Mathematical modeling and analysis of harmful algal blooms in flowing habitats.

In this paper, we survey recent developments of mathematical modeling and analysis of the dynamics of harmful algae in riverine reservoirs. To make the models more realistic, a hydraulic storage zone is incorporated into a flow reactor model and new mathematical challenges arise from the loss of compactness of the solution maps. The key point in the study of the evolution dynamics is to prove the existence of global attractors for the model systems and the principal eigenvalues for the associated linearized systems without compactness.

Extinction and uniform persistence in a microbial food web with mycoloop: limiting behavior of a population model with parasitic fungi.

It is recently known that parasites provide a better picture of an ecosystem, gaining attention in theoretical ecology. Parasitic fungi belong to a food chain between zooplankton and inedible phytoplankton, called mycoloop. We consider a chemostat model that incorporates a single mycoloop, and analyze the limiting behavior of solutions, adding to previous work on steady-state analysis. By way of persistence theory, we establish that a given species survives depending on the food web configuration and the nutrient level. Moreover, we conclude that the model predicts coexistence under bounded nutrient levels.

Single species growth consuming inorganic carbon with internal storage in a poorly mixed habitat.

This paper presents a PDE system modeling the growth of a single species population consuming inorganic carbon that is stored internally in a poorly mixed habitat. Inorganic carbon takes the forms of "CO2" (dissolved CO2 and carbonic acid) and "CARB" (bicarbonate and carbonate ions), which are substitutable in their effects on algal growth. We first establish a threshold type result on the extinction/persistence of the species in terms of the sign of a principal eigenvalue associated with a nonlinear eigenvalue problem. If the habitat is the unstirred chemostat, we add biologically relevant assumptions on the uptake functions and prove the uniqueness and global attractivity of the positive steady state when the species persists.

Numerical proof for chemostat chaos of Shilnikov's type.

A classical chemostat model is considered that models the cycling of one essential abiotic element or nutrient through a food chain of three trophic levels. The long-time behavior of the model was known to exhibit complex dynamics more than 20 years ago. It is still an open problem to prove the existence of chaos analytically. In this paper, we aim to solve the problem numerically. In our approach, we introduce an artificial singular parameter to the model and construct singular homoclinic orbits of the saddle-focus type which is known for chaos generation. From the configuration of the nullclines of the equations that generates the singular homoclinic orbits, a shooting algorithm is devised to find such Shilnikov saddle-focus homoclinic orbits numerically which in turn imply the existence of chaotic dynamics for the original chemostat model.

Algal competition in a water column with excessive dioxide in the atmosphere.

This paper deals with a resource competition model of two algal species in a water column with excessive dioxide in the atmosphere. First, the uniqueness of positive steady state solutions to the single-species model with two resources is established by the application of the degree theory and the strong maximum principle for the cooperative system. Second, some asymptotic behavior of the single-species model is given by comparison principle and uniform persistence theory. Third, the coexistence solutions to the competition system of two species with two substitutable resources are obtained by global bifurcation theory, various estimates and the strong maximum principle for the cooperative system. Numerical simulations are used to illustrate the outcomes of coexistence and competitive exclusion.

Theory of a microfluidic serial dilution bioreactor for growth of planktonic and biofilm populations.

We present the theory of a microfluidic bioreactor with a two-compartment growth chamber and periodic serial dilution. In the model, coexisting planktonic and biofilm populations exchange by adsorption and detachment. The criteria for coexistence and global extinction are determined by stability analysis of the global extinction state. Stability analysis yields the operating diagram in terms of the dilution and removal ratios, constrained by the plumbing action of the bioreactor. The special case of equal uptake function and logistic growth is analytically solved and explicit growth curves are plotted. The presented theory is applicable to generic microfluidic bioreactors with discrete growth chambers and periodic dilution at discrete time points. Therefore, the theory is expected to assist the design of microfluidic devices for investigating microbial competition and microbial biofilm growth under serial dilution conditions.

