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## Mathematical Biosciences & Engineering

2006 , Volume 3 , Issue 1

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2006, 3(1): i-ix
doi: 10.3934/mbe.2006.3.1i

*+*[Abstract](4103)*+*[PDF](330.5KB)**Abstract:**

Zhien Ma's love for mathematics has strongly shaped his educational pursuits. He received formal training from the strong Chinese School of Dynamical System over a period of two years at Peking University and during his later visit to Nanjing University where the internationally renowned professor Yanqian Ye mentored him. Yet, it is well known that Zhien's curiosity and love of challenges have made him his own best teacher. Hence, it is not surprising to see his shift from an outstanding contributor to the field of dynamical systems to a pioneer in the field of mathematical biology. Zhien's vision and courage became evident when he abandoned a promising career in pure mathematics and enthusiastically embraced a career in the field of mathematical biology soon after his first visit to the United States in 1985. His rapid rise to his current role as an international leader and premier mentor to 11 Ph.D. students has facilitated the placement of Chinese scientists and scholars at the forefront of research in the fields of mathematical, theoretical and computational biology.

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2006, 3(1): 1-15
doi: 10.3934/mbe.2006.3.1

*+*[Abstract](2913)*+*[PDF](188.1KB)**Abstract:**

The SARS epidemic of 2002-3 led to the study of epidemic models including management measures and other generalizations of the original 1927 epidemic model of Kermack and McKendrick. We consider some natural extensions of the Kermack-McKendrick model and show that they share the main properties of the original model.

2006, 3(1): 17-36
doi: 10.3934/mbe.2006.3.17

*+*[Abstract](2596)*+*[PDF](1417.0KB)**Abstract:**

In this paper we consider the bifurcations that occur at the trivial equilibrium of a general class of nonlinear Leslie matrix models for the dynamics of a structured population in which only the oldest class is reproductive. Using the inherent net reproductive number n as a parameter, we show that a global branch of positive equilibria bifurcates from the trivial equilibrium at $n=1$ despite the fact that the bifurcation is nongeneric. The bifurcation can be either supercritical or subcritical, but unlike the case of a generic transcritical bifurcation in iteroparous models, the stability of the bifurcating positive equilibria is not determined by the direction of bifurcation. In addition we show that a branch of single-class cycles also bifurcates from the trivial equilibrium at $n=1$. In the case of two population classes, either the bifurcating equilibria or the bifurcating cycles are stable (but not both) depending on the relative strengths of the inter- and intra-class competition. Strong inter-class competition leads to stable cycles in which the two population classes are temporally separated. In the case of three or more classes the bifurcating cycles often lie on a bifurcating invariant loop whose structure is that of a cycle chain consisting of the different phases of a periodic cycle connected by heteroclinic orbits. Under certain circumstances, these bifurcating loops are attractors.

2006, 3(1): 37-50
doi: 10.3934/mbe.2006.3.37

*+*[Abstract](2261)*+*[PDF](248.8KB)**Abstract:**

The dynamics of a differential functional equation system representing an allelopathic competition is analyzed. The delayed allelochemical production process is represented by means of a distributed delay term in a linear quorum-sensing model. Sufficient conditions for local asymptotic stability properties of biologically meaningful steady-state solutions are given in terms of the parameters of the system. A global asymptotic stability result is also proved by constructing a suitable Lyapunov functional. Some simulations confirm the analytical results.

