Objective Bayesian analysis for a capture–recapture model

In this paper, we study a special capture–recapture model, the $M_t$ model, using objective Bayesian methods. The challenge is to find a justified objective prior for an unknown population size $N$ . We develop an asymptotic objective prior for the discrete parameter $N$ and the Jeffreys’ prior for the capture probabilities $\varvec{\theta }$ . Simulation studies are conducted and the results show that the reference prior has advantages over ad-hoc non-informative priors. In the end, two real data examples are presented.     

Equivalent condition for approximately Cohen–Macaulay complexes

We give a necessary and sufficient condition for a simplicial complex to be approximately Cohen–Macaulay. Namely it is approximately Cohen–Macaulay if and only if the ideal associated to its Alexander dual is componentwise linear and generated in two consecutive degrees. This completes the result of J. Herzog and T. Hibi who proved that a simplicial complex is sequentially Cohen–Macaulay if and only if the ideal associated to its Alexander dual is componentwise linear.     

Traveling waves for a reaction–diffusion–advection predator–prey model

In this paper we study a reaction–diffusion–advection predator–prey model in a river. The existence of predator-invasion traveling wave solutions and prey-spread traveling wave solutions in the upstream and downstream directions is established and the corresponding minimal wave speeds are obtained. While some crucial improvements in theoretical methods have been established, the proofs of the existence and nonexistence of such traveling waves are based on Schauder’s fixed-point theorem, LaSalle’s invariance principle and Laplace transform. Based on theoretical results, we investigate the effect of the hydrological and biological factors on minimal wave speeds and hence on the spread of the prey and the invasion of the predator in the river. The linear determinacy of the predator–prey Lotka–Volterra system is compared with nonlinear determinacy of the competitive Lotka–Volterra system to investigate the mechanics of linear and nonlinear determinacy.     

Solvability of a model for monomer–monomer surface reactions

A mathematical model for a monomer–monomer surface reaction is considered taking into account the surface diffusion of adsorbed particles of both reactants. The model is described by a coupled system of parabolic equations where some of them are defined in a domain and the other ones have to be solved on the domain surface. The existence and uniqueness theorem of a classic solution for the time-dependent problem is proved. Non-uniqueness of solutions for the steady-state problem is established.     

Global stability for a nonlocal reaction–diffusion population model

In this paper, we study a nonlocal reaction–diffusion population model. We establish a comparison principle and construct monotone sequences to show the existence and uniqueness of the solution to the model. We then analyze the global stability for the model.     

Optimal harvesting policy for a stochastic predator–prey model

Optimal harvesting of a stochastic predator–prey model is considered in this paper. Sufficient and necessary criteria for the existence of optimal harvesting strategy are obtained. At the same time, the optimal harvest effort and the maximum of sustainable yield are given.     

Homogenization of a Wilson–Cowan model for neural fields

Homogenization of Wilson–Cowan type of nonlocal neural field models is investigated. Motivated by the presence of a convolution term in this type of models, we first prove some general convergence results related to convolution sequences. We then apply these results to the homogenization problem of the Wilson–Cowan-type model in a general deterministic setting. Key ingredients in this study are the notion of algebras with mean value and the related concept of sigma-convergence.     

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.     

Analysis of a Cahn–Hilliard–Brinkman model for tumour growth with chemotaxis

Phase field models recently gained a lot of interest in the context of tumour growth models. Typically Darcy-type flow models are coupled to Cahn–Hilliard equations. However, often Stokes or Brinkman flows are more appropriate flow models. We introduce and mathematically analyse a new Cahn–Hilliard–Brinkman model for tumour growth allowing for chemotaxis. Outflow boundary conditions are considered in order not to influence tumour growth by artificial boundary conditions. Existence of global-in-time weak solutions is shown in a very general setting.     