Dynamics of phytoplankton communities under photoinhibition.

We analyzed a model of phytoplankton competition for light in a well-mixed water column. The model, proposed by Gerla et al. (Oikos 120:519-527, 2011), assumed inhibition of photosynthesis at high irradiance (photoinhibition). We described the global behavior through mathematical analyses, providing a general solution to the multi-species competition for light with photoinhibition. We classified outcomes of 2- and 3-species competitions as examples, and evaluated feasibility of the theoretical predictions using empirical relationships between photosynthetic production and irradiance. Numerical simulations with published p-I curves indicate that photoinhibition may often lead to strong Allee effects and competitive facilitation among species. Hence, our results suggest that photoinhibition may play a major role in organizing phytoplankton communities.

Competition between microorganisms for a single limiting resource with cell quota structure and spatial variation.

Microbial populations compete for nutrient resources, and the simplest mathematical models of competition neglect differences in the nutrient content of individuals. The simplest models also assume a spatially uniform habitat. Here both of these assumptions are relaxed. Nutrient content of individuals is assumed proportional to cell size, which varies for populations that reproduce by division, and the habitat is taken to be an unstirred chemostat where organisms and nutrients move by simple diffusion. In a spatially uniform habitat, the size-structured model predicts competitive exclusion, such that only the species with lowest break-even concentration persists. In the unstirred chemostat, coexistence of two competitors is possible, if one has a lower break-even concentration and the other can grow more rapidly. In all habitats, the calculation of competitive outcomes depends on a principal eigenvalue that summarizes relationships among cell growth, cell division, and cell size.

A Lotka-Volterra competition model with seasonal succession.

A complete classification for the global dynamics of a Lotka-Volterra two species competition model with seasonal succession is obtained via the stability analysis of equilibria and the theory of monotone dynamical systems. The effects of two death rates in the bad season and the proportion of the good season on the competition outcomes are also discussed.

Subfunctionalization reduces the fitness cost of gene duplication in humans by buffering dosage imbalances.

BACKGROUND: Driven essentially by random genetic drift, subfunctionalization has been identified as a possible non-adaptive mechanism for the retention of duplicate genes in small-population species, where widespread deleterious mutations are likely to cause complementary loss of subfunctions across gene copies. Through subfunctionalization, duplicates become indispensable to maintain the functional requirements of the ancestral locus. Yet, gene duplication produces a dosage imbalance in the encoded proteins and thus, as investigated in this paper, subfunctionalization must be subject to the selective forces arising from the fitness bottleneck introduced by the duplication event. RESULTS: We show that, while arising from random drift, subfunctionalization must be inescapably subject to selective forces, since the diversification of expression patterns across paralogs mitigates duplication-related dosage imbalances in the concentrations of encoded proteins. Dosage imbalance effects become paramount when proteins rely on obligatory associations to maintain their structural integrity, and are expected to be weaker when protein complexation is ephemeral or adventitious. To establish the buffering effect of subfunctionalization on selection pressure, we determine the packing quality of encoded proteins, an established indicator of dosage sensitivity, and correlate this parameter with the extent of paralog segregation in humans, using species with larger population -and more efficient selection- as controls. CONCLUSIONS: Recognizing the role of subfunctionalization as a dosage-imbalance buffer in gene duplication events enabled us to reconcile its mechanistic nonadaptive origin with its adaptive role as an enabler of the evolution of genetic redundancy. This constructive role was established in this paper by proving the following assertion: If subfunctionalization is indeed adaptive, its effect on paralog segregation should scale with the dosage sensitivity of the duplicated genes. Thus, subfunctionalization becomes adaptive in response to the selection forces arising from the fitness bottleneck imposed by gene duplication.

Synchronized reproduction promotes species coexistence through reproductive facilitation.

