Global dynamics of a discrete two-species Lottery-Ricker competition model

In this article, we study the global dynamics of a discrete two-dimensional competition model. We give sufficient conditions on the persistence of one species and the existence of local asymptotically stable interior period-2 orbit for this system. Moreover, we show that for a certain parameter range, there exists a compact interior attractor that attracts all interior points except Lebesgue measure zero set. This result gives a weaker form of coexistence which is referred to as relative permanence. This new concept of coexistence combined with numerical simulations strongly suggests that the basin of attraction of the locally asymptotically stable interior period-2 orbit is an infinite union of connected components. This idea may apply to many other ecological models. Finally, we discuss the generic dynamical structure that gives relative permanence.     

Feedback control for chemostat models

 It is shown that a chemostat with two organisms can be made coexistent by means of feedback control of the dilution rate. Remaining freedom in the feedback law can be used to guarantee robustness or improve particular performance indices. Unfortunately a topological property prevents coexistence by feedback control for chemostats with more than two organisms. We apply our results to control bioreactors aimed at producing commercial products through genetically altered organisms. In all our results the coexistence takes its simplest form: a global asymptotically stable equilibrium point in the interior of the non-negative orthant. Received: 1 November 2001 / Revised version: 19 August 2002 / Published online: 19 November 2002 Mathematics Subject Classification (2000): 34H05, 34D23, 93C10 Keywords or phrases: Chemostat – Feedback control – Coexistence     

Stability analysis of pathogen-immune interaction dynamics

The paper considers models of dynamics of infectious disease in vivo from the standpoint of the mathematical analysis of stability. The models describe the interaction of the target cells, the pathogens, and the humoral immune response. The paper mainly focuses on the interior equilibrium, whose components are all positive. If the model ignores the absorption of the pathogens due to infection, the interior equilibrium is always asymptotically stable. On the other hand, if the model does consider it, the interior equilibrium can be unstable and a simple Hopf bifurcation can occur. A sufficient condition that the interior equilibrium is asymptotically stable is obtained. The condition explains that the interior equilibrium is asymptotically stable when experimental parameter values are used for the model. Moreover, the paper considers the models in which uninfected cells are involved in the immune response to pathogens, and are removed by the immune complexes. The effect of the involvement strongly affects the stability of the interior equilibria. The results are shown with the aid of symbolic calculation software.     

Mutualistic relationships between phytoplankton and bacteria caused by carbon excretion from phytoplankton

For the competition system of phytoplankton and bacteria through inorganic phosphorus, our mathematical model showed that mutualistic relationships between them could occur due to production and consumption of extracellular organic carbon by phytoplankton and bacteria. In our model, phytoplankton are limited in their growth by light and phosphorus, and bacteria are limited in their growth by phosphorus and carbon released from phytoplankton. We adopted permanence as a criterion of the coexistence in mathematical analysis, and led necessary conditions of permanence in the model. Under these coexistence conditions, we estimated the strength of total effects of interactions between phytoplankton and bacteria at the steady state by press perturbation method. The results of this estimation indicated the mutualistic interactions between phytoplankton and bacteria. This suggests that mutualistic situation could occur due to the introduction of carbon flow from phytoplankton to bacteria, even if phytoplankton and bacteria compete with each other through common resource, inorganic phosphorus.     

Periodic orbits near heteroclinic cycles in a cyclic replicator system

A species is semelparous if every individual reproduces only once in its life and dies immediately after the reproduction. While the reproduction opportunity is unique per year and the individual’s period from birth to reproduction is just n years, the individuals that reproduce in the ith year (modulo n) are called the ith year class, i = 1, 2, . . . , n. The dynamics of the n year-class system can be described by a differential equation system of Lotka–Volterra type. For the case n = 4, there is a heteroclinic cycle on the boundary as shown in previous works. In this paper, we focus on the case n = 4 and show the existence, growth and disappearance of periodic orbits near the heteroclinic cycle, which is a part of the conjecture by Diekmann and van Gils (SIAM J Appl Dyn Syst 8:1160–1189, 2009). By analyzing the Poincaré map near the heteroclinic cycle and introducing a metric to measure the size of the periodic orbit, we show that (i) when the average competitive degree among subpopulations (year classes) in the system is weak, there exists an asymptotically stable periodic orbit near the heteroclinic cycle which is repelling; (ii) the periodic orbit grows in size when some competitive degree increases, and converges to the heteroclinic cycle when the average competitive degree tends to be strong; (iii) when the average competitive degree is strong, there is no periodic orbit near the heteroclinic cycle which becomes asymptotically stable. Our results provide explanations why periodic solutions expand and disappear and why all but one subpopulation go extinct.     