The generalized Riemann problem for a scalar Chapman–Jouguet combustion model

The generalized Riemann problem for a scalar Chapman–Jouguet combustion model in a neighborhood of the origin (t > 0) on the (x, t) plane is studied. Under the entropy conditions, we obtain the solutions constructively. It is found that, for some cases, the perturbed Riemann solutions are essentially different from the corresponding Riemann solutions. The perturbation may transform a combustion wave CJDT into SDT in the neighborhood of the origin. Especially, it can be observed that burning happens although the corresponding Riemann solution doesn’t contain combustion waves, which exhibits the instability for unburnt states. This work is supported by NSFC 10671120     

Dynamics of a host–parasitoid model with prolonged diapause for parasitoid

In this paper, a host–parasitoid model with prolonged diapause for parasitoid is proposed and analyzed. The asymptotic stability analysis of the system is performed. For a biologically reasonable range of parameter values, the global dynamics of the system have been studied numerically. In particular, the effect of prolonged diapause and parasitism on the system has been investigated. Many forms of complex dynamics are observed. The complexities include: (1) chaotic bands with periodic windows; (2) pitchfork and tangent bifurcations; (3) period-doubling and period-halving cascades; (4) intermittency; (5) supertransients; (6) non-unique dynamics, meaning that several attractors coexist; and (7) attractor crises. Furthermore, the complex dynamic behaviors of the model are confirmed by the largest Lyapunov exponents.     

Dynamics for a diffusive prey–predator model with different free boundaries

To understand the spreading and interaction of prey and predator, in this paper we study the dynamics of the diffusive Lotka–Volterra type prey–predator model with different free boundaries. These two free boundaries, which may intersect each other as time evolves, are used to describe the spreading of prey and predator. We investigate the existence and uniqueness, regularity and uniform estimates, and long time behaviors of global solution. Some sufficient conditions for spreading and vanishing are established. When spreading occurs, we provide the more accurate limits of $(u,v)$ as $t\to \infty$, and give some estimates of asymptotic spreading speeds of $u,v$ and asymptotic speeds of $g,h$. Some realistic and significant spreading phenomena are found.     

On a model for the Navier–Stokes equations using magnetization variables

It is known that in a classical setting, the Navier–Stokes equations can be reformulated in terms of so-called magnetization variables w that satisfy

(1)${?}_{t}w+(\mathbb{P}w?\mathrm{?})w+{(\mathrm{?}\mathbb{P}w)}^{?}w?\mathrm{\Delta }w=0,$

and relate to the velocity u via a Leray projection $u=\mathbb{P}w$. We will prove the equivalence of these formulations in the setting of weak solutions that are also in $L^{\infty }(0,T;H^{1/2})\cap L^2(0,T;H^{3/2})$ on the 3-dimensional torus.Our main focus is the proof of global well-posedness in $H^{1/2}$ for a new variant of (1), where $\mathbb{P}w$ is replaced by w in the second nonlinear term:

(2)${?}_{t}w+(\mathbb{P}w?\mathrm{?})w+\frac{1}{2}\mathrm{?}|w{|}^2?\mathrm{\Delta }w=0.$

This is based on a maximum principle, analogous to a similar property of the Burgers equations.     

Stability and bifurcation analysis for a fractional prey–predator scavenger model

In this study, we consider a fractional prey–predator scavenger model as well as harvesting by a predator and scavenger. We prove the positivity and boundedness of the solutions in this system. The model undergoes a Hopf bifurcation around one of the existing equilibria where the conditions are met for the occurrence of a Hopf bifurcation. The results show that chaos disappears in this biological model. We conclude that the fractional system is more stable compared with the classical case and the stability domain can be extended under fractional order. In addition, a suitable amount of prey harvesting and a fractional order derivative can control the chaotic dynamics and stabilize them. We also present an extended numerical simulation to validate the results.     

Infinite transition solutions for a class of Allen–Cahn model equations

This paper constructs a family of solutions that undergoes an infinite number of spatial transitions for an Allen–Cahn model equation.      

Robust equilibrium reinsurance-investment strategy for a mean–variance insurer in a model with jumps

This paper analyzes the equilibrium strategy of a robust optimal reinsurance-investment problem under the mean–variance criterion in a model with jumps for an ambiguity-averse insurer (AAI) who worries about model uncertainty. The AAI’s surplus process is assumed to follow the classical Cramér–Lundberg model, and the AAI is allowed to purchase proportional reinsurance or acquire new business and invest in a financial market to manage her risk. The financial market consists of a risk-free asset and a risky asset whose price process is described by a jump-diffusion model. By applying stochastic control theory, we establish the corresponding extended Hamilton–Jacobi–Bellman (HJB) system of equations. Furthermore, we derive both the robust equilibrium reinsurance-investment strategy and the corresponding equilibrium value function by solving the extended HJB system of equations. In addition, some special cases of our model are provided, which show that our model and results extend some existing ones in the literature. Finally, the economic implications of our findings are illustrated, and utility losses from ignoring model uncertainty, jump risks and prohibiting reinsurance are analyzed using numerical examples.