Mathematical analysis of a thermo-visco-plastic model with Bodner–Partom constitutive equations

We study a thermo-mechanical model, where the mechanical model of inelastic deformation due to S.R. Bodner and Y. Partom is coupled with a heat equation. The main result is the existence and uniqueness of the solution to the thermo-visco-plastic model.     

Mathematical analysis of an eco-epidemiological predator–prey model with stage-structure and latency

In this paper, an eco-epidemiological predator–prey model with stage structure for the prey and a time delay describing the latent period of the disease is investigated. By analyzing corresponding characteristic equations, the local stability of the trivial equilibrium, the predator-extinction equilibrium, the disease-free equilibrium and the endemic equilibrium is addressed. The existence of Hopf bifurcations at the endemic equilibrium is established. By using Lyapunov functionals and LaSalle’s invariance principle, sufficient conditions are obtained for the global asymptotic stability of the trivial equilibrium, the predator-extinction equilibrium, the disease-free equilibrium and the endemic equilibrium of the model.     

Global behavior of solutions in a Lotka–Volterra predator–prey model with prey-stage structure

In this paper, the global behavior of solutions is investigated for a Lotka–Volterra predator–prey system with prey-stage structure. First, we can see that the stability properties of nonnegative equilibria for the weakly coupled reaction–diffusion system are similar to that for the corresponding ODE system, that is, linear self-diffusions do not drive instability. Second, using Sobolev embedding theorems and bootstrap arguments, the existence and uniqueness of nonnegative global classical solution for the strongly coupled cross-diffusion system are proved when the space dimension is less than 10. Finally, the existence and uniform boundedness of global solutions and the stability of the positive equilibrium point for the cross-diffusion system are studied when the space dimension is one. It is found that the cross-diffusion system is dissipative if the diffusion matrix is positive definite. Furthermore, cross diffusions cannot induce pattern formation if the linear diffusion rates are sufficiently large.     

Existence of piecewise weak solutions of a discrete Cucker–Smale's flocking model with a singular communication weight

We prove existence of global C¹$C^1$ piecewise weak solutions for the discrete Cucker–Smale's flocking model with a non-Lipschitz communication weight ψ(s)=s^−α$\psi (s)=s^{-\alpha }$, 0<α<1$0<\alpha <1$. We also discuss the possibility of finite in time alignment of the velocities of the particles.     

Stability and bifurcation analysis of a prey–predator model with age based predation

It is observed that in large animals only adult predators take part in direct predation while suckling feed on milk of adult predators and juveniles are dependent on the dead prey stock killed by the adult predators. Some parts of the dead prey population is consumed by adult predators and remaining parts are consumed by juveniles and the remaining portion decays naturally. In light of this, a mathematical model is proposed to study the stability and bifurcation behaviour of a prey–predator system with age based predation. All the feasible equilibria of the system are obtained and the conditions for the existence of the interior equilibrium are determined. The local stability analysis of all the feasible equilibria is carried out and the possibility of Hopf-bifurcation of the interior equilibrium is studied. Finally, numerical simulation is conducted to support the analytical results.     

Multiple positive periodic solutions for a generalized delayed population model with an exploited term

In this paper, the existence of two positive periodic solutions for a generalized delayed population model with an exploited term is established by using the continuation theorem of the coincidence degree theory.     

Mathematical study of a bacteria–fish model with level of infection structure

In this paper, we propose a mathematical model to study a bacteria–fish system, based upon the interactions between Clostridium botulinum and tilapia, Oerochromis mossambicus. The fish population is divided into susceptible and infected, and the infected fish population is considered structured by the level of infection. The model is thus a system with the infected fish equation being an evolution equation, while those corresponding to the susceptible fish and bacteria in water are ordinary differential equations. The model is firstly transformed into a system with distributed delay for susceptible fish and bacteria and, further, under some assumptions, into a system with discrete delay. The study of this system gives us some results concerning the existence, uniqueness, positivity and boundedness of solutions; we also discuss the existence and stability of its equilibrium points, including conditions for the appearance of Hopf bifurcation. The theoretical results are illustrated by some numerical simulations.     

Dynamic behavior of a plant–wrack model with spatial diffusion

Banded spatial pattern of plant is one of the center issues in the real ecosystems. As a result, in this paper, a plant–wrack model with spatial diffusion is presented. By both mathematical analysis and numerical simulations, we find that the typical dynamics of plant is the formation of isolated groups, i.e., spotted, labyrinth, and coexistence of stripe-like and spotted. The obtained results may provide help in understanding the dynamics in the real ecosystems.     

