Tumor micro-ecology and competitive interactions

Three nested models describing the growth of individual subpopulations in a heterogeneous environment are described. The models represent the dynamics of two populations which compete, to varying degrees, for common resources. The first model describes growth in a totally non-competitive micro-environment, the second model describes an ecology in which competition is proportional to competitor population size, and the third model ecology extends the model described by Jansson & Revesz (1974), which allows one population to emerge from the other. The critical points for each model are defined using the isoclines derived from the Ordinary Differential Equations (ODE's) describing competitive growth. The critical points for each model are characterized by the signs of the eigenvalues of the variational matrix at each point. The theoretical results of the analysis show that a competitive model ecology with Verhulstian logistics allows four critical points: the origin which is a repeller, two competitive exclusion points, and an equilibrium state (Waltman, 1983). The extended model ecology of Jansson & Revesz (1974), allows three critical points: the origin which is a repeller, competitive exclusion of the first population, and an equilibrium point. Data from a human adenocarcinoma of the colon and murine mammary tumors are used as qualitative measures of the dynamics of the three micro-ecologies. Issues such as stochastic extension to model small populations either for clonal extinction or heterogeneous emergence are discussed.     

Bottom-up derivation of population models for competition involving multiple resources

This paper provides first-principles derivations of population models for competition involving multiple resources with different competition types, based on resource partitioning between individuals. The following two cases are investigated. The first is the case in which the resource competed for and its competition type change depending on life stages from scramble to contest competition, or from contest to scramble competition. The second is the case in which individuals compete for two resources simultaneously with scramble and contest types, respectively. In both cases, population models are derived analytically, and in particular, the Hassell model is derived in the second case. The nature of reproduction curves and the stability properties of three population models derived are compared with each other. These models provide three representative models for competition involving both scramble and contest types.     

Does quasi-local competition lead to pattern formation in metapopulations? An explicit resource competition model

In metapopulations, competitive interactions may extend beyond the confines of the local population such that members of neighbouring habitat patches affect each other adversely (quasi-local competition). We derive a model for quasi-local competition from first principles, assuming that individuals compete for shared resources and members of a population spend a certain fraction of their foraging time in the adjacent populations. Contrary to the results of Doebeli and Killingback [2003. Theor. Popul. Biol. 64, 397-416], our model does not produce spatial patterns of population densities in homogeneous environments. Quasi-local competition nevertheless contributes to pattern formation by amplifying the effect of heterogeneities in the external environment, and this amplification can be extremely strong when dispersal is absent. We discuss why apparently similar models lead to contrasting results.     

代谢互养关系在维持微生物物种多样性中的作用

互养关系（cross-feeding）是微生物物种之间普遍存在的一种相互关系,其中物种利用环境中其他成员的代谢产物以促进自身生长的情形称为代谢互养关系,这种关系对物种间的竞争结果往往有很大影响,甚至会改变种群结构。为了研究代谢互养关系在维持微生物物种多样性中的作用,构建包含不同代谢互养关系的资源竞争模型,这些模型既体现了微生物物种竞争资源时种群密度及资源量的动态,也展示了物种利用其他竞争者的代谢资源对自身生存状况的影响。数值模拟结果显示：（1）考虑微生物中不同的代谢互养关系结构：两物种间单向互养、双向互养以及多物种间的互养,不同的互养关系都可以促进竞争物种稳定共存,竞争中处于劣势的物种通过利用其他竞争成员的代谢产物,打破外界资源量对其生长的限制,改变原本消亡的命运;而处于优势的物种则通过利用其他竞争成员的代谢产物,增大种群密度。（2）多物种竞争同一种有限资源时,不是所有物种都能共存,在四物种模拟中,原本处于最劣势的物种灭绝,其余三者共存。物种产生代谢资源对其本身是"不利"的,如果在模拟中物种利用代谢资源的能力相同,那么物种竞争外界资源的劣势就很可能无法被抵消。通过改变资源利用率发现只有互养关系中代谢资源的利用可以弥补劣势种在竞争外界资源时的不足,多物种才可以全部共存。（3）验证数值模拟结果的普遍性,分析参数变化对共存的影响,结果表明代谢互养关系促进的共存对代谢资源相关参数不敏感,参数的改变只影响平衡态时物种的种群密度。所以,代谢互养关系可以促进相互竞争的微生物物种共存,即微生物之间的互养关系很可能是维持物种多样性的一种机制。     

The Tumor Growth Paradox and Immune System-Mediated Selection for Cancer Stem Cells