Evolutionary stability of coexistence due to the storage effect in a two-season model

The question of species coexistence has been central to ecology since its founding. Ever-present environmental variation may be one answer to that question. Previous models have demonstrated that species can exploit this variation to coexist with competitors by having different environmental responses (the storage effect). When traits governing species’ environmental response can evolve, however, coexistence is not assured. In this study, we use a continuous time, two-season model to determine the evolutionary outcome of competing species evolving in their seasonal performance trait. We extend the competitive exclusion principle to show that the storage effect can allow no more than N species to coexist on N discrete seasons with no relative nonlinearity. We find a broad region of parameter space where coexistence is evolutionarily stable. The size of this region depends on the period of fluctuations relative to the individual lifespan. Relatively long period fluctuations yield a large coexistence region, but as the period decreases, the region narrows and disappears asymptotically. Finally, we cast our adaptive dynamics technique in terms of Chesson’s concept of equalizing and stabilizing mechanisms to demonstrate that the breakdown in coexistence at short periods is due to loss of the stabilizing covariance between the environment and competition.     

Evolutionary coexistence in a fluctuating environment by specialization on resource level

Microbial communities in fluctuating environments, such as oceans or the human gut, contain a wealth of diversity. This diversity contributes to the stability of communities and the functions they have in their hosts and ecosystems. To improve stability and increase production of beneficial compounds, we need to understand the underlying mechanisms causing this diversity. When nutrient levels fluctuate over time, one possibly relevant mechanism is coexistence between specialists on low and specialists on high nutrient levels. The relevance of this process is supported by the observations of coexistence in the laboratory, and by simple models, which show that negative frequency dependence of two such specialists can stabilize coexistence. However, as microbial populations are often large and fast growing, they evolve rapidly. Our aim is to determine what happens when species can evolve; whether evolutionary branching can create diversity or whether evolution will destabilize coexistence. We derive an analytical expression of the invasion fitness in fluctuating environments and use adaptive dynamics techniques to find that evolutionarily stable coexistence requires a special type of trade-off between growth at low and high nutrients. We do not find support for the necessary evolutionary trade-off in data available for the bacterium Escherichia coli and the yeast Saccharomyces cerevisiae on glucose. However, this type of data is scarce and might exist for other species or in different conditions. Moreover, we do find evidence for evolutionarily stable coexistence of the two species together. Since we find this coexistence in the scarce data that are available, we predict that specialization on resource level is a relevant mechanism for species diversity in microbial communities in fluctuating environments in natural settings.     

Rainfall variability maintains grass‐forb species coexistence

Environmental variability can structure species coexistence by enhancing niche partitioning. Modern coexistence theory highlights two fluctuation‐dependent temporal coexistence mechanisms —the storage effect and relative nonlinearity – but empirical tests are rare. Here, we experimentally test if environmental fluctuations enhance coexistence in a California annual grassland. We manipulate rainfall timing and relative densities of the grass Avena barbata and forb Erodium botrys, parameterise a demographic model, and partition coexistence mechanisms. Rainfall variability was integral to grass–forb coexistence. Variability enhanced growth rates of both species, and early‐season drought was essential for Erodium persistence. While theoretical developments have focused on the storage effect, it was not critical for coexistence. In comparison, relative nonlinearity strongly stabilised coexistence, where Erodium experienced disproportionately high growth under early‐season drought due to competitive release from Avena. Our results underscore the importance of environmental variability and suggest that relative nonlinearity is a critical if underappreciated coexistence mechanism.     

