Global dynamics of discrete mathematical models of tuberculosis

In this paper, we develop discrete models of Tuberculosis (TB). This includes SEI endogenous and exogenous models without treatment. These models are then extended to a SEIT model with treatment. We develop two types of net reproduction numbers, one is the traditional R 0 which is based on the disease-free equilibrium, and a new net reproduction number R 0 ( E ∗ ) based on the endemic equilibrium. It is shown that the disease-free equilibrium is globally asymptotically stable if R 0 ≤   1 and unstable if R 0 > 1 . Moreover, the endemic equilibrium is locally asymptotically stable if R 0 ( E ∗ ) < 1 < R 0 .

The effects of evolution on the stability of competing species

Based on evolutionary game theory and Darwinian evolution, we propose and study discrete-time competition models of two species where at least one species has an evolving trait that affects their intra-specific, but not their inter-specific competition coefficients. By using perturbation theory, and the theory of the limiting equations of non-autonomous discrete dynamical systems, we obtain global stability results. Our theoretical results indicate that evolution may promote and/or suppress the stability of the coexistence equilibrium depending on the environment. This relies crucially on the speed of evolution and on how the intra-specific competition coefficient depends on the evolving trait. In general, equilibrium destabilization occurs when α>2, when the speed of evolution is sufficiently slow. In this case, we conclude that evolution selects against complex dynamics. However, when evolution proceeds at a faster pace, destabilization can occur when α<2. In this case, if the competition coefficient is highly sensitive to changes in the trait v, destabilization and complex dynamics occur. Moreover, destabilization may lead to either a period-doubling bifurcation, as in the non-evolutionary Ricker equation, or to a Neimark-Sacker bifurcation.

A continuous-time mathematical model and discrete approximations for the aggregation of β-Amyloid

Alzheimer's disease is a degenerative disorder characterized by the loss of synapses and neurons from the brain, as well as the accumulation of amyloid-based neuritic plaques. While it remains a matter of contention whether β-amyloid causes the neurodegeneration, β-amyloid aggregation is associated with the disease progression. Therefore, gaining a clearer understanding of this aggregation may help to better understand the disease. We develop a continuous-time model for β-amyloid aggregation using concepts from chemical kinetics and population dynamics. We show the model conserves mass and establish conditions for the existence and stability of equilibria. We also develop two discrete-time approximations to the model that are dynamically consistent. We show numerically that the continuous-time model produces sigmoidal growth, while the discrete-time approximations may exhibit oscillatory dynamics. Finally, sensitivity analysis reveals that aggregate concentration is most sensitive to parameters involved in monomer production and nucleation, suggesting the need for good estimates of such parameters.

Discrete evolutionary population models: a new approach

In this paper, we apply a new approach to a special class of discrete time evolution models and establish a solid mathematical foundation to analyse them. We propose new single and multi-species evolutionary competition models using the evolutionary game theory that require a more advanced mathematical theory to handle effectively. A key feature of this new approach is to consider the discrete models as non-autonomous difference equations. Using the powerful tools and results developed in our recent work [E. D'Aniello and S. Elaydi, The structure of ω-limit sets of asymptotically non-autonomous discrete dynamical systems, Discr. Contin. Dyn. Series B. 2019 (to appear).], we embed the non-autonomous difference equations in an autonomous discrete dynamical systems in a higher dimension space, which is the product space of the phase space and the space of the functions defining the non-autonomous system. Our current approach applies to two scenarios. In the first scenario, we assume that the trait equations are decoupled from the equations of the populations. This requires specialized biological and ecological assumptions which we clearly state. In the second scenario, we do not assume decoupling, but rather we assume that the dynamics of the trait is known, such as approaching a positive stable equilibrium point which may apply to a much broader evolutionary dynamics.

Effects of Cholesterol in Stress-Related Neuronal Death-A Statistical Analysis Perspective

The association between plasma cholesterol levels and the development of dementia continues to be an important topic of discussion in the scientific community, while the results in the literature vary significantly. We study the effect of reducing oxidized neuronal cholesterol on the lipid raft structure of plasma membrane. The levels of plasma membrane cholesterol were reduced by treating the intact cells with methyl-ß-cyclodextrin (MßCD). The relationship between the cell viability with varying levels of MßCD was then examined. The viability curves are well described by a modified form of the empirical Gompertz law of mortality. A detailed statistical analysis is performed on the fitting results, showing that increasing MßCD concentration has a minor, rather than significant, effect on the cellular viability. In particular, the dependence of viability on MßCD concentration was found to be characterized by a ~25% increase per 1 μM of MßCD concentration.