Cancer stem cells (CSCs) drive tumor progression, metastases, treatment resistance, and recurrence. Understanding CSC kinetics and interaction with their nonstem counterparts (called tumor cells, TCs) is still sparse, and theoretical models may help elucidate their role in cancer progression. Here, we develop a mathematical model of a heterogeneous population of CSCs and TCs to investigate the proposed “tumor growth paradox”—accelerated tumor growth with increased cell death as, for example, can result from the immune response or from cytotoxic treatments. We show that if TCs compete with CSCs for space and resources they can prevent CSC division and drive tumors into dormancy. Conversely, if this competition is reduced by death of TCs, the result is a liberation of CSCs and their renewed proliferation, which ultimately results in larger tumor growth. Here, we present an analytical proof for this tumor growth paradox. We show how numerical results from the model also further our understanding of how the fraction of cancer stem cells in a solid tumor evolves. Using the immune system as an example, we show that induction of cell death can lead to selection of cancer stem cells from a minor subpopulation to become the dominant and asymptotically the entire cell type in tumors.     

Competitive speciation in quantitative genetic models

We study sympatric speciation due to competition in an environment with a broad distribution of resources. We assume that the trait under selection is a quantitative trait, and that mating is assortative with respect to this trait. Our model alternates selection according to Lotka-Volterra-type competition equations, with reproduction using the ideas of quantitative genetics. The recurrence relations defined by these equations are studied numerically and analytically. We find that when a population enters a new environment, with a broad distribution of unexploited food sources, the population distribution broadens under a variety of conditions, with peaks at the edge of the distribution indicating the formation of subpopulations. After a long enough time period, the population can split into several subpopulations with little gene flow between them.     

Modelling individual plant growth at a variable mean density or at a specific spatial setting

Neighbouring plants generally compete for the limiting resources in order to grow and reproduce. Some resources, e.g., sun light, may be monopolised by the larger plants and this may lead to asymmetric competition where a plant, which is twice as large, grows more than twice as fast. A previously published individual-based Richards growth model that describes the asymmetric growth of individual plants is here generalised with respect to a variable mean plant density and an explicit spatial setting.     

Single-molecule studies reveal method for tuning the heterogeneous activity of alkaline phosphatase

Single-molecule-enzymology (SME) methods have enabled the observation of heterogeneous catalytic activities within a single enzyme population. Heterogeneous activity is hypothesized to originate from conformational changes in the enzyme that result from changes in the local environment leading to catalytically active substates. Here, we use SME to investigate the mechanisms of heterogeneous activity exhibited by tissue nonspecific alkaline phosphatase (TNSALP), which reveals two subpopulations with different catalytic activities. We show the effect of pH and temperature on the distribution of TNSALP activity and confirm the presence of two subpopulations attributed to half- and fully active TNSALP substates. We provide mechanistic insight about protein structure using molecular dynamic simulations and show pH- and temperature-dependent conformational transitions that corroborate experimentally observed changes in TNSALP activity. These results show the utility of SME to understand heterogeneous enzyme activity and demonstrate a simple approach using pH and temperature to tune catalytic activity within an enzyme population.     

Life not lived due to disequilibrium in heterogeneous age-structured populations

Three models of age-structured populations with demographically heterogeneous subpopulations are analyzed. In the first model, each subpopulation has its own age-specific vital rates which are fixed in time. In the second model, the vital rates of each subpopulation are uniformly inhibited by increasing total numbers of individuals. In the third, the vital rates of groups of subpopulations are inhibited by the total numbers of individuals in other groups of subpopulations with an intensity that depends on the interacting pair of groups. Three functions are defined to measure disequilibrium in the subpopulation frequencies, subpopulation age structures, and total population size. For the first model, we show that disequilibrium will shift the trajectory of the total numbers of individuals forward or backward in time by an asymptotic constant that is proportional to the sum of the disequilibrium measures. For the second model, we establish sufficient conditions for the existence of a globally stable equilibrium and we show that disequilibrium will result in a finite loss or gain in life which is proportional to the sum of the disequilibrium measures. For the last model, we show that the loss or gain in life for each group of subpopulations is a linear combination over all groups of the sums of the three disequilibrium measures. We illustrate these results with numerical examples and give possible biological interpretations of the models. We relate these new results to previous work on the cost of natural selection and measures of demographic disequilibrium.     