Permanence of discrete-time Kolmogorov systems for two species and saturated fixed points

This paper considers the dynamics of a discrete-time Kolmogorov system for two-species populations. In particular, permanence of the system is considered. Permanence is one of the concepts to describe the species coexistence. By using the method of an average Liapunov function, we have found a simple sufficient condition for permanence of the system. That is, nonexistence of saturated boundary fixed points is enough for permanence of the system under some appropriate convexity or concavity properties for the population growth rate functions. Numerical investigations show that for the system with population growth rate functions without such properties, the nonexistence of saturated boundary fixed points is not sufficient for permanence, actually a boundary periodic orbit or a chaotic orbit can be attractive despite the existence of a stable coexistence fixed point. This result implies, in particular, that existence of a stable coexistence fixed point is not sufficient for permanence.     

Evolution of cooperation under -person snowdrift games

In the animal world, performing a given task which is beneficial to an entire group requires the cooperation of several individuals of that group who often share the workload required to perform the task. The mathematical framework to study the dynamics of collective action is game theory. Here we study the evolutionary dynamics of cooperators and defectors in a population in which groups of individuals engage in N-person, non-excludable public goods games. We explore an N-person generalization of the well-known two-person snowdrift game. We discuss both the case of infinite and finite populations, taking explicitly into consideration the possible existence of a threshold above which collective action is materialized. Whereas in infinite populations, an N-person snowdrift game (NSG) leads to a stable coexistence between cooperators and defectors, the introduction of a threshold leads to the appearance of a new interior fixed point associated with a coordination threshold. The fingerprints of the stable and unstable interior fixed points still affect the evolutionary dynamics in finite populations, despite evolution leading the population inexorably to a monomorphic end-state. However, when the group size and population size become comparable, we find that spite sets in, rendering cooperation unfeasible.     

The information capacity of hypercycles

Hypercycles are information integration systems which are thought to overcome the information crisis of prebiotic evolution by ensuring the coexistence of several short templates. For imperfect template replication, we derive a simple expression for the maximum number of distinct templates n_m that can coexist in a hypercycle and show that it is a decreasing function of the length L of the templates. In the case of high replication accuracy we find that the product n_mL tends to a constant value, limiting thus the information content of the hypercycle. Template coexistence is achieved either as a stationary equilibrium (stable fixed point) or a stable periodic orbit in which the total concentration of functional templates is nonzero. For the hypercycle system studied here we find numerical evidence that the existence of an unstable fixed point is a necessary condition for the presence of periodic orbits.     

Global Dynamics of an In-host Viral Model with Intracellular Delay

The dynamics of a general in-host model with intracellular delay is studied. The model can describe in vivo infections of HIV-I, HCV, and HBV. It can also be considered as a model for HTLV-I infection. We derive the basic reproduction number R ₀ for the viral infection, and establish that the global dynamics are completely determined by the values of R ₀. If R ₀≤1, the infection-free equilibrium is globally asymptotically stable, and the virus are cleared. If R ₀>1, then the infection persists and the chronic-infection equilibrium is locally asymptotically stable. Furthermore, using the method of Lyapunov functional, we prove that the chronic-infection equilibrium is globally asymptotically stable when R ₀>1. Our results shows that for intercellular delays to generate sustained oscillations in in-host models it is necessary have a logistic mitosis term in target-cell compartments.     

Relative impact of earlier diagnosis and improved treatment on survival for colorectal cancer: A US database study among elderly patients

