Shair Ahmad

On the nonautonomous Volterra-Lotka competition equations

A nonautonomous competitive Lotka-Volterra system of two equations is considered. It is shown that if the coefficients are continuous and satisfy certain inequalities, then any solution that is positive at some point has the property that one of its components vanishes while the other approaches a certain solution of the logistic equation

Extinction of species in nonautonomous Lotka-Volterra systems

A nonautonomous nth order Lotka,Volterra system of differential equations is considered. It is shown that if the coefficients satisfy certain inequalities, then any solution with positive components at some point will have all of its last n 1 components tend to zero, while the first one will stabilize at a certain solution of a logistic equation.

Extinction in nonautonomous T-periodic competitive Lotka-Volterra system

A nonautonomous T-periodic competitive Lotka-Volterra system of n species is considered. It is shown that if the coefficients are T-periodic, continuous and satisfy certain inequalities, then any solution with strictly positive initial conditions has the property that all but one of its components vanish while the remaining component approaches the canonical solution of a certain logistic differential equation.

Almost necessary and sufficient conditions for survival of species

Abstract We consider a nonautonomous competitive Lotka–Volterra system of two species, satisfying two inequalities involving averages of the growth rates and the interaction coefficients, which imply persistence. We introduce a third species and give a third inequality, involving the average of the growth rate of the third species and solutions of a linear algebraic system, which guarantees persistence of the system. It is also shown that reversing this inequality implies non-persistence; more specifically, extinction of the third species with small positive initial values, in the autonomous case. Our conditions are simple and computable.

A resonance problem in which the nonlinearity may grow linearly

The purpose of this paper is to study a semilinear two point boundary value problem of resonance type in which the nonlinear perturbation may grow linearly. A significant improvement of a recent result due to Cesari and Kannan is given.

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

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

Nonselfadjoint resonance problems with unbounded perturbations

Soit Ω un domaine borne de R N (N≥1) a frontiere de classe C 2+ δ pour δ∈(0,1). Soit Lu=−Σ i=1 N Σ j=1 N a idj (x)∂ 2 u/∂x i ∂x j +Σ i=1 N b i (x)∂u/∂x i +a 0 (x)u ou a ij (x)=a ji (x), 1≤i, j≤N, a 0 (x)≥0 sur Ω et Σ i=1 N Σ j=1 N a ij (x)ξ i ξ j >0 pour x∈Ω et ξ∈R N −{0}. On considere le probleme Lu=λ 1 u+g(u)−h, u/∂Ω=0, ou λ 1 est la valeur propre principale de L, h∈Cδ(Ω), et g est localement lipschitzien sur (−∞,∞)

POSITIVE OPERATORS AND STURMIAN THEORY OF NONSELFADJOINT SECOND-ORDER SYSTEMS

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 .

Lotka-Volterra and related systems : recent developments in population dynamics

This book facilitates research in the general area of population dynamics by presenting some of the recent developments involving theories, methods and application in this important area of research. The underlying common feature of the studies included in the book is that they are related, either directly or indirectly, to the well-known Lotka-Volterra systems which offer a variety of mathematical concepts from both theoretical and application points of view.

On an Extension of the Sturm Comparison Theorem

The main purpose of this paper is to extend the Sturm comparison theorem for second-order differential equations to the equation