Theories for species coexistence often emphasize niche differentiation and temporal segregation of recruitment to avoid competition. Recent work on mutualism suggested that plant species sharing pollinators provide mutual facilitation when exhibit synchronized reproduction. The facilitation on reproduction may enhance species persistence and coexistence. Theoretical ecologists paid little attention to such indirect mutualistic systems by far. We propose a new model for a two-species system using difference equations. The model focuses on adult plants and assumes no resource competition between these well-established individuals. Our formulas include demographic parameters, such as mortality and recruitment rates, and functions of reproductive facilitation. Both recruitment and facilitation effects reach saturation levels when flower production is at high levels. We conduct mathematical analyses to assess conditions of coexistence. We establish demographical conditions permitting species coexistence. Our analyses suggest a "rescue" effect from a "superior" species to a "weaker" species under strong recruitment enhancement effect when the later is not self-sustainable. The facilitation on rare species may help to overcome Allee effect.

Competition and coexistence in flowing habitats with a hydraulic storage zone.

This paper examines a model of a flowing water habitat with a hydraulic storage zone in which no flow occurs. In this habitat, one or two microbial populations grow while consuming a single nutrient resource. Conditions for persistence of one population and coexistence of two competing populations are derived from eigenvalue problems, the theory of bifurcation and the theory of monotone dynamical systems. A single population persists if it can invade the trivial steady state of an empty habitat. Under some conditions, persistence occurs in the presence of a hydraulic storage zone when it would not in an otherwise equivalent flowing habitat without such a zone. Coexistence of two competing species occurs if each can invade the semi-trivial steady state established by the other species. Numerical work shows that both coexistence and enhanced persistence due to a storage zone occur for biologically reasonable parameters.

Non-periodicity in chemostat equations: a multi-dimensional negative Bendixson-Dulac criterion.

We establish conditions which exclude periodic solutions in a simple chemostat with a single nutrient and N competing species. Growth rates are not required to be proportional to food uptake. Instead of a Lyapunov function approach, we develop and apply a multi-dimensional Bendixson-Dulac type exclusion principle based on differential forms.

On the role of asymptomatic infection in transmission dynamics of infectious diseases.

We propose a compartmental disease transmission model with an asymptomatic (or subclinical) infective class to study the role of asymptomatic infection in the transmission dynamics of infectious diseases with asymptomatic infectives, e.g., influenza. Analytical results are obtained using the respective ratios of susceptible, exposed (incubating), and asymptomatic classes to the clinical symptomatic infective class. Conditions are given for bistability of equilibria to occur, where trajectories with distinct initial values could result in either a major outbreak where the disease spreads to the whole population or a lesser outbreak where some members of the population remain uninfected. This dynamic behavior did not arise in a SARS model without asymptomatic infective class studied by Hsu and Hsieh (SIAM J. Appl. Math. 66(2), 627-647, 2006). Hence, this illustrates that depending on the initial states, control of a disease outbreak with asymptomatic infections may involve more than simply reducing the reproduction number. Moreover, the presence of asymptomatic infections could result in either a positive or negative impact on the outbreak, depending on different sets of conditions on the parameters, as illustrated with numerical simulations. Biological interpretations of the analytical and numerical results are also given.

Impact of quarantine on the 2003 SARS outbreak: a retrospective modeling study.

During the 2003 Severe Acute Respiratory Syndrome (SARS) outbreak, traditional intervention measures such as quarantine and border control were found to be useful in containing the outbreak. We used laboratory verified SARS case data and the detailed quarantine data in Taiwan, where over 150,000 people were quarantined during the 2003 outbreak, to formulate a mathematical model which incorporates Level A quarantine (of potentially exposed contacts of suspected SARS patients) and Level B quarantine (of travelers arriving at borders from SARS affected areas) implemented in Taiwan during the outbreak. We obtain the average case fatality ratio and the daily quarantine rate for the Taiwan outbreak. Model simulations is utilized to show that Level A quarantine prevented approximately 461 additional SARS cases and 62 additional deaths, while the effect of Level B quarantine was comparatively minor, yielding only around 5% reduction of cases and deaths. The combined impact of the two levels of quarantine had reduced the case number and deaths by almost a half. The results demonstrate how modeling can be useful in qualitative evaluation of the impact of traditional intervention measures for newly emerging infectious diseases outbreak when there is inadequate information on the characteristics and clinical features of the new disease-measures which could become particularly important with the looming threat of global flu pandemic possibly caused by a novel mutating flu strain, including that of avian variety.