2006, 3(1): 51-65
doi: 10.3934/mbe.2006.3.51

*+*[Abstract](2076)*+*[PDF](272.3KB)**Abstract:**

A heterogenous environment usually impacts, and sometimes determines, the structure and function of organisms in a population. We simulate the effects of a chemical on a population in a spatially heterogeneous environment to determine perceived stressor and spatial effects on dynamic behavior of the population. The population is assumed to be physiologically structured and composed of individuals having both sessile and mobile life history stages, who utilize energetically-controlled, resource-directed, chemical-avoidance advective movements and are subjected to random or density dependent diffusion. From a modeling perspective, the presence of a chemical in the environment requires introduction of both an exposure model and an effects module. The spatial location of the chemical stressor determines the exposure levels and ultimately the effects on the population while the relative location of the resource and organism determines growth. We develop a mathematical model, the numerical analysis for this model, and the simulation techniques necessary to solve the problem of population dynamics in an environment where heterogeneity is generated by resource and chemical stressor. In the simulations, the chemical is assumed to be a nonpolar narcotic and the individuals respond to the chemical via both physiological response and by physical movement. In the absence of a chemical stressor, simulation experiments indicate that despite a propensity to move to regions of higher resource density, organisms need not concentrate in the vicinity of high levels of resource. We focus on the dynamical variations due to advection induced by the toxicant. It is demonstrated that the relationship between resource levels and toxicant concentrations is crucial in determining persistence or extinction of the population.

2006, 3(1): 67-77
doi: 10.3934/mbe.2006.3.67

*+*[Abstract](2579)*+*[PDF](221.2KB)**Abstract:**

In this paper we outline some methods of finding limit cycles for planar autonomous systems with small parameter perturbations. Three ways of studying Hopf bifurcations and the method of Melnikov functions in studying Poincaré bifurcations are introduced briefly. A new method of stability-changing in studying homoclinic bifurcation is described along with some interesting applications to polynomial systems.

2006, 3(1): 79-87
doi: 10.3934/mbe.2006.3.79

*+*[Abstract](2125)*+*[PDF](180.0KB)**Abstract:**

For a reaction-diffusion model of microbial flow reactor with two competing populations, we show the coexistence of weakly coupled traveling wave solutions in the sense that one organism undergoes a population growth while another organism remains in a very low population density in the first half interval of the space line; the population densities then exchange the position in the next half interval. This type of traveling wave can occur only if the input nutrient slightly exceeds the maximum carrying capacity for these two populations. This means, lacking an adequate nutrient, two competing organisms will manage to survive in a more economical way.

2006, 3(1): 89-100
doi: 10.3934/mbe.2006.3.89

*+*[Abstract](2385)*+*[PDF](213.4KB)**Abstract:**

We formulate differential susceptibility and differential infectivity models for disease transmission in this paper. The susceptibles are divided into

*n*groups based on their susceptibilities, and the infectives are divided into

*m*groups according to their infectivities. Both the standard incidence and the bilinear incidence are considered for different diseases. We obtain explicit formulas for the reproductive number. We define the reproductive number for each subgroup. Then the reproductive number for the entire population is a weighted average of those reproductive numbers for the subgroups. The formulas for the reproductive number are derived from the local stability of the infection-free equilibrium. We show that the infection-free equilibrium is globally stable as the reproductive number is less than one for the models with the bilinear incidence or with the standard incidence but no disease-induced death. We then show that if the reproductive number is greater than one, there exists a unique endemic equilibrium for these models. For the general cases of the models with the standard incidence and death, conditions are derived to ensure the uniqueness of the endemic equilibrium. We also provide numerical examples to demonstrate that the unique endemic equilibrium is asymptotically stable if it exists.

2006, 3(1): 101-109
doi: 10.3934/mbe.2006.3.101

*+*[Abstract](3223)*+*[PDF](169.0KB)**Abstract:**

In this paper, an SIR epidemic model for the spread of an infectious disease transmitted by direct contact among humans and vectors (mosquitoes) which have an incubation time to become infectious is formulated. It is shown that a disease-free equilibrium point is globally stable if no endemic equilibrium point exists. Further, the endemic equilibrium point (if it exists) is globally stable with respect to a ''weak delay''. Some known results are generalized.

2006, 3(1): 111-123
doi: 10.3934/mbe.2006.3.111

*+*[Abstract](2457)*+*[PDF](307.2KB)**Abstract:**

A competition model of the chemostat with an external inhibitor is considered. This inhibitor is lethal to one competitor and results in the decrease of growth rate of this competitor. The existence and stability of the extinction equilibria are discussed by using Liapunov function. The necessary and sufficient condition guaranteeing the existence of the interior equilibrium is given. It is found by numerical simulation that the system may be globally stable or have a stable limit cycle if the interior equilibrium exists.