PurposesTo estimate what proportion of improvement in relative survival was attributable to smaller stage/size due to early detection and what proportion was attributable to cancer chemotherapy in patients with colorectal cancer (CRC).MethodsWe studied 69,718 patients with CRC aged ≥66 years in 1992–2009 from Surveillance, Epidemiology, and End Results registries. Study periods were categorized into three periods according to the major changes or advances in screening and chemotherapy regimens: (1) Period-1 (1992–1995), during which there was no evidence-based recommendation for routine CRC screening and 5-fluorouracil was the mainstay for chemotherapy; (2) Period-2 (1996–2000), during which evidences and guidelines supported the use of fecal occult blood test (FOBT) and sigmoidoscopy for routine CRC screening; and (3) Period-3 (2001–2009), during which Medicare Program added the full coverage for colonoscopy screening to average-risk individuals, and several newly developed chemotherapy regimens were approved. Outcome variables included the likelihood of being diagnosed at an early stage or with a small tumor size, and improvement in relative survival.ResultsCompared to period-1, likelihood of being diagnosed with early stage CRC increased by 20% in period-2 (odds ratio = 1.2, 95%CI: 1.1–1.2) and 30% in period-3 (1.3, 1.2–1.4); and likelihood of being diagnosed with small-size CRC increased by 60% in period-2 and 110% in period-3. Similarly, 5-year overall relative survival increased from 51% in period-1 to 56% in period-2 and 60% in period-3. Increase in survival attributable to migration in stage/size was 9% in period-2 and 20% in period-3, while the remaining survival improvement during period-2 and period-3 were largely attributable to more effective chemotherapy regimens (≥71.6%) and other treatment factors (≤25%).ConclusionsImprovements in CRC screening resulted in a migration of CRC toward earlier tumor stage and smaller size, which contributed to ≤20% of survival increase. Survival improvement over the past 2 decades was largely explained by more effective chemotherapy regimens (≥71.6%).     

Multiple limit cycles in the standard model of three species competition for three essential resources

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. This research was partially supported by NSF grant DMS 0211614. KY 40292, USA. This author's research was supported in part by NSF grant DMS 0107160     

具有年龄结构的食蚜蝇-蚜虫捕食模型

现有的具有年龄结构的捕食-食饵模型总是假设只有成年捕食者捕食猎物,这与实际情况不符。建立了一个幼年捕食者捕食食饵的具有年龄结构的食蚜蝇-蚜虫模型,应用微分方程定性理论,讨论了系统平衡点及其稳定性:其中平衡点E1(0,0,0)为不稳定的;满足一定条件时,边界平衡点E2(K,0,0)及正平衡点E3(x*,y1*,y2*)为局部渐近稳定的;且应用一致持续生存理论得到了系统永久持续生存的条件,为有害生物综合治理提供了理论依据。     

Coexistence of plant species in a biodiversity hotspot is stabilized by competition but not by seed predation

Understanding the mechanisms of species coexistence is a key task for ecology. Recent theory predicts that both competition and predation (which causes apparent competition among prey) can either promote or limit species coexistence. Both mechanisms cause negative interactions between individuals, and each mechanism promotes stable coexistence if it causes negative interactions to be stronger between conspecifics than between heterospecifics. However, the relative importance of competition and predation for coexistence in natural communities is poorly known. Here, we study how competition and apparent competition via pre‐dispersal seed predators affect the long‐term fecundity of Protea shrubs in the fire‐prone Fynbos biome (South Africa). These shrubs store all viable seeds produced since the last fire in fire‐proof cones. Competitive effects on cone number and pre‐dispersal seed predation reduce their fecundity and can thus limit recruitment after the next fire. In 27 communities comprising 49 990 shrubs of 22 Protea species, we measured cone number and per‐cone seed predation rate of 2154 and 1755 focal individuals, respectively. Neighbourhood analyses related these measures to individual‐based community maps. We found that conspecific neighbours had stronger competitive effects on cone number than heterospecific neighbours. In contrast, apparent competition via seed predators was comparable between conspecifics and heterospecifics. This indicates that competition stabilizes coexistence of Protea species, whereas pre‐dispersal seed predation does not. Larger neighbours had stronger competitive effects and neighbours with large seed crops exerted stronger apparent competition. For 97% of the focal plants, competition reduced fecundity more than apparent competition. Our results show that even in communities of closely related and ecologically similar species, intraspecific competition can be stronger than interspecific competition. On the other hand, apparent competition through seed predators need not have such a stabilizing effect. These findings illustrate the potential of ‘community demography’, the demographic study of multiple interacting species, for understanding plant coexistence.     

Ecology of sessile animals on sublittoral hard substrata: The need to measure variation

This paper concerns assemblages of sessile animals occupying shaded, commonly vertical hard substrata in the shallow subtidal zone. We are interested particularly in questions about the coexistence of species and about what influences their joint dynamics. We propose a conceptual model which focuses on variation in characteristics such as birth and death rates, competitive interactions and dispersal. We argue that, qualitatively, this model appears to be a satisfactory representation of the important characteristics of certain sessile assemblages. Further, we suggest that it may explain the coexistence of a large number of ecologically similar species in assemblages that appear in a sense ‘stable’ (but where assemblages at different sites differ in detail). There is support for this assertion from formal theoretical work on simpler versions of the model. It is not sufficient merely to argue that the model seems satisfactory or plausible, so we finally consider what kind of data is needed for the further development and testing of this kind of model.     

