Diffusive Logistic Equations with Degenerate Boundary Conditions

The purpose of this paper is to provide a careful and accessible exposition of static bifurcation theory for a class of degenerate boundary value problems for diffusive logistic equations with indefinite weights that model population dynamics in environments with spatial heterogeneity. We discuss the changes that occur in the structure of the positive solutions as a parameter varies near the first eigenvalue of the linearized problem, and prove that the most favorable situations will occur if there is a relatively large favorable region (with good resources and without crowding effects) located some distance away from the boundary of the environment.Dedicated to Professor Mitsuru Ikawa on the occasion of his 60th birthday     

Maximization of the total population in a reaction–diffusion model with logistic growth

This paper is concerned with a nonlinear optimization problem that naturally arises in population biology. We consider the effect of spatial heterogeneity on the total population of a biological species at a steady state, using a reaction–diffusion logistic model. Our objective is to maximize the total population when resources are distributed in the habitat to control the intrinsic growth rate, but the total amount of resources is limited. It is shown that under some conditions, any local maximizer must be of “bang–bang” type, which gives a partial answer to the conjecture addressed by Ding et al. (Nonlinear Anal Real World Appl 11(2):688–704, 2010). To this purpose, we compute the first and second variations of the total population. When the growth rate is not of bang–bang type, it is shown in some cases that the first variation becomes nonzero and hence the resource distribution is not a local maximizer. When the first variation becomes zero, we prove that the second variation is positive. These results implies that the bang–bang property is essential for the maximization of total population.     

Bifurcations in a predator–prey model with general logistic growth and exponential fading memory

A multiparameter predator–prey system generalizing the model introduced in [6] is considered. The system studied in this paper corresponds to the type of models with exponential fading memory where the logistic per capita rate growth of the prey is given by an arbitrary function of class C^k, k ≥ 3. We prove that the model has a Hopf bifurcation and that there exist open sets in the parameter space such that the system exhibits singular attractors and asymptotically stable limit cycles. A numerical simulation is conducted in order to show the existence of critical attractor elements.As pointed out by Ayala et al. in [14], the Lotka–Volterra model of interspecific competition, which is based on the logistic theory of population growth and assumes that the intra and interspecific competitive interactions between species are linear, does not explain satisfactorily the population dynamics of some species. This is due to fact that the model does not take into account some important features of the population, which affect its dynamics. The model introduced in this paper provides independent conditions of these facts, for the existence of a Hopf bifurcation and the asymptotically stable limit cycles.     

Global well-posedness for the diffusion equation of population genetics

In this paper, we study the forward diffusion equation of population genetics. We prove the global existence of smooth solutions if the initial value is smooth. We also show that if the initial value is singular on the boundary, in a weighted Sobolev space, the diffusion equation exists a unique weak solution which is a probability density function. Moreover, we investigate the asymptotic behavior of the weak solution by the entropy method.     

总人口规模变化的年龄结构MSEIR流行病模型的再生数

在总人口规模变化和疾病影响死亡率的假设下,讨论了带二次感染和接种疫苗的年龄结构MSEIR流行病模型.首先给出再生数R(ψ,λ)(这里ψ(a)是接种疫苗率,λ是总人口的增长指数)的显式表达式.其次,证明了当R(ψ,λ)<1时,系统的无病平衡态是稳定的;当R(ψ,λ)>1时,无病平衡态是不稳定的.     

Competition in patchy space with cross-diffusion and toxic substances

The main goal of this paper is to continue our investigations of the important system (see [S. Aly, M. Farkas, Competition in patchy environment with cross diffusion, Nonlinear Analysis: Real World Applications 5 (2004) 589–595]), by considering a Lotka–Volterra competitive system affected by toxic substances in two patches in which the per capita migration rate of each species is influenced not only by its own but also by the other one’s density, i.e. there is cross-diffusion present and it is assumed that the individuals of a particular species will initiate toxin production at a rate proportional not only to its own but also to the other one’s density. In the absence of diffusion, we study the conditions of the existence and stability properties of the equilibrium point with toxic substances. For the full general model (with both toxic substances and diffusion) we show that at a critical value of the bifurcation parameter of diffusion the system undergoes a Turing bifurcation and numerical studies show that if the bifurcation parameter of diffusion is increased through a critical value the spatially homogeneous equilibrium loses its stability and two new stable equilibria emerge, i.e., the cross-migration response is an important factor that should not be ignored when a pattern emerges.     