Quantitative Characterization of Cellular Membrane-Receptor Heterogeneity through Statistical and Computational Modeling

Cell population heterogeneity can affect cellular response and is a major factor in drug resistance. However, there are few techniques available to represent and explore how heterogeneity is linked to population response. Recent high-throughput genomic, proteomic, and cellomic approaches offer opportunities for profiling heterogeneity on several scales. We have recently examined heterogeneity in vascular endothelial growth factor receptor (VEGFR) membrane localization in endothelial cells. We and others processed the heterogeneous data through ensemble averaging and integrated the data into computational models of anti-angiogenic drug effects in breast cancer. Here we show that additional modeling insight can be gained when cellular heterogeneity is considered. We present comprehensive statistical and computational methods for analyzing cellomic data sets and integrating them into deterministic models. We present a novel method for optimizing the fit of statistical distributions to heterogeneous data sets to preserve important data and exclude outliers. We compare methods of representing heterogeneous data and show methodology can affect model predictions up to 3.9-fold. We find that VEGF levels, a target for tuning angiogenesis, are more sensitive to VEGFR1 cell surface levels than VEGFR2; updating VEGFR1 levels in the tumor model gave a 64% change in free VEGF levels in the blood compartment, whereas updating VEGFR2 levels gave a 17% change. Furthermore, we find that subpopulations of tumor cells and tumor endothelial cells (tEC) expressing high levels of VEGFR (>35,000 VEGFR/cell) negate anti-VEGF treatments. We show that lowering the VEGFR membrane insertion rate for these subpopulations recovers the anti-angiogenic effect of anti-VEGF treatment, revealing new treatment targets for specific tumor cell subpopulations. This novel method of characterizing heterogeneous distributions shows for the first time how different representations of the same data set lead to different predictions of drug efficacy.     

Stability of Multispecies Bacterial Communities: Signaling Networks May Stabilize Microbiomes

Multispecies bacterial communities can be remarkably stable and resilient even though they consist of cells and species that compete for environmental resources. In silico models suggest that common signals released into the environment may help selected bacterial species cluster at common locations and that sharing of public goods (i.e. molecules produced and released for mutual benefit) can stabilize this coexistence. In contrast, unilateral eavesdropping on signals produced by a potentially invading species may protect a community by keeping invaders away from limited resources. Shared bacterial signals, such as those found in quorum sensing systems, may thus play a key role in fine tuning competition and cooperation within multi-bacterial communities. We suggest that in addition to metabolic complementarity, signaling dynamics may be important in further understanding complex bacterial communities such as the human, animal as well as plant microbiomes.     

Grazers and Diggers: Exploitation Competition and Coexistence among Foragers with Different Feeding Strategies on a Single Resource

A mathematical model is presented that describes a system where two consumer species compete exploitatively for a single renewable resource. The resource is distributed in a patchy but homogeneous environment; that is, all patches are intrinsically identical. The two consumer species are referred to as diggers and grazers, where diggers deplete the resource within a patch to lower densities than grazers. We show that the two distinct feeding strategies can produce a heterogeneous resource distribution that enables their coexistence. Coexistence requires that grazers must either move faster than diggers between patches or convert the resources to population growth much more efficiently than diggers. The model shows that the functional form of resource renewal within a patch is also important for coexistence. These results contrast with theory that considers exploitation competition for a single resource when the resource is assumed to be well mixed throughout the system.     

The population consequences of territorial behaviour

Many organisms compete for space, or for resource that are linked to space. Territorial behavior in animals is one expression of competition for space. Models of competition for space seek to predict how the arrangement of individuals in a population changes as new individuals appear, others die, and neighbors interact with each other; studies of territorial behaviour examine how neighbor interactions lead animals to establish and maintain their use of space. In recent work on compition for space and on territorial behaviour, there has been a shift from simple, general models to ones that incorporate heterogeneity in the spatial and temporal distribution of resources, and in the ways individuals use resources.     

The Population Dynamics and Community Ecology of Root Hemiparasitic Plants

Root hemiparasitic plants and their host plants interact directly, through parasitism, as well as indirectly, through scramble competition for resources. To understand the population dynamics and community ecology of root hemiparasitic plants and their hosts, models of resource-based competition have been extended to include resource parasitism. Parasitism provides a mechanism for parasitic plants to overcome deficits in their ability to compete for soil resources. The interaction ranges from competitive to exploiter-victim, depending on whether the benefits of parasitism overshadow the costs of competition. These models predict that as productivity in the system increases, parasitic plants should become more abundant. In diverse host communities, differences in the impact that parasites have on their hosts and the benefits that they receive from parasitizing different hosts may lead to nontransitive competitive relationships and a sort of apparent competition. The possible dynamics include paper-rock-scissors oscillations and indirect mutualisms between parasitic plants and their hosts that allow them to form coalitions that can exclude competitive dominants.     

Accounting for interspecific competition and age structure in demographic analyses of density dependence improves predictions of fluctuations in population size

Understanding species coexistence has long been a major goal of ecology. Coexistence theory for two competing species posits that intraspecific density dependence should be stronger than interspecific density dependence. Great tits and blue tits are two bird species that compete for food resources and nesting cavities. On the basis of long‐term monitoring of these two competing species at sites across Europe, combining observational and manipulative approaches, we show that the strength of density regulation is similar for both species, and that individuals have contrasting abilities to compete depending on their age. For great tits, density regulation is driven mainly by intraspecific competition. In contrast, for blue tits, interspecific competition contributes as much as intraspecific competition, consistent with asymmetric competition between the two species. In addition, including age‐specific effects of intra‐ and interspecific competition in density‐dependence models improves predictions of fluctuations in population size by up to three times.     

