# American Institute of Mathematical Sciences

## Journals

MBE
Symmetric evolutionary games, i.e., evolutionary games with symmetric fitness matrices, have important applications in population genetics, where they can be used to model for example the selection and evolution of the genotypes of a given population. In this paper, we review the theory for obtaining optimal and stable strategies for symmetric evolutionary games, and provide some new proofs and computational methods. In particular, we review the relationship between the symmetric evolutionary game and the generalized knapsack problem, and discuss the first and second order necessary and sufficient conditions that can be derived from this relationship for testing the optimality and stability of the strategies. Some of the conditions are given in different forms from those in previous work and can be verified more efficiently. We also derive more efficient computational methods for the evaluation of the conditions than conventional approaches. We demonstrate how these conditions can be applied to justifying the strategies and their stabilities for a special class of genetic selection games including some in the study of genetic disorders.
keywords: evolutionary games genetic selection population genetics evolutionary stability. Evolutionary biology generalized knapsack problems
PROC
The authors study a type of nonlinear fractional boundary value problem with nonlocal boundary conditions. An associated Green's function is constructed. Then a criterion for the existence of at least one positive solution is obtained by using fixed point theory on cones.
keywords: fractional calculus. positive solution Green's function
PROC
In this paper, by using the dual least action principle, the authors investigate the existence of solutions to a multi-point boundary value problem for a second-order differential system with $p$-Laplacian.
keywords: dual least action principle. Multi-point boundary value problem critical point theory
PROC
Sufficient conditions are obtained for the existence of multiple solutions to the discrete fourth order periodic boundary value problem \begin{equation*} \begin{array}{l} \Delta^4 u(t-2)-\Delta(p(t-1)\Delta u(t-1))+q(t) u(t)=f(t,u(t)),\quad t\in [1,N]_{\mathbb{Z}},\\ \Delta^iu(-1)=\Delta^iu(N-1),\quad i=0, 1,2, 3. \end{array} \end{equation*} Our analysis is mainly based on the variational method and critical point theory. One example is included to illustrate the result.
keywords: fourth order variational methods. critical points Discrete boundary value problem multiple solutions
DCDS-B
The reaction-diffusion system for an $SIR$ epidemic model with a free boundary is studied. This model describes a transmission of diseases. The existence, uniqueness and estimates of the global solution are discussed first. Then some sufficient conditions for the disease vanishing are given. With the help of investigating the long time behavior of solution to the initial and boundary value problem in half space, the long time behavior of the susceptible population $S$ is obtained for the disease vanishing case.
keywords: long time behavior. Reaction-diffusion systems SIR model dynamics free boundary
DCDS-B
In this paper, we consider a mathematical model for a prey-predator dynamical system with diffusion and trophic interactions of three levels. In this model, a general trophic function based on the ratio between the prey and a linear function of the predator is used at each level. At the two limits of this trophic function, one recovers the classical prey-dependent and ratio-dependent predation models, respectively. We offer a complete discussion of the dynamical behavior of the model under the homogeneous Neumann boundary condition (the same behavior is also seen in the absence of diffusion). We also discuss existence, uniqueness, stability and bifurcation of equilibrium behavior corresponding to positive steady state solutions under the homogeneous Dirichlet boundary condition. Finally, we give interpretations of some of these results in the context of different predation models.
keywords: trophic function positive steady state solution. Prey-predator model
PROC
We study an elliptic system arising from a prey-predator model, where cross diffusions are included to reflect the influences of density gradients of prey and predator toward the fluxes of underlying populations. We establish existence and non{existence of non{constant positive solutions, or the possible pattern of popula- tion distribution.
keywords: elliptic system Steady states prey-predator mode.
JIMO
This paper develops a model to determine an optimal replenishment policy with defective items and shortage backlogging under conditions of permissible delay of payments. It is assumed that 100% of each lot are screened to separate good and defective items which are classified as imperfect quality and scrap items. Difference between unit selling price and unit purchase cost is also included in our mathematical model and analysis. Under this assumption, we model the retailer's inventory system as an expected profit maximization problem to determine the retailer's optimal inventory cycle time and optimal order quantity. Then, a theorem is established to describe the optimal replenishment policy for the retailer. Finally, numerical examples are given to illustrate the theorem and obtain some managerial phenomena.
keywords: Backlogging Inventory Permissible delay of payments. Defective items Shortages
DCDS
Existence, uniqueness, and stability of Heaviside function like solutions of a Keller and Segel's minimal chemotaxis model are established when a chemotaxis parameter is large enough. Asymptotic expansions of the solution in terms of the large chemotaxis parameter are also derived.
keywords: chemotaxis. asymptotic expansion stability uniqueness Existence eigenvalue
PROC
Using the variational method and critical point theory, the authors study the existence of infinitely many homoclinic solutions to the difference equation \begin{equation*} -\Delta \big(a(k)\phi_p(\Delta u(k-1))\big)+b(k)\phi_p(u(k))=\lambda f(k,u(k))),\quad k\in\mathbb{Z}, \end{equation*} where $p>1$ is a real number, $\phi_p(t)=|t|^{p-2}t$ for $t\in\mathbb{R}$, $\lambda>0$ is a parameter, $a, b:\mathbb{Z}\to (0,\infty)$, and $f: \mathbb{Z}\times\mathbb{R}\to\mathbb{R}$ is continuous in the second variable. Related results in the literature are extended.
keywords: homoclinic solutions Diff erence equations infi nitely many homoclinic solutions variational method.