Hierarchical competition models with the Allee effect III: multispecies

A general notion of the Allee effect for higher dimensional triangular maps is proposed. A global dynamics theory is established. The theory is applied to multi-species hierarchical models. Then we provide a detailed study of the global dynamics of three-species Ricker competition models with the Allee effect. Regions of extinction, exclusion and coexistence are identified.

A discrete mathematical model for the aggregation of β-Amyloid

Dementia associated with the Alzheimer's disease is thought to be correlated with the conversion of the β − Amyloid (Aβ) peptides from soluble monomers to aggregated oligomers and insoluble fibrils. We present a discrete-time mathematical model for the aggregation of Aβ monomers into oligomers using concepts from chemical kinetics and population dynamics. Conditions for the stability and instability of the equilibria of the model are established. A formula for the number of monomers that is required for producing oligomers is also given. This may provide compound designers a mechanism to inhibit the Aβ aggregation.

A discrete-time host-parasitoid model with an Allee effect

We introduce a discrete-time host-parasitoid model with a strong Allee effect on the host. We adapt the Nicholson-Bailey model to have a positive density dependent factor due to the presence of an Allee effect, and a negative density dependence factor due to intraspecific competition. It is shown that there are two scenarios, the first with no interior fixed points and the second with one interior fixed point. In the first scenario, we show that either both host and parasitoid will go to extinction or there are two regions, an extinction region where both species go to extinction and an exclusion region in which the host survives and tends to its carrying capacity. In the second scenario, we show that either both host and parasitoid will go to extinction or there are two regions, an extinction region where both species go to extinction and a coexistence region where both species survive.

Hierarchical competition models with the Allee effect II: the case of immigration

This is part II of an earlier paper that dealt with hierarchical models with the Allee effect but with no immigration. In this paper, we greatly simplify the proofs in part I and provide a proof of the global dynamics of the non-hyperbolic cases that were previously conjectured. Then, we show how immigration to one of the species or to both would, drastically, change the dynamics of the system. It is shown that if the level of immigration to one or to both species is above a specified level, then there will be no extinction region where both species go to extinction.

Hierarchical competition models with Allee effects

We consider a two-species hierarchical competition model with a strong Allee effect. The Allee effect is assumed to be caused by predator saturation. Moreover, we assume that there is a 'silverback' species x that gets first choice of the resources and where growth is limited by its own intraspecific competition, while the second 'inferior' species y gets whatever is left. Both species x and y are assumed to have the property of strong Allee effect. In this paper we determine the impact of the presence of the Allee effect on the global dynamics of both species.

Stability and invariant manifolds of a generalized Beddington host-parasitoid model

We will investigate the stability and invariant manifolds of a new discrete host-parasitoid model. It is a generalization of the Beddington-Nicholson-Bailey model. Our study establishes analytically, for the first time, the stability of the coexistence fixed point.

General Allee effect in two-species population biology

The main objective of this work is to present a general framework for the notion of the strong Allee effect in population models, including competition, mutualistic, and predator-prey models. The study is restricted to the strong Allee effect caused by an inter-specific interaction. The main feature of the strong Allee effect is that the extinction equilibrium is an attractor. We show how a 'phase space core' of three or four equilibria is sufficient to describe the essential dynamics of the interaction between two species that are prone to the Allee effect. We will introduce the notion of semistability in planar systems. Finally, we show how the presence of semistable equilibria increases the number of possible Allee effect cores.

Periodic difference equations, population biology and the Cushing–Henson conjectures

We show that for a k-periodic difference equation, if a periodic orbit of period r is globally asymptotically stable (GAS), then r must be a divisor of k. Moreover, if r divides k we construct a non-autonomous dynamical system having minimum period k and which has a GAS periodic orbit with minimum period r. Our method uses the technique of skew-product dynamical systems. Our methods are then applied to prove two conjectures of Cushing and Henson concerning a non-autonomous Beverton-Holt equation which arises in the study of the response of a population to a periodically fluctuating environmental force such as seasonal fluctuations in carrying capacity or demographic parameters like birth or death rates. We give an equality linking the average population with the growth rates and carrying capacities (in the 2-periodic case) which shows that out-of-phase oscillations in these quantities always have a deleterious effect on the average population. We give an example where in-phase oscillations cause the opposite to occur.