Qualitative analysis on a diffusive SIS epidemic system with logistic source and spontaneous infection in a heterogeneous environment

In this paper, we consider an SIS epidemic reaction–diffusion model with spontaneous infection and logistic source in a heterogeneous environment. The uniform bounds of solutions are established, and the global asymptotic stability of the constant endemic equilibrium is discussed in the case of homogeneous environment. This paper aims to analyze the asymptotic profile of endemic equilibria (when it exists) as the diffusion rate of the susceptible or infected population is small or large. Our results on this new model reveal that varying total population and spontaneous infection can enhance persistence of infectious disease, which may provide some implications on disease control and prediction.     

一类非线性人口系统的稳定性和振荡

周期现象和振荡是非保守系统中存在着的极为普遍的现象,K.E.Swick 和 M.E.Gurtin & R.c.MacCamy 就几种描述生态演化的非线性动力模型讨论了系统周期解的存在性.由于周期解是系统的一种特殊的平衡态,对这种现象的研究无论从理论上还是实践上都有明显的意义,本文的目的就是在研究一类非线性人口发展方程稳定性的     

应急物流能力突变模型研究

运用燕尾突变理论,以物流能力为状态变量,物流流量变化率、流速变化率和时间变化率为控制变量,建立应急物流能力突变模型,运用势函数确定了分岐点集,讨论了应急物流能力的突变临界点及稳定性,并用算例分析模型应用的可行性.最后得出三点结论:根据实测及调查数据可以确定三个控制变量的值,从而确定控制点在分歧点集的区域;通过计算分析可以确定应急物流能力在分岐点集各区域的奇点个数和性质;其突变方向和可能性随之可以确定:控制点从奇点多的区域向奇点少的区域移动,应急物流能力发生突变的可能性大,反之可能性小,甚至不发生突变.因此把握控制点在分岐点集中的变化方向和规律,采取相应措施改变相应控制变量,可以提升或稳定应急物流能力.     

The challenge in managing new financial risks: adopting an heuristic or theoretical approach

The financial crisis began with the collapse of Lehman Brothers and the subprime asset backed securities debacle. Credit risk was turned into liquidity risk, resulting in a lack of confidence among financial institutions. In this article, we will propose a way to model liquidity risk and the credit risk in best practices. We will show that liquidity risk is a new type of risk and the current way to deal with it is based solely on observed variables without any theoretical link. We propose an heuristic approach to combine the numerous liquidity risk indicators with a logistic regression for the first time. In regards to credit risk, several articles prove that the best practice is to use an option model to appreciate this risk. We will present our methodology using stochastic diffusion for the interest rate because currently the yield curves aren’t liquid. This approach is more relevant because the basis model in prior publications has a constant interest rate or a forward rate. Both models allow a better understanding of liquidity and credit risks and the further development of research deals with the link between these two financial risks.     

On a quasilinear parabolic–elliptic chemotaxis system with logistic source

This paper deals with a quasilinear parabolic–elliptic chemotaxis system with logistic source, under homogeneous Neumann boundary conditions in a smooth bounded domain. For the case of positive diffusion function, it is shown that the corresponding initial boundary value problem possesses a unique global classical solution which is uniformly bounded. Moreover, if the diffusion function is zero at some point, or a positive diffusion function and the logistic damping effect is rather mild, we proved that the weak solutions are global existence. Finally, it is asserted that the solutions approach constant equilibria in the large time for a specific case of the logistic source.     

A population explosion in an evolutionary game in spatial economics: Blow up radial solutions to the initial value problem for the replicator equation whose growth rate is determined by the continuous Dixit–Stiglitz–Krugman model in an urban setting

We consider a spatially continuous evolutionary game whose payoff is defined as the density of real wages that is determined by the continuous Dixit–Stiglitz–Krugman model in an urban setting. This evolutionary game is expressed by the initial value problem for the replicator equation whose growth rate contains an operator which acts on an unknown function that denotes the density of workers. We prove that this initial value problem has a unique global solution, and that if workers are distributed radially in space and concentrated in the neighborhood of a city at the initial time, then all workers will move toward the center of the city in such a way that the density of workers converges to the Dirac delta function in the sense of distribution. In the real world this result describes a population explosion caused by concentration of workers motivated by the disparity in real wages.     

Population dynamical behavior of a single-species nonlinear diffusion system with random perturbation

We consider a single-species stochastic logistic model with the population’s nonlinear diffusion between two patches. We prove the system is stochastically permanent and persistent in mean, and then we obtain sufficient conditions for stationary distribution and extinction. Finally, we illustrate our conclusions through numerical simulation.     