2006, 3(1): 125-135
doi: 10.3934/mbe.2006.3.125

*+*[Abstract](2882)*+*[PDF](1741.2KB)**Abstract:**

By using the theory of planar dynamical systems to the travelling wave equation of a higher order nonlinear wave equations of KdV type, the existence of smooth solitary wave, kink wave and anti-kink wave solutions and uncountably infinite many smooth and non-smooth periodic wave solutions are proved. In different regions of the parametric space, the sufficient conditions to guarantee the existence of the above solutions are given. In some conditions, exact explicit parametric representations of these waves are obtain.

2006, 3(1): 137-144
doi: 10.3934/mbe.2006.3.137

*+*[Abstract](2542)*+*[PDF](167.2KB)**Abstract:**

The permanence of the following Lotka-Volterra system with time delays

$\dot{x}_ 1(t) = x_1(t)[r_1 - a_1x_1(t) + a_11x_1(t - \tau_11) + a_12x_2(t - \tau_12)]$,

$\dot{x}_ 2(t) = x_2(t)[r_2 - a_2x_2(t) + a_21x_1(t - \tau_21) + a_22x_2(t - \tau_22)]$,

is considered. With intraspecific competition, it is proved that in competitive case, the system is permanent if and only if the interaction matrix of the system satisfies condition (C1) and in cooperative case it is proved that condition (C2) is sufficient for the permanence of the system.

2006, 3(1): 145-160
doi: 10.3934/mbe.2006.3.145

*+*[Abstract](2547)*+*[PDF](1181.9KB)**Abstract:**

A patchy model for the spatial spread of West Nile virus is formulated and analyzed. The basic reproduction number is calculated and compared for different long-range dispersal patterns of birds, and simulations are carried out to demonstrate discontinuous or jump spatial spread of the virus when the birds' long-range dispersal dominates the nearest neighborhood interaction and diffusion of mosquitoes and birds.

2006, 3(1): 161-172
doi: 10.3934/mbe.2006.3.161

*+*[Abstract](3299)*+*[PDF](207.1KB)**Abstract:**

In this paper we derive threshold conditions for eradication of diseases that can be described by seasonally forced susceptible-exposed-infectious-recovered (SEIR) models or their variants. For autonomous models, the basic reproduction number $\mathcal{R}_0 < 1$ is usually both necessary and sufficient for the extinction of diseases. For seasonally forced models, $\mathcal{R}_0$ is a function of time $t$. We find that for models without recruitment of susceptible individuals (via births or loss of immunity), max$_t{\mathcal{R}_0(t)} < 1$ is required to prevent outbreaks no matter when and how the disease is introduced. For models with recruitment, if the latent period can be neglected, the disease goes extinct if and only if the basic reproduction number $\bar{\mathcal{R}}$ of the time-average systems (the autonomous systems obtained by replacing the time-varying parameters with their long-term time averages) is less than 1. Otherwise, $\bar{\mathcal{R}} < 1$ is sufficient but not necessary for extinction. Thus, reducing $\bar{\mathcal{R}}$ of the average system to less than 1 is sufficient to prevent or curtail the spread of an endemic disease.

2006, 3(1): 173-187
doi: 10.3934/mbe.2006.3.173

*+*[Abstract](2876)*+*[PDF](3553.9KB)**Abstract:**

We consider the following Lotka-Volterra predator-prey system with two delays:

$x'(t) = x(t) [r_1 - ax(t- \tau_1) - by(t)]$

$y'(t) = y(t) [-r_2 + cx(t) - dy(t- \tau_2)]$ (E)

We show that a positive equilibrium of system (E) is globally asymptotically stable for small delays. Critical values of time delay through which system (E) undergoes a Hopf bifurcation are analytically determined. Some numerical simulations suggest an existence of subcritical Hopf bifurcation near the critical values of time delay. Further system (E) exhibits some chaotic behavior when $tau_2$ becomes large.

