establishing because interference competition is inherently costly and so cannot fully compensate for limitations in the primary resource. However, we also show that there is a tradeoff in performance among functional groups between small and large resource gaps. Our results suggest we modify the way we view and manage species invasion in systems dominated by interference competition.     

Distinct Ca2+ channels maintain a high motility state of the sperm that may be needed for penetration of egg jelly of the newt,Cynops pyrrhogaster

Activation state of sperm motility named “hyperactivation” enables mammalian sperm to progress through the oviductal matrix, although a similar state of sperm motility is unknown in non‐mammalian vertebrates at fertilization. Here, we found a high motility state of the sperm in the newt Cynops pyrrhogaster. It was predominantly caused in egg jelly extract (JE) and characterized by a high wave velocity of the undulating membrane (UM) that was significantly higher at the posterior midpiece. An insemination assay suggested that the high motility state might be needed for sperm to penetrate the egg jelly, which is the accumulated oviductal matrix. Specific characteristics of the high motility state were completely abrogated by a high concentration of verapamil, which blocks the L‐type and T‐type voltage‐dependent Ca²⁺ channels (VDCCs). Mibefradil, a dominant blocker of T‐type VDCCs, suppressed the wave of the UM at the posterior midpiece with separate wave propagation from both the anterior midpiece and the posterior principal piece. In addition, nitrendipine, a dominant L‐type VDCC blocker, weakened the wave of the UM, especially in the anterior midpiece. Live Ca²⁺ imaging showed that, compared with the intact sperm in the JE, the relative intracellular Ca²⁺ level changed especially in the anterior and posterior ends of the midpiece of the blocker‐treated sperm. These suggest that different types of Ca²⁺ channels mediate the intracellular Ca²⁺ level predominantly in the anterior and posterior ends of the midpiece to maintain the high motility state of the newt sperm.     

Multi-scale methods predict invasion speeds in variable landscapes

Spread rates of invasive plant species depend heavily on variable seed/seedling survivorships over various habitat types as well as on variability in seed dispersal induced by rapid transport of propagules in open areas and slow transport in vegetated areas. The ability to capture spatial variability in seed survivorship and dispersal is crucial to accurately predict the rate of spread of plants in real world landscapes. However, current analytic methods for predicting spread rates are not suited for arbitrary, spatially heterogeneous systems. Here, we analyze invasion rates of the invasive plant Phragmites australis (common reed) over variable wetland landscapes. Phragmites is one of the most pervasive perennial grasses, outcompeting native vegetation, providing poor wildlife habitat, and proving difficult to eradicate across its invasive range in North America. Phragmites spreads sexually via seeds and asexually via underground (rhizomes) and aboveground (stolons) stems. We construct a structured integrodifference equation model of the Phragmites life cycle capturing variable seed survivorship in a seed bank, sexual and asexual recruitment into a juvenile age class, and differential competition among all classes with adults. The demographic model is coupled with a homogenized ecological diffusion/settling seed dispersal model that allows for seed deposition that varies with habitat type. The dispersal kernel we develop does not require local normalization and can be implemented efficiently using standard computational techniques. The model generates a traveling wave of isolated patches, establishing only in suitable habitats. We use the method of multiple scales to predict invasion speed as a solvability condition at large scales and test the predictions numerically. Accurate predictions are generated for a wide range of landscape parameters, indicating that invasion speeds can be understood in landscapes of arbitrary structure using this approach.     

Species packing and the competition function with illustrations from coral reef fish

This paper concerns the contrast between guilds whose species show resource partitioning and those that show extensive overlap. Using a Lotka-Volterra model, the ease of invasion by a third species into a guild already containing two species is examined for various shapes of resource utilization curves. I show that (a) a guild is more easily invasible and allows tighter packing if its member species have leptokurtic (thick-tailed) resource utilization curves than if they have platykurtic (thin-tailed) curves; (b) the distribution of niche separation distances is bimodal in a “thin-tailed” guild and is unimodal in a “thick-tailed” guild; (c) there are three-species guilds such that removal of one particular species leaves a two-species system in which one of the remaining species excludes the other. In this context, competition pressure is a force maintaining species diversity.Groupers (Serranidae) appear to be a thin-tailed guild, and Parrotfish (Scaridae) and Surgeonfish (Acanthuridae) together appear to be a thick-tailed guild, and these guilds show many properties predicted by the model. I conjecture that thick-tailed guilds form when the constituent species are selected to be generalists and apply this idea to tropical fruit- and flower-feeding birds.     