The final size of a SARS epidemic model without quarantine.

In this article, we present the continuing work on a SARS model without quarantine by Hsu and Hsieh [Sze-Bi Hsu, Ying-Hen Hsieh, Modeling intervention measures and severity-dependent public response during severe acute respiratory syndrome outbreak, SIAM J. Appl. Math. 66 (2006) 627-647]. An "acting basic reproductive number" ψ is used to predict the final size of the susceptible population. We find the relation among the final susceptible population size S ∞ , the initial susceptible population S 0 , and ψ. If ψ > 1 , the disease will prevail and the final size of the susceptible, S ∞ , becomes zero; therefore, everyone in the population will be infected eventually. If ψ < 1 , the disease dies out, and then S ∞ > 0 which means part of the population will never be infected. Also, when S ∞ > 0 , S ∞ is increasing with respect to the initial susceptible population S 0 , and decreasing with respect to the acting basic reproductive number ψ.

SARS outbreak, Taiwan, 2003.

We studied the severe acute respiratory syndrome (SARS) outbreak in Taiwan, using the daily case-reporting data from May 5 to June 4 to learn how it had spread so rapidly. Our results indicate that most SARS-infected persons had symptoms and were admitted before their infections were reclassified as probable cases. This finding could indicate efficient admission, slow reclassification process, or both. The high percentage of nosocomial infections in Taiwan suggests that infection from hospitalized patients with suspected, but not yet classified, cases is a major factor in the spread of disease. Delays in reclassification also contributed to the problem. Because accurate diagnostic testing for SARS is currently lacking, intervention measures aimed at more efficient diagnosis, isolation of suspected SARS patients, and reclassification procedures could greatly reduce the number of infections in future outbreaks.

A ratio-dependent food chain model and its applications to biological control.

While biological controls have been successfully and frequently implemented by nature and human, plausible mathematical models are yet to be found to explain the often observed deterministic extinctions of both pest and control agent in such processes. In this paper we study a three trophic level food chain model with ratio-dependent Michaelis-Menten type functional responses. We shall show that this model is rich in boundary dynamics and is capable of generating such extinction dynamics. Two trophic level Michaelis-Menten type ratio-dependent predator-prey system was globally and systematically analyzed in details recently. A distinct and realistic feature of ratio-dependence is its capability of producing the extinction of prey species, and hence the collapse of the system. Another distinctive feature of this model is that its dynamical outcomes may depend on initial populations levels. Theses features, if preserved in a three trophic food chain model, make it appealing for modelling certain biological control processes (where prey is a plant species, middle predator as a pest, and top predator as a biological control agent) where the simultaneous extinctions of pest and control agent is the hallmark of their successes and are usually dependent on the amount of control agent. Our results indicate that this extinction dynamics and sensitivity to initial population levels are not only preserved, but also enriched in the three trophic level food chain model. Specifically, we provide partial answers to questions such as: under what scenarios a potential biological control may be successful, and when it may fail. We also study the questions such as what conditions ensure the coexistence of all the three species in the forms of a stable steady state and limit cycle, respectively. A multiple attractor scenario is found.

Plasmid-bearing, plasmid-free organisms competing for two complementary nutrients in a chemostat.

A model of competition for two complementary nutrients between plasmid-bearing and plasmid-free organisms in a chemostat is proposed. A rigorous mathematical analysis of the global asymptotic behavior of the model is presented. The work extends the model of competition for a single-limited nutrient studied by Stephanopoulos and Lapidus [Chem. Engng. Sci. 443 (1988) 49] and Hsu, Waltman and Wolkowicz [J. Math. Biol. 32 (1994) 731].