2006, 3(1): 189-204
doi: 10.3934/mbe.2006.3.189

*+*[Abstract](2739)*+*[PDF](220.6KB)**Abstract:**

The nonlinear $L^2$-stability (instability) of the equilibrium states of two-species population dynamics with dispersal is studied. The obtained results are based on (i) the rigorous reduction of the $L^2$-nonlinear stability to the stability of the zero solution of a linear binary system of ODEs and (ii) the introduction of a particular Liapunov functional V such that the sign of $\frac{dV}{dt}$ along the solutions is linked directly to the eigenvalues of the linear problem.

2006, 3(1): 205-218
doi: 10.3934/mbe.2006.3.205

*+*[Abstract](3100)*+*[PDF](243.9KB)**Abstract:**

The goal of this paper is to study the global spread of SARS. We propose a multiregional compartmental model using medical geography theory (central place theory) and regarding each outbreak zone (such as Hong Kong, Singapore, Toronto, and Beijing) as one region. We then study the effect of the travel of individuals (especially the infected and exposed ones) between regions on the global spread of the disease.

2006, 3(1): 219-235
doi: 10.3934/mbe.2006.3.219

*+*[Abstract](3047)*+*[PDF](217.4KB)**Abstract:**

The frequency-dependent (standard) form of the incidence is used for the transmission dynamics of an infectious disease in a competing species model. In the global analysis of the SIS model with the birth rate independent of the population size, a modified reproduction number $\mathbf{R}_1$ determines the asymptotic behavior, so that the disease dies out if $\mathbf{R}_1 \leq 1$ and approaches a globally attractive endemic equilibrium if $\mathbf{R}_1 > 1$. Because the disease- reduced reproduction and disease-related death rates are often different in two competing species, a shared disease can change the outcome of the competition. Models of SIR and SIRS type are also considered. A key result in all of these models with the frequency-dependent incidence is that the disease must either die out in both species or remain endemic in both species.

2006, 3(1): 237-248
doi: 10.3934/mbe.2006.3.237

*+*[Abstract](2654)*+*[PDF](211.9KB)**Abstract:**

Based on some important experimental dates, in this paper we shall introduce time delays into Mehrs's non-linear differential system model which is used to describe proliferation, differentiation and death of T cells in the thymus (see, for example, [3], [6], [7] and [9]) and give a revised nonlinear differential system model with time delays. By using some classical analysis techniques of functional differential equations, we also consider local and global asymptotic stability of the equilibrium and the permanence of the model.

2006, 3(1): 249-266
doi: 10.3934/mbe.2006.3.249

*+*[Abstract](3747)*+*[PDF](347.7KB)**Abstract:**

Ecstasy has gained popularity among young adults who frequent raves and nightclubs. The Drug Enforcement Administration reported a 500 percent increase in the use of ecstasy between 1993 and 1998. The number of ecstasy users kept growing until 2002, years after a national public education initiative against ecstasy use was launched. In this study, a system of differential equations is used to model the peer-driven dynamics of ecstasy use. It is found that backward bifurcations describe situations when sufficient peer pressure can cause an epidemic of ecstasy use. Furthermore, factors that have the greatest influence on ecstasy use as predicted by the model are highlighted. The effect of education is also explored, and the results of simulations are shown to illustrate some possible outcomes.

2006, 3(1): 267-279
doi: 10.3934/mbe.2006.3.267

*+*[Abstract](3531)*+*[PDF](434.1KB)**Abstract:**

Epidemic models with behavior changes are studied to consider effects of protection measures and intervention policies. It is found that intervention strategies decrease endemic levels and tend to make the dynamical behavior of a disease evolution simpler. For a saturated infection force, the model may admit a stable disease-free equilibrium and a stable endemic equilibrium at the same time. If we vary a recovery rate, numerical simulations show that the boundaries of the region for the persistence of the disease undergo the changes from the separatrix of a saddle to an unstable limit cycle. If the inhibition effect from behavior changes is weak, we find two limit cycles and obtain bifurcations of the model as the population size changes. We also find that the disease may die out although there are two endemic equilibria.

2018
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