Effect of <Emphasis Type="Italic">Diachasmimorpha longicaudata</Emphasis> releases on the native parasitoid guild attacking <Emphasis Type="Italic">Anastrepha</Emphasis> spp. larvae in disturbed zones of Chiapas,Mexico

We evaluated the effect of augmentative releases of Diachasmimorpha longicaudata (Ashmead), on the native parasitoid guild of Anastrepha spp. over a two year period in zones adjacent to mango commercial orchards in Chiapas, Mexico. We chose two 15 ha working zones, 15 km apart, harbouring fruit fly hosts of varying densities without chemical control. In 2013, parasitoids were released in zone “A” while zone “B” served as control. In 2014 zones were exchanged. As expected, releases of D. longicaudata significantly increased total parasitism, from around 0.5–5% to over 22%, but annual parasitism by native parasitoids was only significantly affected in zone “A”. The numbers of native parasitoids were higher in zone “A” in both years, and diversity (H′) was not affected by D. longicaudata releases in both zones. Our results suggest that releases of D. longicaudata affect the relative abundance but not the species richness of native parasitoids.     

Coexistence and displacement in consumer-resource systems with local and shared resources

Competition for local and shared resources is widespread. For example, colonial waterbirds consume local prey in the immediate vicinity of their colony, as well as shared prey across multiple colonies. However, there is little understanding of conditions facilitating coexistence vs. displacement in such systems. Extending traditional models based on type I and type II functional responses, we simulate consumer-resource systems in which resources are “substitutable,” “essential,” or “complementary.” It is shown that when resources are complementary or essential, a small increase in carrying capacity or decrease in handling time of a local resource may displace a spatially separate consumer species, even when the effect on shared resources is small. This work underscores the importance of determining both the nature of resource competition (substitutable, essential, or complementary) and appropriate scale-dependencies when studying metacommunities. We discuss model applicability to complex systems, e.g., urban wildlife that consume natural and anthropogenic resources which may displace rural competitors by depleting shared prey.     

Dispersal and competition models for plants

New models for seed dispersal and competition between plant species are formulated and analyzed. The models are integrodifference equations, discrete in time and continuous in space, and have applications to annual and perennial species. The spread or invasion of a single plant species into a geographic region is investigated by studying the travelling wave solutions of these equations. Travelling wave solutions are shown to exist in the single-species models and are compared numerically. The asymptotic wave speed is calculated for various parameter values. The single-species integrodifference equations are extended to a model for two competing annual plants. Competition in the two-species model is based on a difference equation model developed by Pakes and Maller [26]. The two-species model with competition and dispersal yields a system of integrodifference equations. The effects of competition on the travelling wave solutions of invading plant species is investigated numerically.     

SPERM COMPETITION AND THE EVOLUTION OF SEMINAL FLUID COMPOSITION

Male ejaculates include large amounts of seminal fluid proteins (Sfps) that influence male sperm competitive success. In spite of their diverse proximate functions, Sfps involved in sperm competition increase male fitness in one of three ways: (1) “avoidance” proteins help males avoid sperm competition, (2) “defense” proteins help males defend their sperm from displacement by the female's subsequent mate, and (3) “offense” proteins aid males in displacing sperm of preceding males. Here, we present a population genetic model of the evolution of allocation of finite resources by males to the three kinds of Sfps. We analyze the influence of relative efficiencies of different Sfps, of plasticity in resource allocation, and of differences in viability costs of Sfps. We find that in absence of plasticity or different viability costs, equal investment in defense and offense Sfps evolves, irrespective of their relative efficiency. In all cases, males evolve to invest more in avoidance when avoidance proteins are increasingly efficient, and when offense is more efficient than defense. Differences in viability costs result in lower investment in costly proteins, whereas plasticity has complex effects, influencing both the optimal seminal fluid composition and maintenance of variation in investment in these proteins across populations.     

