Junping Shi

Exact Multiplicity of Positive Solutions for a Class of Semilinear Problem, II

We consider the positive solutions to the semilinear problem: { Δ u + λ f ( u ) = 0 , in B n , u = 0 , on ∂ B n . . where Bn is the unit ball in Rn, n ⩾ 1, and λ is a positive parameter. It is well known that if ƒ is a smooth function, then any positive solution to the equation is radially symmetric, and all solutions can be parameterized by their maximum values. We develop a unified approach to obtain the exact multiplicity of the positive solutions for a wide class of nonlinear functions ƒ(u), and the precise shape of the global bifurcation diagrams are rigorously proved. Our technique combines the bifurcation analysis, stability analysis, and topological methods. We show that the shape of the bifurcation curve depends on the shape of the graph of function ƒ(u)/u as well as the growth rale of ƒ.

On a singular nonlinear semilinear elliptic problem

where K(x)μC2,b(V9 ), a, pμ(0, 1) and l is a real parameter. Such singular elliptic problems arise in the contexts of chemical heterogeneous catalysts, nonNewtonian fluids and also the theory of heat conduction in electrically conducting materials, see [3, 5, 8, 9] for a detailed discussion. Obviously (1.1) cannot have a solution uμC2(V9 ) if K(x) is not vanishing near ∂V. However, under various appropriate assumptions on K(x), we will obtain classical solutions of (1.1) for l belonging to a certain range, and we will also obtain some uniqueness criteria. Here a classical solution is a solution u of (1.1) which belongs to C2(V)mC(V9 ) with u>0 in V. We also study the boundary behaviour of solutions of (1.1), and we will show that the solution u of (1.1) lies in a certain Holder class. The special case when K(x) is negative and l=0 has been studied by several authors. The existence and uniqueness of the solution were established by Crandall, Rabinowitz and Tartar [6], Del Pino [7], Gomes [10] and Lazer and McKenna

Predator-prey system with strong Allee effect in prey.

Global bifurcation analysis of a class of general predator–prey models with a strong Allee effect in prey population is given in details. We show the existence of a point-to-point heteroclinic orbit loop, consider the Hopf bifurcation, and prove the existence/uniqueness and the nonexistence of limit cycle for appropriate range of parameters. For a unique parameter value, a threshold curve separates the overexploitation and coexistence (successful invasion of predator) regions of initial conditions. Our rigorous results justify some recent ecological observations, and practical ecological examples are used to demonstrate our theoretical work.

Diffusive logistic equation with constant yield harvesting, I: Steady States

We consider a reaction-diffusion equation which models the constant yield harvesting to a spatially heterogeneous population which satisfies a logistic growth. We prove the existence, uniqueness and stability of the maximal steady state solutions under certain conditions, and we also classify all steady state solutions under more restricted conditions. Exact global bifurcation diagrams are obtained in the latter case. Our method is a combination of comparison arguments and bifurcation theory.

Global stability of multigroup epidemic model with group mixing and nonlinear incidence rates

In this paper, we introduce a basic reproduction number for a multigroup SEIR model with nonlinear incidence of infection and nonlinear removal functions between compartments. Then, we establish that global dynamics are completely determined by the basic reproduction number R0. It shows that, the basic reproduction number R0 is a global threshold parameter in the sense that if it is less than or equal to one, the disease free equilibrium is globally stable and the disease dies out; whereas if it is larger than one, there is a unique endemic equilibrium which is globally stable and thus the disease persists in the population. Finally, two numerical examples are also included to illustrate the effectiveness of the proposed result.

Allee effect and bistability in a spatially heterogeneous predator-prey model

A spatially heterogeneous reaction-diffusion system modelling predator-prey interaction is studied, where the interaction is governed by a Holling type II functional response. Existence of multiple positive steady states and global bifurcation branches are examined as well as related dynamical behavior. It is found that while the predator population is not far from a constant level, the prey population could be extinguished, persist or blow up depending on the initial population distributions, the various parameters in the system, and the heterogeneous environment. In particular, our results show that when the prey growth is strong, the spatial heterogeneity of the environment can play a dominant role for the presence of the Allee effect. Our mathematical analysis relies on bifurcation theory, topological methods, various comparison principles and elliptic estimates. We combine these methods with monotonicity arguments to the system through the use of some new auxiliary scalar equations, though the system itself does not keep an order structure as the competition system does. Among other things, this allows us to obtain partial descriptions of the dynamical behavior of the system.

On stationary patterns of a reaction–diffusion model with autocatalysis and saturation law

Understanding of spatial and temporal behaviour of interacting species or reactants in ecological or chemical systems has become a central issue, and rigorously determining the formation of patterns in models from various mechanisms is of particular interest to applied mathematicians. In this paper, we study a bimolecular autocatalytic reaction–diffusion model with saturation law and are mainly concerned with the corresponding steady-state problem subject to the homogeneous Neumann boundary condition. In particular, we derive some results for the existence and non-existence of non-constant stationary solutions when the diffusion rate of a certain reactant is large or small. The existence of non-constant stationary solutions implies the possibility of pattern formation in this system. Our theoretical analysis shows that the diffusion rate of this reactant and the size of the reactor play decisive roles in leading to the formation of stationary patterns.

Existence and instability of spike layer solutions to singular perturbation problems

Abstract An abstract framework is given to establish the existence and compute the Morse index of spike layer solutions of singularly perturbed semilinear elliptic equations. A nonlinear Lyapunov–Schmidt scheme is used to reduce the problem to one on a normally hyperbolic manifold, and the related linearized problem is also analyzed using this reduction. As an application, we show the existence of a multi-peak spike layer solution with peaks on the boundary of the domain, and we also obtain precise estimates of the small eigenvalues of the operator obtained by linearizing at a spike layer solution.

STATIONARY PATTERN OF A RATIO-DEPENDENT FOOD CHAIN MODEL WITH DIFFUSION ∗

In the paper, we investigate a three-species food chain model with diffusion and ratio-dependent predation functional response. We mainly focus on the coexistence of the three species. For this coupled reaction-diffusion system, we study the persistent property of the solution, the stability of the constant positive steady state solution, and the existence and nonexistence of nonconstant positive steady state solutions. Both the general stationary pattern and Turing pattern are observed as a result of diffusion. Our results also exhibit some interesting effects of diffusion and functional responses on pattern formation.

Semilinear Neumann boundary value problems on a rectangle

We consider a semilinear elliptic equation Δu + λf(u) = 0, x ∈ Ω, ∂u/∂n = 0, x ∈ ∂Ω, where Ω is a rectangle (0, a) x (0, b) in R 2 . For balanced and unbalanced f, we obtain partial descriptions of global bifurcation diagrams in (A, u) space. In particular, we rigorously prove the existence of secondary bifurcation branches from the semi-trivial solutions, which is called dimension-breaking bifurcation. We also study the asymptotic behavior of the monotone solutions when λ → ∞. The results can be applied to the Allen-Cahn equation and some equations arising from mathematical biology.