On the dynamics of a predator–prey model with nonconstant death rate and diffusion

The main goal of this paper is to describe the global dynamic of a predator–prey model with nonconstant death rate and diffusion. We obtain necessary and sufficient conditions under which the system is dissipative and permanent. We study the global stability of the nontrivial equilibrium, when it is unique. Finally, we show that there are no nontrivial steady state solutions for certain parameter configuration.     

Boundary layer analysis in the semiclassical limit of a quantum drift–diffusion model

We study a singularly perturbed elliptic second order system in one space variable as it appears in a stationary quantum drift–diffusion model of a semiconductor. We prove the existence of solutions and their uniqueness as minimizers of a certain functional and determine rigorously the principal part of an asymptotic expansion of a boundary layer of those solutions. We prove analytical estimates of the remainder terms of this asymptotic expansion, and confirm by means of numerical simulations that these remainder estimates are sharp.     

Characterization of the Bernoulli–Navier model for a rectangular section beam as the limit of the Kirchhoff–Love model for a plate

In this paper, we compare the Kirchhoff–Love model for a linearly elastic rectangular plate \({\Omega^{t\varepsilon}=(0,L)\times(-t,t)\times(-\varepsilon,\varepsilon)}\) of thickness \({2\varepsilon}\) with the Bernoulli–Navier model for the same solid considered as a linearly elastic beam of length \({L}\) and cross section \({\omega_1^{t\varepsilon}=(-t,t)\times(-\varepsilon,\varepsilon)}\). We assume that the solid is clamped on both ends \({\{0,L\}\times[-t,t]\times[-\varepsilon,\varepsilon]}\). We show that the scaled version of the displacements field \({{\bf{\zeta}}^t}\) in the middle plane, solution of the Kirchhoff–Love model, converges strongly to the unique solution of a one-dimensional problem when the plate width parameter \({t}\) tends to zero. Moreover, after rescaling this limit, we show that, as a matter of fact, it is the solution of the Bernoulli–Navier model for the beam. This means that, under appropriate assumptions on the order of magnitude of the data, the Bernoulli–Navier displacement field is the natural approximation of the Kirchhoff–Love displacement field when the cross section of the plate is rectangular and its width is sufficiently small and homothetic to thickness.     

Homogenization of the Peierls–Nabarro model for dislocation dynamics

This paper is concerned with a result of homogenization of an integro-differential equation describing dislocation dynamics. Our model involves both an anisotropic Lévy operator of order 1 and a potential depending periodically on $u/?$. The limit equation is a non-local Hamilton–Jacobi equation, which is an effective plastic law for densities of dislocations moving in a single slip plane.     

Global dynamics and zero-diffusion limit of a parabolic–elliptic–parabolic system for ion transport networks

This paper is concerned with a parabolic–elliptic–parabolic system arising from ion transport networks. It shows that for any properly regular initial data, the corresponding initial–boundary value problem associated with Neumann–Dirichlet boundary conditions possesses a global classical solution in one-dimensional setting, which is uniformly bounded and converges to a trivial steady state, either in infinite time with a time-decay rate or in finite time. Moreover, by taking the zero-diffusion limit of the third equation of the problem, the global weak solution of its partially diffusive counterpart is established and the explicit convergence rate of the solution of the fully diffusive problem toward the solution of the partially diffusive counterpart, as the diffusivity tends to zero, is obtained.     

Global dynamics of the Kirschner–Panetta model for the tumor immunotherapy

In this paper we examine the global dynamics of the Kirschner–Panetta model describing the tumor immunotherapy. We give upper and lower ultimate bounds for densities of cell populations involved in this model. We demonstrate for this dynamics that there is a positively invariant polytope in the positive orthant. We present sufficient conditions on model parameters and treatment parameters under which all trajectories in the positive orthant tend to the tumor-free equilibrium point. We compare our results with Kirschner–Tsygvintsev results and concern biological implications of our assertions.     

Global dynamics of a predator–prey model with time delay and stage structure for the prey

A stage-structured predator–prey system with Holling type-II functional response and time delay due to the gestation of predator is investigated. By analyzing the characteristic equations, the local stability of each of feasible equilibria of the system is discussed and the existence of a Hopf bifurcation at the coexistence equilibrium is established. By means of the persistence theory on infinite dimensional systems, it is proven that the system is permanent if the coexistence equilibrium exists. By using Lyapunov functionals and LaSalle invariant principle, it is shown that the trivial equilibrium is globally stable when both the predator-extinction equilibrium and the coexistence equilibrium are not feasible, and that the predator-extinction equilibrium is globally asymptotically stable if the coexistence equilibrium does not exist, and sufficient conditions are derived for the global stability of the coexistence equilibrium. Numerical simulations are carried out to illustrate the main theoretical results.     

