Sophia R.-J. Jang

Allee effects in a discrete-time host-parasitoid model

A discrete-time Beverton-Holt stock recruitment model with Allee effects is proposed and studied. Its asymptotic dynamics are then used to investigate a host-parasitoid model with Allee effects occurring in the host population. There exists a population threshold of which initial host population abundance must exceed so that the host population may persist. Both populations are more likely to become extinct when Allee effects are incorporated into this host-parasitoid interaction.

Nutrient-phytoplankton-zooplankton models with a toxin

Models of nutrient-plankton interaction with a toxic substance that inhibits either the growth rate of phytoplankton, zooplankton or both trophic levels are proposed and studied. For simplicity, it is assumed that both nutrient and the toxin have the same constant input and washout rate as the chemostat system. The effects of the toxin upon the existence, magnitude, and stability of the steady states are examined. Numerical simulations demonstrate that the system can have multiple attractors when the phytoplanktons nutrient uptake rate is inhibited by the toxin.

A host-parasitoid interaction with Allee effects on the host

We explore the addition of Allee effects to single-species discrete-time models with overcompensatory density dependence. When the intrinsic growth rate of the population, r, is large, the population bifurcates into chaos. The population goes extinct if r is either below a threshold level or very large. The model is then used to study host-parasitoid interactions with and without Allee effects in the host. The coexistence of the host and parasitoid populations both depends on threshold levels of r, threshold levels of the host population size, and the parasitoid potential, which is the product of the searching efficiency of the parasitoid for the host, and the fecundity of the parasitoid. The addition of Allee effects has a negative impact on the coexistence of both populations.

Discrete-time host–parasitoid models with Allee effects: Density dependence versus parasitism

We present two general discrete-time host–parasitoid models with Allee effects on the host. In the first model, it is assumed that parasitism occurs prior to density dependence, while in the second model we assume that density dependence operates first followed by parasitism. It is shown that both models have similar asymptotic behaviour. The parasitoid population will definitely go extinct if the maximal growth rate of the host population is less than or equal to one, independent of whether density dependence or parasitism occurs first. The fate of the population is initial condition dependent if this maximal growth rate exceeds one. In particular, there exists a host population threshold, the Allee threshold, below which the host population goes extinct and so does the parasitoid. This threshold is the same for both models. Numerical examples with different functions are simulated to illustrate our analytical results.

A discrete two-stage population model: continuous versus seasonal reproduction

A discrete two-stage model which describes the dynamics of a population where juveniles and adults compete for different resources is developed. A motivating example is the green tree frog (Hyla cinerea) where tadpoles and adult frogs feed on separate resources. First, continuous breeding is assumed and the asymptotic behavior of the resulting autonomous model is fully analyzed. It is shown that the unique interior equilibrium is globally asymptotically stable when the inherent net reproductive number is greater than one. However, when the inherent net reproductive number is less than one, the population becomes extinct. Then a seasonal breeding described by a periodic birth rate with period 2 is assumed. It is proved that for this nonautonomous model a period two solution is globally asymptotically stable when the inherent net reproductive number is greater than one and when the inherent net reproductive number is less than one the population becomes extinct. Finally, the advantage (in terms of maximizing the number of juveniles and adults in the population over a fixed time period) of having a seasonal breeding is studied by comparing the average of the juvenile and adult numbers of the periodic solution for the nonautonomous model to the equilibrium solution of the autonomous model. Our results indicate that for high birth rates the equilibrium of the autonomous model is higher than the average of the two cycle solution. Therefore, all other factors being equal, seasonal breeding appears to be deleterious to populations with high birth rates. However, for low birth rates seasonal breeding can be beneficial. It is also shown that for a range of birth rates the nonautnomous model is persistent while the solution to the autonomous model goes to extinction.

On a discrete West Nile epidemic model

A West Nile epidemic model in discrete-time is proposed. The model consists of two interacting populations, the vector and the avian populations. The avian population is classified into susceptible, infective, and recovered classes while an individual vector is either susceptible or infective. The transmission of the disease is assumed only by mosquitoes bites and vertical transmission in the vector population. The model behavior depends on a lumpedparameter R0. The disease-free equilibrium is locally asymptotically stable if R0 1. Consequently, the disease can persist in the populations if R0 > 1.

A nutrient-prey-predator model with intratrophic predation

A simple food chain which consists of nutrient, prey and predator with intratrophic predation of the predator is proposed and analyzed. The dynamics of the model depend on the basic reproductive numbers of the prey and predator. Intratrophic predation can have impact on the system only if the basic reproductive number of the predator is greater than 1. In this case, the system may have multiple coexisting equilibria. However, intratrophic predation can stabilize the system when such an equilibrium is unique. Moreover, it can elevate the magnitude of the prey population and diminish the level of nutrient concentration of any coexisting equilibrium.

A simple food chain with a growth inhibiting nutrient

We study the dynamics of a simple food chain consisting of a nutrient, prey and predator, where the nutrient is growth inhibiting at high concentrations. The Michaelis-Menten-Monod form for the nutrient uptake rate is generalized to a nonmonotone uptake rate. It is shown that the positive equilibrium is a global attractor for low initial concentrations of the nutrient, i.e., when there is no inhibition effect. However, the behavior of the system can be initial condition dependent at high initial concentrations of the nutrient; persistence may not occur. Several thresholds are defined in terms of the model parameters which determine the global dynamics of the system.

Dynamics of phytoplankton-zooplankton systems with toxin producing phytoplankton

We propose periodic systems of phytoplankton-zooplankton interactions with toxin producing phytoplankton to study the effects of TPP upon extinction and persistence of the populations. Using the concept of uniform persistence, it is proved that the two populations can coexist for certain parameter regimes. The numerical investigation demonstrates that toxin producing phytoplankton may promote survival of zooplankton population on one hand and may destabilize the interactions on the other hand. Moreover, passive diffusion of both populations can simplify the dynamics of the interactions and exhibit plankton patchiness.

Deterministic and stochastic nutrient-phytoplankton- zooplankton models with periodic toxin producing phytoplankton

Deterministic and stochastic nutrient-phytoplankton-zooplankton models with toxin producing phytoplankton are studied.The input nutrient concentration and the toxin liberation rate play critical roles in the plankton dynamics.Periodic toxin liberation by phytoplankton can complicate plankton interactions.Harmful algal blooms can be diminished if the maximal toxin liberation rate is large. Deterministic and stochastic models of nutrient-phytoplankton-zooplankton interaction are proposed to investigate the impact of toxin producing phytoplankton upon persistence of the populations. The toxin liberation by phytoplankton is modeled periodically in the deterministic system. We derive two thresholds in terms of the parameters for which both plankton populations can persist if these thresholds are positive. We construct stochastic models with Ito differential equations to model variability in the environment. It is concluded that the input nutrient concentration along with the toxin liberation rate play critical roles in the dynamics of the planktonic interaction. In particular, toxin producing phytoplankton can terminate harmful algal blooms and the planktonic interaction is more stable if either the input nutrient concentration is smaller or if the toxin production rate is larger.