Spatial Dynamics of a Lattice Lotka-Volterra Competition Model with a Shifting Habitat

In this paper, we concern with the spatial dynamics of the lattice Lotka-Volterra competition system in a shifting habitat. We study the impact of the environmental deterioration rate on the population density under the strong competition condition. Our results show that if the environment deteriorates rapidly, both species will become extinct. However, when the environmental degradation rate is not so fast, the species with slow diffusion will go extinct, while those with fast diffusion will survive. The extinction of species with slow diffusion can be divided into two situations: one is the extinction caused by environmental deterioration faster than its own diffusion speed, the other is the extinction caused by slow diffusion speed under the influence of strong competition.     

The Oberbeck-Boussinesq problem modified by a thermo-absorption term

We consider the Oberbeck-Boussinesq problem with an extra coupling, establishing a suitable relation between the velocity and the temperature. Our model involves a system of equations given by the transient Navier-Stokes equations modified by introducing the thermo-absorption term. The model involves also the transient temperature equation with nonlinear diffusion. For the obtained problem, we prove the existence of weak solutions for any N?2 and its uniqueness if N=2. Then, considering a low range of temperature, but upper than the phase changing one, we study several properties related with vanishing in time of the velocity component of the weak solutions. First, assuming the buoyancy forces field extinct after a finite time, we prove the velocity component will extinct in a later finite time, provided the thermo-absorption term is sublinear. In this case, considering a suitable buoyancy forces field which vanishes at some instant of time, we prove the velocity component extinct at the same instant. We prove also that for non-zero buoyancy forces, but decaying at a power time rate, the velocity component decay at analogous power time rates, provided the thermo-absorption term is superlinear. At last, we prove that for a general non-zero bounded buoyancy force, the velocity component exponentially decay in time whether the thermo-absorption term is sub or superlinear.     

A moving boundary problem with space-fractional diffusion logistic population model and density-dependent dispersal rate

In this article, we discuss a space-fractional diffusion logistic population model with Caputo fractional derivative and density-dependent dispersal rate. The numerical solution of the problem is obtained by using a finite difference scheme. The consistency and stability of the scheme for our solution to the problem are also discussed. The effect of the density-dependent dispersal rate and order of the space-fractional derivative are analyzed for the population density and expanding front (moving boundary).     

Dynamics of a Stochastic SIR Epidemic Model with Logistic Growth

In this paper, a stochastic SIR epidemic model with saturated treatment function, non-monotone incidence rate and logistic growth is studied. First, we prove that the stochastic model has a unique global positive solution. Next, by constructing a suitable Lyapunov function, we can show that there exists an ergodic stationary distribution in the random SIR model. Then, we show that a sufficient condition can make the disease tend to extinction. Finally, some numerical simulations are used to prove our analytical result.     

Dynamics of a population in two patches with dispersal

A two-dimensional discrete system of a species in two patches proposed by Newman et al. is studied. It is shown that the unique interior steady state is globally asymptotically stable if the active population has a Beverton–Holt type growth rate. If the population is also subject to Allee effects, then the system has two interior steady states whenever the density-independent growth rate is large. In addition, the model has period-two solutions if the symmetric dispersal exceeds a critical threshold. For small dispersal, populations may either go extinct or eventually stabilize. However, populations are oscillating over time if dispersal is beyond the critical value and the initial populations are large.     

医疗资源影响下具有潜伏期的一类传染病模型建立及性态分析

建立了医疗资源影响下的考虑疾病具有潜伏期的一类传染病模型,并分析了模型的动力学性态.发现疾病流行与否由基本再生数和医院病床数共同决定,并得到了病床数的阈值条件.当基本再生数R_0大于1时,系统只存在惟一正平衡点,且通过构造Dulac函数证明了正平衡点只要存在一定是全局渐近稳定的;当R_01,我们得到系统存在两个正平衡点及无正平衡点的条件,且只有当医院的病床数小于阈值时,系统会经历后向分支.因此,可根据实际情况使医院病床的投入量不低于阈值条件,不仅有利于疾病的控制而且不会出现医疗资源过剩的现象.     

A numerical method for nonlinear age-structured population models with finite maximum age

We propose a new numerical method for the approximation of solutions to a non-autonomous form of the classical Gurtin-MacCamy population model with a mortality rate that is the sum of an intrinsic age-dependent rate that becomes unbounded as the age approaches its maximum value, plus a non-local, non-autonomous, bounded rate that depends on some weighted population size. We prove that our new quadrature based method converges to second-order and we show the results of several numerical simulations.