Allelopathy prevents competitive exclusion and promotes phytoplankton biodiversity

It has been hypothesized that allelopathy can prevent competitive exclusion and promote phytoplankton diversity in aquatic ecosystems, where numerous species coexist on a limited number of resources. However, experimental proof‐of‐principle is not available to support this hypothesis. Here we present the first experimental evidence to support this hypothesis by demonstrating that allelopathy promotes the coexistence of two phytoplankton species, Ankistrodesmus falcatus and Oscillatoria sp., that compete for a single limiting nutrient. By performing long‐term competition experiments in nitrate‐limited continuous cultures, and by describing the population dynamics using a mechanistic model, we demonstrate that when allelopathy comes into play, one of the following outcomes is possible depending on the relative initial abundances of the species: dominance of the stronger competitor for nitrate (the non‐allelopathic species), oscillatory coexistence, or dominance of the weaker competitor (the allelopathic species). Our model analysis revealed that sustained oscillatory coexistence of the two species would be a common outcome of this experiment. Our study confirms for the first time, based on laboratory experiments combined with mechanistic models, that allelopathy can alter the predicted outcome of inter‐specific competition in a nutrient‐limited environment and increase the potential for the coexistence of more species than resources, thereby contributing to the identification of endogenous mechanisms that explain the extreme diversity of phytoplankton communities.     

Revising the Role of Species Mobility in Maintaining Biodiversity in Communities with Cyclic Competition

One of the most crucial tasks faced by biologists today is revealing the mechanisms which account for biodiversity, yet we are still far from a full understanding of these mechanisms, and in particular the role of spatially heterogeneous population distributions. Recently, the spatially heterogeneous coexistence seen in cyclic competition models-in which species compete as in the game rock-paper-scissors-has brought them to the fore as a paradigm for biodiversity. Research into cyclic competition has so far been focused almost exclusively on stochastic lattice models with discrete space, which ignore several key dynamical aspects. In particular, such models usually assume that species disperse at the same speed. This paper aims to extend our understanding of cyclic competition by applying a reaction-diffusion Lotka-Volterra scheme to the problem, which allows us to vary the mobility of each species, and lets us take into account cyclic competition with more complex underlying mechanisms. In this paper we reveal an entirely new kind of cyclic competition-'conditional' cyclic competition, with a different underlying mechanism to 'classic' cyclic competition-and we show that biodiversity in communities with cyclic competition in fact depends heavily on the ratios between the species mobilities. Furthermore, we show that this dependence can be completely different for conditional and classic cyclic competition. We also present a wide range of spatiotemporal patterns which are formed in the system, including spiral and target waves, spiralling patches, and irregular chaotic patches. We show that the previously unknown case of conditional cyclic competition is host to a scenario of patchy co-invasion, where the spread of the population front takes place via the formation, splitting and propagation of patches of high species density. This is also an example of invasional meltdown because one competitor facilitates the invasion of the other, but unlike well-known cases of invasional meltdown the co-invaders in this system are not mutualists but antagonistic competitors, and the overall result mitigates, rather than amplifies, the damage done to the native ecosystem.     

Interspecific Resource Competition Effects on Fisheries Revenue

In many fisheries multiple species are simultaneously caught while stock assessments and fishing quota are defined at species level. Yet species caught together often share habitat and resources, resulting in interspecific resource competition. The consequences of resource competition on population dynamics and revenue of simultaneously harvested species has received little attention due to the historical single stock approach in fisheries management. Here we present the results of a modelling study on the interaction between resource competition of sole (Solea solea) and slaice (Pleuronectus platessa) and simultaneous harvesting of these species, using a stage-structured population model. Three resources were included of which one is shared with a varied competition intensity. We find that plaice is the better competitor of the two species and adult plaice are more abundant than adult sole. When competition is high sole population biomass increases with increasing fishing effort prior to plaice extinction. As a result of this increase in the sole population, the revenue of the stocks combined as function of effort becomes bimodal with increasing resource competition. When considering a single stock quota for sole, its recovery with increasing effort may result in even more fishing effort that would drive the plaice population to extinction. When sole and plaice compete for resources the highest revenue is obtained at effort levels at which plaice is extinct. Ignoring resource competition promotes overfishing due to increasing stock of one species prior to extinction of the other species. Consequently, efforts to mitigate the decline in one species will not be effective if increased stock in the other species leads to increased quota. If a species is to be protected against extinction, management should not only be directed at this one species, but all species that compete with it for resource as well.