以三个典型教学案例,分别从"知其根源,知其应用,知其发展"三方面,依据学生知识储备、专业特性和学情状况选择不同的教学模式,以学生为主体,以教师为主导,引入思政教学点,引导学生主动学习、思考、讨论、探索.在学习相关微积分知识的同时,培养学生的应用能力、创新能力、团队协作精神、工匠精神和严谨求实的科学精神,培养学生的家国情怀.

Under the shrinking curvature flow with inner normal velocity V = kα(α > 1), it is shown that highly symmetric, locally convex initial curves evolve into a point asymptotically like an multi‐circles. The proof relies on a crucial use of Bonnensen inequality for highly symmetric, locally convex curves. Copyright © 2016 John Wiley & Sons, Ltd.

微积分课程的教学对高校创新型人才的培养具有重要的作用,其教材建设受到普遍关注.本文以中美两部具有代表性的微积分教材为比较对象,着重对两本教材在数学内容、编排、数学在其他领域的应用、与计算机的结合、知识呈现等各方面进行了比较.力图从中展示出中美高等教育在教学思想和表现形式上的某些不同之处,由此得到一些有益的启示.

The dynamics of biological models have received intensive study,and it has been an important aspect in the field of nonlinear partial differential equations.In the specific predator-prey model,a key factor is the response function.In this paper,we present a predator-prey model with Sigmoidal functional response under homogeneous Dirichlet boundary.First,we give a priori estimates of positive steady-states solutions by using the method of upper and lower solutions,and then by means of the topological degree theory in cones,combing with maximum principle,we obtain some results on the existence of positive steady states.Moveover,a necessary and sufficient condition for the existence of positive solutions is given.Besides,the bifurcation of positive solutions is investigated with bifurcation theory.

本文介绍了美国两所高等工科学校先进的教学理念:里海大学的新生研讨课与伍斯特理工学院的基于项目式的教学模式,从中可以体验到启发式、探究式、讨论式、参与式、团队式和基于项目的研讨式教学的重要性,并吸取教学经验用于自身教学中。

A Lotka-Volterra competition model with cross-diffusions under homogeneous Dirichlet boundary condition is considered, where cross-diffusions are included in such a way that the two species run away from each other because of the competition between them. Using the method of upper and lower solutions, sufficient conditions for the existence of positive solutions are provided when the cross-diffusions are sufficiently small. Furthermore, the investigation of nonexistence of positive solutions is also presented.

An eco-epidemiological model with an epidemic in the predator and with a Holling type Ⅱ function is considered. A system with diffusion under the homogeneous Neumann boundary condition is studied. The existence for a positive solution of the corresponding steady state problem is mainly discussed. First, a prior estimates (positive upper and lower bounds) of the positive steady states of the reaction-diffusion system is given by the maximum principle and the Harnack inequation. Then, the nonexistence of non-constant positive steady states by using the energy method is given. Finally, the existence of non-constant positive steady states is obtained by using the topological degree.

The traveling wave solution of a hyperbolic model for chemotaxis in one space dimension is studied in this paper. By using some transformations of dependent variables and independent variables, we apply the tanh method and improved tanh method to the model, from which some traveling wave solutions in explicit form are presented.

A prey-predator model with Beddington-DeAngelis and Leslie functional response is studied.The uniqueness and stability for the coexistence solutions are given by the implicit function theorem and perturbation technique.It turns out that when a is larger than λ1+a2/k and b is larger than λ1,the uniqueness and stability of positive steady states are obtained with the parameter 0<a21,m1,or k1.What′s more,if 2ka1>ma2,the same conclusion is obtained with 0<h1.

This paper concerns a diffusive predator-prey model with revised Holling type Ⅱ response and no-flux boundary condition. The large time behavior of the solution including dissipation, persistence and stability is studied by use of the comparison principle. In addition by energy method, the nonexistence of the nonconstant positive steady solution is also obtained.

This paper concerns a diffusive predator-prey model with revised Holling typeⅡresponse and no-flux boundary condition.The large time behavior of the solution including dissipation,persistence and stability is studied by use of the comparison principle.In addition by energy method,the nonexistence of the nonconstant positive steady solution is also obtained.

A predator-prey model with Modified Holling-Type Ⅱ Schemes under homogeneous Dirichlet boundary condition is considered.By analyzing the asymptotic behaviors of positive solutions to the corresponding reaction-diffusion model,a necessary condition for the existence of positive solutions is given.Moreover,by virtue of the topological degree theory in cones,it turns out that the necessary condition is sufficient.Besides,the uniqueness of positive solutions in one dimension space is described.

In this paper,the steady-states of a predator-prey system with diffusion are considered.First,the authors give a priori estimates of positive solutions,and then study the non-existence,existence of non-constant positive solutions as some parameters are varied by using the energy methods and the topological degree theory,respectively.

In this paper, a nonlinear predator reproduction and prey competition model with diffusion is discussed. Some existence and non-existence results concerning non-constant positive steady-states are presented using topological degree argument and the energy method, respectively.

This paper is concerned with a strongly coupled prey-predator model subject to the homogeneous Neumann boundary condition. The existence of positive non-constant solutions is discussed.

A strongly coupled competitive model under homogeneous Dirichlet boundary conditions is discussed.Using the upper and lower solution method,we come to a conclusion that the positive solutions exist if the cross-diffusion is weak in some degree.

A predator-prey system with Holling type Ⅱ functional response under homogeneous Neumann boundary condition is considered.The globally asymptotically stability of semi-trivial solution to this system is firstly considered.Secondly,the globally asymptotically stability and locally asymptotically stability of the positive steady-states are discussed by constructing Lyapunov function and using Routh-Hurwitz theorem respectively.Then an priori estimates(positive upper and lower bounders) of positive steady-states is given.Finally,the existence of non-constant positive steady-states of the problem is obtained by using topology degree.

A predator-prey system with Holling type II functional response under homogeneous Neumann boundary condition is considered. The globally asymptotically stability of semi-trivial solution to this system is firstly considered. Secondly, the globally asymptotically stability and locally asymptotically stability of the positive steady-states are discussed by constructing Lyapunov function and using Routh-Hurwitz theorem respectively. Then an priori estimates (positive upper and lower bounders) of positive steady-states is given. Finally, the existence of non-constant positive steady-states of the problem is obtained by using topology degree.

This paper studies positive steady-state solutions of the diffusive Holling-Tanner prey-predator model in heterogeneous environments subject to the homogeneous Neumann boundary condition. In particular, it investigates the appearance of sharp spatial patterns arising from degeneracies of the model.

In this paper, we study the positive steady states of a prey-predator model with diffusion throughout and a non-monotone conversion rate under the homogeneous Dirichlet boundary condition. We obtain some results of the existence and non-existence of positive steady states. The stability and uniqueness of positive steady states are also discussed.