Qualitative Analysis of a Single-Species Model with Distributed Delay and Nonlinear Harvest

In this paper, a single-species population model with distributed delay and Michaelis-Menten type harvesting is established. Through an appropriate transformation, the mathematical model is converted into a two-dimensional system. Applying qualitative theory of ordinary differential equations, we obtain sufficient conditions for the stability of the equilibria of this system under three cases. The equilibrium A1 of system is globally asymptotically stable when br−c>0 and η<0. Using Poincare-Bendixson theorem, we determine the existence and stability of limit cycle when br−c>0 and η>0. By computing Lyapunov number, we obtain that a supercritical Hopf bifurcation occurs when η passes through 0. High order singularity of the system, such as saddle node, degenerate critical point, unstable node, saddle point, etc, is studied by the theory of ordinary differential equations. Numerical simulations are provided to verify our main results in this paper.

Hopf Bifurcation in a Delayed Single Species Network System

In this paper, we focus on a network system which describes spatiotemporal dynamics of single species population at different patches since species can have different features in various life stages and different behaviors in various spatial environments. With the effect of time delay and spatial dispersion, homogenous, periodic and spatiotemporally nonhomogeneous distributions are identified. The stability analysis is carried out for the discrete-space and continuous-time network on single species with time delay and the Hopf bifurcation of the single species population model in a network is explored. Formulas for determining the direction of Hopf bifurcation are derived by using the center manifold method and the normal form theorem. It is found that the network can generate spatial patterns only when time delay is present. Finally, numerical simulations are performed which agree well with our theoretical result, i.e. this discrete-space and continuous-time model admits regular temporal patterns since the delay induces Hopf bifurcations with network structure.

Survival analysis of a stochastic delay single-species system in polluted environment with psychological effect and pulse toxicant input

We propose and study a stochastic delay single-species population system in polluted environment with psychological effect and pulse toxicant input. We establish sufficient conditions for the extinction, nonpersistence in the mean, weak persistence, and strong persistence of the single-species population and obtain the threshold value between extinction and weak persistence. Finally, we confirm the efficiency of the main results by numerical simulations.

Streamlined and Abundant Bacterioplankton Thrive in Functional Cohorts

ABSTRACT While fastidious microbes can be abundant and ubiquitous in their natural communities, many fail to grow axenically in laboratories due to auxotrophies or other dependencies. To overcome auxotrophies, these microbes rely on their surrounding cohort. A cohort may consist of kin (ecotypes) or more distantly related organisms (community) with the cooperation being reciprocal or nonreciprocal and expensive (Black Queen hypothesis) or costless (by-product). These metabolic partnerships (whether at single species population or community level) enable dominance by and coexistence of these lineages in nature. Here we examine the relevance of these cooperation models to explain the abundance and ubiquity of the dominant fastidious bacterioplankton of a dimictic mesotrophic freshwater lake. Using both culture-dependent (dilution mixed cultures) and culture-independent (small subunit [SSU] rRNA gene time series and environmental metagenomics) methods, we independently identified the primary cohorts of actinobacterial genera “Candidatus Planktophila” (acI-A) and “Candidatus Nanopelagicus” (acI-B) and the proteobacterial genus “Candidatus Fonsibacter” (LD12). While “Ca. Planktophila” and “Ca. Fonsibacter” had no correlation in their natural habitat, they have the potential to be complementary in laboratory settings. We also investigated the bifunctional catalase-peroxidase enzyme KatG (a common good which “Ca. Planktophila” is dependent upon) and its most likely providers in the lake. Further, we found that while ecotype and community cooperation combined may explain “Ca. Planktophila” population abundance, the success of “Ca. Nanopelagicus” and “Ca. Fonsibacter” is better explained as a community by-product. Ecotype differentiation of “Ca. Fonsibacter” as a means of escaping predation was supported but not for overcoming auxotrophies. IMPORTANCE This study examines evolutionary and ecological relationships of three of the most ubiquitous and abundant freshwater bacterial genera: “Ca. Planktophila” (acI-A), “Ca. Nanopelagicus” (acI-B), and “Ca. Fonsibacter” (LD12). Due to high abundance, these genera might have a significant influence on nutrient cycling in freshwaters worldwide, and this study adds a layer of understanding to how seemingly competing clades of bacteria can coexist by having different cooperation strategies. Our synthesis ties together network and ecological theory with empirical evidence and lays out a framework for how the functioning of populations within complex microbial communities can be studied.

