Yang Kuang

Uniqueness of limit cycles in Gause-type models of predator-prey systems

This paper deals with the question of uniqueness of limit cycles in predator-prey systems of Gause type. By utilizing several transformations, these systems are reduced to a generalized Lienard system as discussed by Cherkas and Zhilevich and by Zhang. As a consequence, criteria for the uniqueness of limit cycles are derived, which include results of Cheng and is related to results in Liou and Cheng. Several examples are given to illustrate our results.

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.

Predator-Prey Dynamics in Models of Prey Dispersal in Two-Patch Environments*

Models are presented for a single species that disperses between two patches of a heterogeneous environment with barriers between patches and a predator for which the dispersal between patches does not involve a barrier. Conditions are established for the existence, uniform persistence, and local and global stability of positive steady states. In particular, an example that demonstrates both the stabilizing and destabilizing effects of dispersion is presented. This example indicates that a stable migrating predator-prey system can be made unstable by changing the amount of migration in both directions.

Wavefronts and global stability in a time-delayed population model with stage structure

We formulate and study a one–dimensional single–species diffusive–delay population model. The time delay is the time taken from birth to maturity. Without diffusion, the delay differential model extends the well–known logistic differential equation by allowing delayed constant birth processes and instantaneous quadratically regulated death processes. This delayed model is known to have simple global dynamics similar to that of the logistic equation. Through the use of a sub/supersolution pair method, we show that the diffusive delay model continues to generate simple global dynamics. This has the important biological implication that quadratically regulated death processes dramatically simplify the growth dynamics. We also consider the possibility of travelling wavefront solutions of the scalar equation for the mature population, connecting the zero solution of that equation with the positive steady state. Our main finding here is that our fronts appear to be all monotone, regardless of the size of the delay. This is in sharp contrast to the frequently reported findings that delay causes a loss of monotonicity, with the front developing a prominent hump in some other delay models.

Prevention and Control of Zika as a Mosquito-Borne and Sexually Transmitted Disease: A Mathematical Modeling Analysis

The ongoing Zika virus (ZIKV) epidemic in the Americas poses a major global public health emergency. While ZIKV is transmitted from human to human by bites of Aedes mosquitoes, recent evidence indicates that ZIKV can also be transmitted via sexual contact with cases of sexually transmitted ZIKV reported in Argentina, Canada, Chile, France, Italy, New Zealand, Peru, Portugal, and the USA. Yet, the role of sexual transmission on the spread and control of ZIKV infection is not well-understood. We introduce a mathematical model to investigate the impact of mosquito-borne and sexual transmission on the spread and control of ZIKV and calibrate the model to ZIKV epidemic data from Brazil, Colombia, and El Salvador. Parameter estimates yielded a basic reproduction number 0 = 2.055 (95% CI: 0.523–6.300), in which the percentage contribution of sexual transmission is 3.044% (95% CI: 0.123–45.73). Our sensitivity analyses indicate that 0 is most sensitive to the biting rate and mortality rate of mosquitoes while sexual transmission increases the risk of infection and epidemic size and prolongs the outbreak. Prevention and control efforts against ZIKV should target both the mosquito-borne and sexual transmission routes.

Growth and neutral lipid synthesis in green microalgae: A mathematical model

Many green microalgae significantly increased their cellular neutral lipid content when cultured in nitrogen limited or high light conditions. Due to their lipid production potential, these algae have been suggested as promising feedstocks for biofuel production. However, no models for algal lipid synthesis with respect to nutrient and light have been developed to predict lipid production and to help improve the production process. A mathematical model is derived describing the growth dynamics and neutral lipid production of green microalgae grown in batch cultures. The model assumed that as the nitrogen was depleted, photosynthesis became uncoupled from growth, resulting in the synthesis and accumulation of neutral lipids. Simulation results were compared with experimental data for the green microalgae Pseudochlorococcum sp. For growth media with low nitrogen concentration, the model agreed closely with the data; however, with high nitrogen concentration the model overestimated the biomass. It is likely that additional limiting factors besides nitrogen could be responsible for this discrepancy.

