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PROC
Sufficient conditions are given such that the discrete time competition models constructed by applying nonstandard finite difference (NSFD) schemes for the Lotka-Volterra competition models are dynamically consistent. The derived discrete models preserve the
positivity of solutions, local stability conditions, boundedness, and the monotonicity of the continuous Lotka-Volterra system. In other words, we are able to construct discrete-time competition models that behave just like the continuous-time Lotka-Volterra competition models.
PROC
Discrete-time $SI$ and
$SIS$ epidemic models are constructed by applying the nonstandard finite difference (NSFD) schemes to the differential equation models. The difference equation systems are dynamically consistent with their analog continuous-time models.
The basic standard incidence $SI$ and $SIS$ models without births and deaths, with births and deaths, and with immigrations, are considered. The continuous models possess either the conservation law that the total population is a constant or the total population $N$ satisfies $N'(t)=\lambda-\mu N$ and so that $N$ approaches a constant $\lambda/\mu$ as $t$ approaches infinity. The difference equation systems via NSFD schemes preserve all properties including the positivity of solutions, the conservation law, and the local and some of the global stability of the equilibria. They are said to be dynamically consistent with the continuous models with respect to these properties.
We show that a simple criterion for choosing a certain NSFD scheme such that the positivity solutions are preserved is usually an indication of an appropriate NSFD scheme.
DCDS-B
We analyzed the local dynamics of a three-dimensional Ricker type
discrete-time competition model that is analogous to the May-Leonard
(M-L) differential equation model. The symmetric discrete M-L model
is mentioned by Hofbauer et al. [[7] J. Math. Biol.,
25:553--570,1987] as "perhaps one of the most difficult
three species problems''. Both of the discrete and the continuous
M-L models have similar local dynamics. However, the discrete model
is not dynamically consistent with the continuous model. Unlike the
continuous M-L model, the discrete Hopf bifurcations (Neimark-Sacker
bifurcations) of the discrete M-L model are not degenerate. The
continuous M-L model is the limiting case of the discrete model.
DCDS-B
Discrete-time Lotka-Volterra competition models are obtained by applying nonstandard finite difference (NSFD) schemes to the continuous-time counterparts of the model.
The NSFD methods are noncanonical symplectic numerical schemes when applying to the predator-prey model $x'=x-xy$ and $y'=-y+xy$.
The local dynamics of the discrete-time model are analyzed and compared with the continuous model. We find the NSFD schemes that preserve the local dynamics of the continuous model. The local stability criteria are exactly the same between the continuous model and the discrete model independent of the step size. Two specific discrete-time Lotka-Volterra competition models by NSFD schemes that preserve positivity of solutions and monotonicity of the system are also given. The two discrete-time models are dynamically consistent with their continuous counterpart.
MBE
Tuberculosis (TB) is the leading cause of death among individuals infected with the human immunodeficiency virus (HIV). The study of the joint dynamics of HIV and TB present formidable mathematical challenges due to the fact that the models of transmission are quite distinct. Furthermore, although there is overlap in the populations at risk of HIV and TB infections, the magnitude of the proportion of individuals at risk for both diseases is not known. Here, we consider a highly simplified deterministic model that incorporates the joint dynamics of TB and HIV, a model that is quite hard to analyze. We compute independent reproductive numbers for TB ($\R_1$) and HIV ($\R_2$) and the overall reproductive number for the system, $\R =\max \{\R_1, \R_2\}$. The focus is naturally (given the highly simplified nature of the framework) on the qualitative analysis of this model. We find that if $\R <1$ then the disease-free equilibrium is locally asymptotically stable.
The TB-only equilibrium $E_T$ is locally asymptotically
stable if $\R_1>1$ and $\R_2<1$. However, the symmetric condition, $\R_1<1$ and $\R_2>1$,
does not necessarily guarantee the stability of the HIV-only equilibrium $E_H$,
and it is possible that TB can coexist with HIV when $\R_2>1$. In other words, in the case when $\R_1<1$ and $\R_2>1$ (or when $\R_1>1$ and $\R_2>1$), we are able to find a stable HIV/TB coexistence equilibrium.
Moreover, we show that the prevalence level of TB increases with $\R_2>1$ under
certain conditions. Through simulations, we find that
i) the increased progression rate from latent to active TB in co-infected individuals may play a significant role in the rising prevalence of TB; and ii)
the increased progression rates from HIV to AIDS have not only increased the prevalence level of HIV while decreasing TB prevalence, but also generated damped oscillations in the system.
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