No evidence of sperm conjugate formation in an Australian mouse bearing sperm with three hooks

Sperm conjugation occurs when two or more sperm physically unite for motility or transport through the female reproductive tract. In many muroid rodent species, sperm conjugates have been shown to form by a single, conspicuous apical hook located on the sperm head. These sperm “trains” have been reported to be highly variable in size and, despite all the heads pointing in roughly the same direction, exhibit a relatively disordered arrangement. In some species, sperm “trains” have been shown to enhance sperm swimming speed, and thus have been suggested to be advantageous in sperm competition. Here, we assessed the behavior of sperm in the sandy inland mouse (Pseudomys hermannsburgensis), a muroid rodent that bears sperm with three apical hooks. First, we accrued genetic evidence of multiple paternity within “wild” litters to unequivocally show that sperm competition does occur in this species. Following this we utilized both in vitro and in vivo methodologies to determine whether sandy inland mouse sperm conjugate to form motile trains. Our observations of in vitro preparations of active sperm revealed that sandy inland mouse sperm exhibit rapid, progressive motility as individual cells only. Similarly, histological sections of the reproductive tracts of mated females revealed no in vivo evidence of sperm conjugate formation. We conclude that the unique, three‐hooked morphology of the sandy inland mouse sperm does not facilitate the formation of motile conjugates, and discuss our findings in relation to the different hypotheses for the evolution of the muroid rodent hook/s.     

Quantitative studies of bacterial chemotaxis and microbial population dynamics

Although there is a long history of conjecture regarding the role and significance of bacterial chemotaxis in microbial ecology, only recently has a significant body of work appeared attempting to address this issue. The purpose of this paper is to provide a concise overview of this work, which combined mathematical modeling of bacterial population migration and experimental measurement of the model parameters with modeling of competitive microbial population dynamics in a nonmixed environment. Predictions from the population dynamics models, based on experimental estimates of the various motility and growth parameter values, are related to the small number of experimental observations available to date dealing with the effects of bacterial motility on competition in a nonmixed environment. Current results indicate that cell motility and chemotaxis properties can be as important to population dynamics as cell growth kinetic properties, so that greater attention to this aspect of microbial behavior is warranted in future studies of microbial ecology.     

The Enemy of My Enemy Hypothesis: Why Coexisting with Grasses May Be an Adaptive Strategy for Savanna Trees

Savannas are characterized by the coexistence of trees and flammable grasses. Yet, tree–grass coexistence has been labeled as paradoxical—how do these two functional groups coexist over such an extensive area, despite being generally predisposed to excluding each other? For instance, many trees develop dense canopies that limit grass growth, and many grasses facilitate frequent/intense fires, increasing tree mortality. This study revisits tree–grass coexistence with a model of hierarchical competition between pyrogenic grasses, “forest trees” adapted to closed-canopy competition, and “savanna trees” that are inferior competitors in closed-canopy communities, but more resistant to fire. The assumptions of this model are supported by empirical observations, including a systematic review of savanna and forest tree community composition reported here. In general, the model simulations show that when savanna trees exert weaker competitive effects on grasses, a self-reinforcing grass community is maintained, which limits forest tree expansion while still allowing savanna trees to persist (albeit as a subdominant to grasses). When savanna trees exert strong competitive effects on grasses, savanna trees cover increases initially, but as grasses decline their inhibitory effect on forest trees weakens, allowing forest trees to expand and exclude grasses and savanna trees. Rather than paradoxical, these results suggest that having weaker competitive effects on grasses may be advantageous for savanna trees, leading to greater long-term abundance and stability. We label this the “enemy of my enemy hypothesis,” which might apply to species coexistence in communities defined by hierarchical competition or with species capable of generating strong ecological feedbacks.     