Global existence of periodic solutions in a class of delayed Gause-type predator-prey systems

where x(t), y(t) denote the population density of prey and predator at time t, respectively. g(e), p(e) and /z(s) are assumed to satisfy appropriate conditions. v, a positive constant, stands for the death rate of predator y in the absence of prey x. We may think of this system as herbivores (y) grazing upon vegetation (x), which take time r to recover. In view of the fact that many predator-prey systems display sustained fluctuations (for examples, see Freedman [l]), it is thus desirable to construct predator-prey models capable of producing nonconstant periodic solutions. When r = 0, (1.1) reduces to the well-known undelayed Gause-type or Kolmogorov predator-prey systems (again, see Freedman [l] and the references cited theorein). Under some assumptions, the resulting systems can have a unique limit cycle in the positive cone (see Kuang and Freedman [2] and the references cited therein). To the best of our knowledge about delayed population interaction models, the existing literature establishes the existence of a periodic solution by Hopf bifurcation argument, or reduces the delayed systems to undelayed systems of higher dimensions. Therefore, these existence results are generally local. Most of the existing results on the global existence of nonconstant periodic solutions in delayed equations deal with scalar equations with single discrete delay, such as 13-101

Dynamics of a delay differential equation model of hepatitis B virus infection

We formulate and systematically study the global dynamics of a simple model of hepatitis B virus in terms of delay differential equations. This model has two important and novel features compared to the well-known basic virus model in the literature. Specifically, it makes use of the more realistic standard incidence function and explicitly incorporates a time delay in virus production. As a result, the infection reproduction number is no longer dependent on the patient liver size (number of initial healthy liver cells). For this model, the existence and the component values of the endemic steady state are explicitly dependent on the time delay. In certain biologically interesting limiting scenarios, a globally attractive endemic equilibrium can exist regardless of the time delay length.

Competition and stoichiometry: coexistence of two predators on one prey.

The competitive exclusion principle (CEP) states that no equilibrium is possible if n species exploit fewer than n resources. This principle does not appear to hold in nature, where high biodiversity is commonly observed, even in seemingly homogenous habitats. Although various mechanisms, such as spatial heterogeneity or chaotic fluctuations, have been proposed to explain this coexistence, none of them invalidates this principle. Here we evaluate whether principles of ecological stoichiometry can contribute to the stable maintenance of biodiverse communities. Stoichiometric analysis recognizes that each organism is a mixture of multiple chemical elements such as carbon (C), nitrogen (N), and phosphorus (P) that are present in various proportions in organisms. We incorporate these principles into a standard predator-prey model to analyze competition between two predators on one autotrophic prey. The model tracks two essential elements, C and P, in each species. We show that a stable equilibrium is possible with two predators on this single prey. At this equilibrium both predators can be limited by the P content of the prey. The analysis suggests that chemical heterogeneity within and among species provides new mechanisms that can support species coexistence and that may be important in maintaining biodiversity.

Modeling and analysis of a marine bacteriophage infection with latency period

In a previous paper [4] the authors proposed a simple model to describe the epidemics induced by bacteriophages in marine bacteria populations like cyanobacteria and heterotrophic bacteria where the environment is the thermoclinic layer of the sea within which bacteriophages and bacteria are assumed to be homogeneously distributed. The main model simpli cation was in modeling the latent period of infected bacteria in order to describe the model with three nonlinear ordinary di erential equations. However, modeling of the latent period by suitable delay terms looks to be biologically reasonable and mathematically challenging, the ndings of which can be interesting to compare with the outcomes of our previous model [4]. Hence, in the following we rst recall the biological justi cation for the model and then introduce the model itself comparing it with other models on the same topic. The experimental evidence of the bacteriophage infection of marine bacteria can be found, for example, in the papers by Sieburth [14], Moebus [11], Bergh et al. [3]; Proctor and Fuhrman [12]. It is