Using stable isotopes to test for trophic niche partitioning: a case study with stream salamanders and fish

1. Stream salamanders and fish often co‐occur even though fish prey on and outcompete salamanders. However, the mechanisms that allow palatable salamanders to coexist with fish are unknown. 2. We tested mechanisms in the field that promote coexistence between Idaho giant salamanders (Dicamptodon aterrimus) and stream salmonid fishes in headwater streams. Previous research in this system indicated that salamander dispersal did not promote coexistence with fish. We tested the hypothesis that D. aterrimus shift their diet when they occur with fish, facilitating coexistence through local niche partitioning. 3. We used nitrogen and carbon stable isotopes to describe the trophic niche of D. aterrimus and fish in three co‐occurring populations of salamanders and fish and three populations of salamanders without fish. We used two approaches to quantify trophic niche partitioning with stable isotopes: 95% kernel density estimators and isotopic mixing models. 4. We found that salamanders and fish were generalists that consumed aquatic invertebrates primarily, but both species were also cannibalistic and predatory on one another. We also found no support for trophic niche partitioning as a coexistence mechanism because there were no differences in the trophic niche metrics among salamander populations with and without fish. 5. Although we did not identify mechanisms that facilitate salamander and fish coexistence, our empirical data and use of novel approaches to describe the trophic niche did yield important insights on the role of predator–prey interactions and cannibalism as alternative coexistence mechanisms. In addition, we found that 95% kernel estimators are a simple and robust method to describe population‐level measure of trophic structure.     

Coevolution of competing Callosobruchus species does not stabilize coexistence

Interspecific resource competition is expected to select for divergence in resource use, weakening interspecific relative to intraspecific competition, thus promoting stable coexistence. More broadly, because interspecific competition reduces fitness, any mechanism of interspecific competition should generate selection favoring traits that weaken interspecific competition. However, species also can adapt to competition by increasing their competitive ability, potentially destabilizing coexistence. We reared two species of bean beetles, the specialist Callosobruchus maculatus and the generalist C. chinensis, in allopatry and sympatry on a mixture of adzuki beans and lentils, and assayed mutual invasibility after four, eight, and twelve generations of evolution. Contrary to the expectation that coevolution of competitors will weaken interspecific competition, the rate of mutual invasibility did not differ between sympatry and allopatry. Rather, the invasion rate of C. chinensis, but not C. maculatus, increased with duration of evolution, as C. chinensis adapted to lentils without experiencing reduced adaptation to adzuki beans, and regardless of the presence or absence of C. maculatus. Our results highlight that evolutionary responses to interspecific competition promote stable coexistence only under specific conditions that can be difficult to produce in practice.     

Dynamics of an epidemic model with non-local infections for diseases with latency over a patchy environment

In this paper, with the assumptions that an infectious disease in a population has a fixed latent period and the latent individuals of the population may disperse, we formulate an SIR model with a simple demographic structure for the population living in an n-patch environment (cities, towns, or countries, etc.). The model is given by a system of delay differential equations with a fixed delay accounting for the latency and a non-local term caused by the mobility of the individuals during the latent period. Assuming irreducibility of the travel matrices of the infection related classes, an expression for the basic reproduction number R₀{\mathcal{R}_0} is derived, and it is shown that the disease free equilibrium is globally asymptotically stable if R₀ < 1{\mathcal{R}_0 < 1} , and becomes unstable if ${\mathcal{R}_0 > 1}${\mathcal{R}_0 > 1} . In the latter case, there is at least one endemic equilibrium and the disease will be uniformly persistent. When n = 2, two special cases allowing reducible travel matrices are considered to illustrate joint impact of the disease latency and population mobility on the disease dynamics. In addition to the existence of the disease free equilibrium and interior endemic equilibrium, the existence of a boundary equilibrium and its stability are discussed for these two special cases.