Alan C. Lazer

An application of topological degree to the periodic competing species problem

We consider the Volterra-Lotka equations for two competing species in which the right-hand sides are periodic in time. Using topological degree, we show that conditions recently given by K. Gopalsamy, which imply the existence of a periodic solution with positive components, also imply the uniqueness and asymptotic stability of the solution. We also give optimal upper and lower bounds for the components of the solution.

On an abstract competition model and applications

is a map such that the coordinates of F(x,, x2) represent the population or population densities which have evolved from the state (xi, x2) at a fixed time later. Setting (yi, y2) = F(x,, x2), it is natural to suppose that an increase in x, coupled with a decrease in x2 results in an increase in y, and a decrease in y2 and, symmetrically, a decrease in x1 coupled with an increase in x2 results in a decrease in yr and an increase in yz. We define an order in E, x E2 as follows: If x,, xi E E,, x2, xi E E,, x1 I xi, and xi I x2, then we write (x1 7

The permanence and global attractivity in a nonautonomous Lotka–Volterra system

1 5 (4 9 x;>.

On the Nonautonomous N-competing Species Problems

Abstract In this paper, we consider a nonautonomous Lotka–Volterra system. By means of Ahmad and Lazers definitions of lower and upper averages of a function, we give the averaged conditions for the permanence and global attractivity of this system. It is shown that our averaged conditions are generalization of that of Ahmad and Lazer.

An N-Dimensional Extension of the Sturm Separation and Comparison Theory to a Class of Nonselfadjoint Systems

We consider a nonautonomous system of ordinary differential equations which models competition among n species. Conditions are given under which there exists a unique solution with components bounded above and below by positive constants on (-∞, ∞) and which attracts all other solutions with positive components. In the case where this system is periodic or almost periodic in time, this unique solution is periodic or almost periodic. Our proofs make use of a combination of techniques from [6] and [8] and improve the results of these papers. An example is given to illustrate this improvement and another example shows that certain conditions which imply existence do not imply the conditions guaranteeing uniqueness and stability.

Monotone scheme for finite difference equations concerning steady-state prey-predator interactions

Sturmian theory is extended to nonselfadjoint second order linear homogeneous systems. Almost all the results obtained are new even in the selfadjoint case.

Multiplicity of solutions of nonlinear boundary value problems

Abstract In this article a system of semilinear elliptic partial differential equations is studied. This system determines the equilibria of the Volterra-Lotka equations describing prey-predator interactions with diffusion. To analyze the system, a new monotone scheme is presented. A rigorous foundation is given for numerical calculations by adapting a suitable finite difference method to the new monotone scheme. Earlier theories in finite differences are not successful in solving the system without this scheme.

POSITIVE OPERATORS AND STURMIAN THEORY OF NONSELFADJOINT SECOND-ORDER SYSTEMS

Sharp results for the number of solutions of a one-dimensional nonlinear Neumann boundary value problem are given, in terms of the range of its linearization, and the projection of the source term onto the principle eigenfunction.

Preliminary communication: Nontrivial solutions of operator equations and morse indices of critical points of min-max type

This chapter discusses positive operators and Sturmian theory of nonself adjoint second-order systems. It presents a theorem that states that let A be linear, positive, and completely continuous. If there exists u ∈ E such that – u K , u=v–w with v, w ∈ K , and there exists a number c > 0 and an integer p such that cA p u > u . Then A has a characteristics vector x0 ∈ K : xo = λo A xo where the positive characteristic value λo satisfies λo ≤ci/p (c) 1/p . A cone K ⊂ E is called solid if it contains interior points. If A: E → E is linear, A will be called strictly positive with respect to the solid cone K , if x∈ K and x≠0 implies that A x is in the interior of K .

On an Extension of the Sturm Comparison Theorem

On montre que si les hypotheses du theoreme de point selle sont satisfaites, si dim Y≥2, et si f −1 ({c}) contient seulement des points critiques non degeneres, alors f −1 ({c}) contient au moins un point critique u 0 tel que 2≤indice de Morse u 0 ≤dim Y