Collective dynamics of cancer cells confined in a confluent monolayer of normal cells

Tumorigenesis often involves specific changes in cell motility and intercellular adhesion. Understanding the collective cancer cell behavior associated with these specific changes could facilitate the detection of malignant characteristics during tumor growth and invasion. In this study, a cellular vertex model is developed to investigate the collective dynamics of a disk-like aggregate of cancer cells confined in a confluent monolayer of normal cells. The effects of intercellular adhesion and cell motility on tumor progression are examined. It is found that the stresses in both the cancer cells and the normal cells increase with tumor growth, resulting in a crowded environment and enhanced cell apoptosis. The intercellular adhesion between cancer cells and normal cells is revealed to promote tumor growth and invasion. The tumor invasion dynamics hinges on the motility of cancer cells. The cancer cells could orchestrate into different collective migration modes, e.g., directional migration and rotational oscillations, dictated by the competition between cell persistence and local coordination. Phase diagrams are established to reveal the competitive mechanisms. This work highlights the role of mechanics in regulating tumor growth and invasion.     

Interactions between root and shoot competition vary among species

Understanding how the competition varies with productivity is essential for differentiating among alternative models of plant community organization. Prior attempts to explain shifts in root and shoot competition along gradients have generally assumed an additive interaction between the two competitive forms, using an experimental design which does not fully separate both above‐ and belowground processes. At the most basic level, few field studies have separated root and shoot competition, and we have limited knowledge about both the relative importance of these processes, and how they interact to affect plant growth in the field. Presented here are findings from a field study in which root and shoot competition were experimentally separated by using root exclusion tubes and neighbor tiebacks in an early successional community. Individuals of four species (Abutilon theophrasti, Amaranthus retroflexus, Rumex crispus, and Plantago lanceolata) were grown at two levels of fertilization with full competition, aboveground competition only, belowground competition only, or neither above‐ nor belowground competition. Competition was measured as competitive response, which is the natural log of the relative biomass of a target plant grown with competition compared to growth without competition. In contrast to predictions from current models of productivity‐competition relationships, but in agreement with other experimental studies, there was no change in the strengths or root, shoot, or total competition with a modest increase in productivity. Despite no effect of fertilization on the strength of competition, the form of interaction between root and shoot competition varied both as a function of species identity and fertilization. For both of the rosette forming species, the combined effects of root and shoot competition were less than predicted assuming no interaction (a “negative interaction”), with one species switching from a negative to an additive interaction with fertilization. The fact that fertilization caused a shift in the root‐shoot interaction, but not in the total strength of root and shoot competition, suggests that the root‐shoot interaction is itself a highly labile variable. If root‐shoot interactions are common in natural systems, then simply measuring the strength of one form of competition in no way provides any information about the overall importance of that competitive form to plant growth.     

Effect of the spatial context along the invasion process: “Hierarchical spatial” or “Host-switching spatial” hypotheses?

Very little is known about how spatial effects influence invasive species throughout the invasion sequence. We propose here two mechanisms to explain the changes in spatial effects throughout the stages of invasion, using the soybean aphid (Aphis glycines) as a model. First, the “hierarchical spatial effect” hypothesis, based on a change in the relative importance of the spatial scales throughout the invasion process, with main effect at broad scale during the first years of invasion, and main effect at local scale during the subsequent years. Second, the “host-switching spatial effect” hypothesis, stating that the spatial effect is driven by a switch in the effect of the host/habitat throughout the invasion process, from effect of main summer host/habitat during the first years of invasion to effect of overwintering host/habitat during the subsequent years. Data from governmental archives and field samplings enabled to investigate the spatial effects on aphid density at three scales (regional, landscape, local) during a 7 year period (2006–2012). Our results demonstrate that the hierarchical spatial effect hypothesis is not an adequate model for the soybean aphid, aphid density being more affected by landscape-scale factors irrespective of years. In contrast, our results are in accordance with the host-switching spatial hypothesis, with positive effect of the main summer host/habitat (soybean) during the first steps of invasion (2006–2008), followed by a positive effect of overwintering habitats (buckthorn, woodland) during the subsequent years (2010–2012). Overall, investigating these hypotheses in other systems would determine whether the same tendency is observed for other invasive species.