Global stability of a DS–DI epidemic model with age of infection

A model with differential susceptibility, differential infectivity (DS–DI), and age of infection is formulated in this paper. The susceptibles are divided into n groups according to their susceptibilities. The infectives are divided into m groups according to their infectivities. The total population size is assumed constant. Formula for the reproductive number is derived so that if the reproduction number is less than one, the infection-free equilibrium is locally stable, and unstable otherwise. Furthermore, if the reproductive number is less than one, the infection-free equilibrium is globally asymptotically stable. If the reproductive number is greater than one, it is shown that there exists a unique endemic equilibrium which is globally asymptotically stable. This result is obtained through a Lyapunov function.     

Mathematical properties of a discontinuous Cournot–Stackelberg model

The object of this work is to perform the global analysis of a recent duopoly model which couples the two points of view of Cournot and Stackelberg [17], [18]. The Cournot model is assumed with isoelastic demand function and unit costs. The coupling leads to discontinuous reaction functions, whose bifurcations, mainly border collision bifurcations, are investigated as well as the global structure of the basins of attraction. In particular, new properties are shown, associated with the introduction of horizontal branches, which differ significantly when the constant value is zero or positive and small. The good behavior of the model with positive constant is proved, leading to stable cycles of any period.     

On the asymptotic behavior of solutions of Emden–Fowler equations on time scales

Consider the Emden-Fowler dynamic equation $$ x^{\Delta\Delta}(t)+p(t)x^\alpha(t)=0,\:\:\alpha >0 , \qquad \qquad \qquad \qquad (0.1) $$ where ${p\in C_{rd}([t_0,\infty)_{\mathbb{T}},\mathbb{R}), \alpha}$ is the quotient of odd positive integers, and ${\mathbb{T}}$ denotes a time scale which is unbounded above and satisfies an additional condition (C) given below. We prove that if ${\int^\infty_{t_0}t^\alpha |p(t)|\Delta t<\infty}$ (and when ???=?1 we also assume lim _t???? tp(t)??(t)?=?0), then (0.1) has a solution x(t) with the property that $$ \lim_{t\rightarrow\infty} \frac{x(t)}{t}=A\neq 0.$$      

Complicated asymptotic behavior of solutions for the Cauchy problem of Cahn–Hilliard equation

In this paper, we study the Cauchy problem of the Cahn–Hilliard equation, and first reveal that the complicated asymptotic behavior of solutions can happen in high-order parabolic equation.     

Local solutions of the fast–slow model of an optoelectronic oscillator with delay

Theoretical and Mathematical Physics - We study a difference–differential model of an optoelectronic oscillator that is a modification of the Ikeda equation with delay. We analyze the...     

Bifurcation analysis of an autonomous epidemic predator–prey model with delay

In this paper, a class of an autonomous epidemic predator–prey model with delay is considered. Its linear stability and Hopf bifurcation are investigated. Applying the normal form theory and center manifold theory, the explicit formulas for determining the stability and the direction of the Hopf bifurcation periodic solutions are derived. Some numerical simulations for justifying the theoretical analysis are also provided. Finally, main conclusions are included.     

Maximal regularity and asymptotic behavior of solutions for the Cahn–Hilliard equation with dynamic boundary conditions

This paper deals with the Cahn–Hilliard equation subject to the boundary conditions and the initial condition ψ(0,x) = ψ₀(x) where J = (0,∞), and Ω ⊂ ℝ ⁿ is a bounded domain with smooth boundary Γ = ∂ G, n≤ 3, and Γ _s ,σ _s ,g _s > 0, h are constants. This problem has already been considered in the recent paper of R. Racke and S. Zheng (The Cahn–Hilliard equation with dynamic boundary conditions. Adv. Diff. Eq. 8, 83–110, 2003), where global existence and uniqueness were obtained. In this paper we first obtain results on the maximal L _p -regularity of the solution. We then study the asymptotic behavior of the solution of this problem and prove the existence of a global attractor. Mathematics Subject Classification (2000) 82C26, 35B40, 35B65, 35Q99     

Optimal control of effort of a stage structured prey–predator fishery model with harvesting

This paper describes a prey–predator fishery model with stage structure for prey. The adult prey and predator populations are harvested in the proposed system. The dynamic behavior of the model system is discussed. It is observed that singularity induced bifurcation phenomenon is appeared when variation of the economic interest of harvesting is taken into account. We have incorporated state feedback controller to stabilize the model system in the case of positive economic interest. Fishing effort used to harvest the adult prey and predator populations is used as a control to develop a dynamic framework to investigate the optimal utilization of the resource, sustainability properties of the stock and the resource rent earned from the resource. Pontryagin’s maximum principle is used to characterize the optimal control. The optimal system is derived and then solved numerically using an iterative method with Runge–Kutta fourth-order scheme. Simulation results show that the optimal control scheme can achieve sustainable ecosystem.