Spatial segregation limit of a non-autonomous competition–diffusion system

This paper is concerned with the spatial behavior of the non-autonomous competition–diffusion system arising in population ecology. The limiting profile of the system is given as the competition rate tends to infinity. Our result shows that two competing species spatially segregate as the competition rates become large. Moreover, for the case of the same non-autonomous terms, we obtain the uniform convergence result.     

Bifurcation and stability of a two-species reaction–diffusion–advection competition model

This paper is concerned with the dynamics of a two-species reaction–diffusion–advection competition model subject to the no-flux boundary condition in a bounded domain. By the signs of the associated principal eigenvalues, we derive the existence and local stability of the trivial and semi-trivial steady-state solutions. Moreover, the nonexistence and existence of the coexistence steady-state solutions stemming from the two boundary steady states are obtained as well. In particular, we describe the feature of the coincidence of bifurcating coexistence steady-state solution branches. At the same time, the effect of advection on the stability of the bifurcating solution is also investigated, and our results suggest that the advection term may change the stability. Finally, we point out that the methods we applied here are mainly based on spectral analysis, perturbation theory, comparison principle, monotone theory, Lyapunov–Schmidt reduction, and bifurcation theory.     

Spectral methods for bivariate Markov processes with diffusion and discrete components and a variant of the Wright–Fisher model

The aim of this paper is to study differential and spectral properties of the infinitesimal operator of two dimensional Markov processes with diffusion and discrete components. The infinitesimal operator is now a second-order differential operator with matrix-valued coefficients, from which we can derive backward and forward equations, a spectral representation of the probability density, study recurrence of the process and the corresponding invariant distribution. All these results are applied to an example coming from group representation theory which can be viewed as a variant of the Wright–Fisher model involving only mutation effects.     

Asymptotic behavior and stability switch for a mature–immature model of cell differentiation

We consider a nonlinear age-structured model, inspired by hematopoiesis modelling, describing the dynamics of a cell population divided into mature and immature cells. Immature cells, that can be either proliferating or non-proliferating, differentiate in mature cells, that in turn control the immature cell population through a negative feedback. We reduce the system to two delay differential equations, and we investigate the asymptotic stability of the trivial and the positive steady states. By constructing a Lyapunov function, the trivial steady state is proven to be globally asymptotically stable when it is the only equilibrium of the system. The asymptotic stability of the positive steady state is related to a delay-dependent characteristic equation. Existence of a Hopf bifurcation and stability switch for the positive steady state is established. Numerical simulations illustrate the stability results.     

Uniform estimates and uniqueness of stationary solutions to the drift–diffusion model for semiconductors

We study the stationary problem of the drift–diffusion model with a mixed boundary condition. For this problem, the existence of solutions was established in general settings, while the uniqueness was investigated only in some special cases which do not entirely cover situations that semiconductor devices are used in integrated circuits. In this paper, we prove the uniqueness in a physically relevant situation. The key to the proof is to derive two-sided uniform estimates for the densities of the electron and hole. We establish a new technique to show the lower bound. This together with the Moser iteration method leads to the upper bound.     

The effect of diffusion for the multispecies Lotka–Volterra competition model

The purpose of this paper is to study the effect of diffusion in the existence of non-constant steady states for the Lotka–Volterra competition-diffusion system with three species, under Neumann boundary conditions. It will be shown that two large diffusion rates prevent the appearance of non-constant steady states, while if just one species diffuses fast non-constant equilibria may arise. The existence is shown by two methods, degree theory and bifurcation techniques. The stability of bifurcating steady states will be established.     

Consequences of refuge and diffusion in a spatiotemporal predator–prey model

In this investigation, we offer and examine a predator–prey interacting model with prey refuge in proportion to both the species and Beddington–DeAngelis functional response. We first prove the well-posedness of the temporal and spatiotemporal models which are restricted in a positive invariant region. Then for the temporal model, we analyse its temporal dynamics including uniform boundedness, permanence, stability of all feasible non-negative equilibria and show that refugia can induce periodic oscillation via Hopf bifurcation around the unique positive equilibrium; for the spatiotemporal model, we not only investigate its permanence, stability of non-negative constant steady states and Turing instability but also study the existence and non-existence of non-constant positive steady states by Leray–Schauder degree theory. The key observation is that the coefficient of refuge cooperates a significant part in modifying the dynamics of the current system and mediates the population permanence, stability of coexisting equilibrium and even the Turing instability parameter space. Finally, general numerical simulation consequences are given to illustrate the validity of the theoretical results. Through numerical simulations, one observes that the model dynamics shows prey refugia and self-diffusion control spatiotemporal pattern growth to spots, stripe–spot mixtures and stripes reproduction. The outcomes assign that the dynamics of the model with prey refuge is not simple, but rich and complex. Additionally, numerical simulations show that the other model parameters have an important effect on species’ spatially inhomogeneous distribution, which results in the formation of spots pattern, mixture of spots and stripes pattern, mixture of spots, stripes and rings pattern and anti-spot pattern. This may improve the model dynamics of the prey refuge on the reaction–diffusion predator–prey system.