Coarse, Medium or Fine? A Quantum Mechanics Approach to Single Species Population Dynamics

Standard heuristic mathematical models of population dynamics are often constructed using ordinary differential equations (ODEs). These deterministic models yield pre-dictable results which allow researchers to make informed recommendations on public policy. A common immigration, natural death, and fission ODE model is derived from a quantum mechanics view. This macroscopic ODE predicts that there is only one stable equilibrium point . We therefore presume that as t → ∞, the expected value should be . The quantum framework presented here yields the same standard ODE model, however with very unexpected quantum results, namely . The obvious questions are: why isn’t , why are the probabilities ≈ 0.37, and where is the missing probability of 0.26? The answer lies in quantum tunneling of probabilities. The goal of this paper is to study these tunneling effects that give specific predictions of the uncertainty in the population at the macroscopic level. These quantum effects open the possibility of searching for “black–swan” events. In other words, using the more sophisticated quantum approach, we may be able to make quantitative statements about rare events that have significant ramifications to the dynamical system.

STUDY OF SINGLE-SPECIES POPULATION MODELS

In this research paper one dimensional population models developed centuries ago shows that growth and/decay of single homogeneous populations But environmental effects spatial heterogeneity or age-structure deterministic models prevailing single species population models.

Streamlined freshwater bacterioplankton Nanopelagicales (acI) and “Ca. Fonsibacter” (LD12) thrive in functional cohorts

ABSTRACTWhile fastidious microbes can be abundant and ubiquitous in their natural communities, many fail to grow axenically in laboratories due to auxotrophies or other dependencies. To overcome auxotrophies these microbes rely on their surrounding cohort. A cohort may consist of kin (ecotypes) or more distantly related organisms (community) with the cooperation being reciprocal or non-reciprocal, and expensive (Black Queen hypothesis) or costless (byproduct). These metabolic partnerships (whether at single species population or community level) enable dominance by and coexistence of these lineages in nature. Here we examine the relevance of these cooperation models to explain the abundance and ubiquity of the dominant fastidious bacterioplankton of a dimictic mesotrophic freshwater lake. Using both culture dependent (minimalist mixed cultures) and culture independent (SSU rRNA gene time series and environmental metagenomics) methods we independently identified the primary cohorts of Actinobacterial genera “Ca. Planktophila” (acI-A) and “Ca. Nanopelagicus” (acI-B), and the Proteobacterial genus “Ca. Fonsibacter” (LD12). While “Ca. Planktophila” and “Ca. Fonsibacter” had no correlation in their natural habitat, they have the potential to be complementary in laboratory settings. We also investigated the bi-functional catalase-peroxidase enzyme KatG (a common good which “Ca. Planktophila” is dependent upon) and its most likely providers in the lake. Further we found that while ecotype and community cooperation combined may explain “Ca. Planktophila” population abundance, the success of “Ca. Nanopelagicus” and “Ca. Fonsibacter” is better explained as a community byproduct. Ecotype differentiation of “Ca. Fonsibacter” as a means of escaping predation was supported but not for overcoming auxotrophies.IMPORTANCEThis study examines evolutionary and ecological relationships of three of the most ubiquitous and abundant freshwater bacterial genera: “Ca. Planktophila” (acI-A), “Ca. Nanopelagicus” (acI-B), and “Ca. Fonsibacter” (LD12). Due to high abundance, these genera might have a significant influence on nutrient cycling in freshwaters worldwide and this study adds a layer of understanding to how seemingly competing clades of bacteria can co-exist by having different cooperation strategies. Our synthesis ties together network and ecological theory with empirical evidence and lays out a framework for how the functioning of populations within complex microbial communities can be studied.