Anthony W. Leung

On the existence and uniqueness of positive steady states in the volterra-lotka ecological models with diffusion

Using the theory of quasimonotone increasing systems developed by P. J. McKenna and W. Walter, we give a rather detailed analysis of the steady state solutions for the Volterra-Lotka model of two cooperating species, and prove some new nonexistence results for the competing species case. We indicate generalizations to the case of n > 2 species

Optimal harvesting-coefficient control of steady-state prey-predator diffusive Volterra-Lotka systems

This article considers the optimal control of the harvesting of a prey-predator system in an environment. The species are assumed to be in steady state under diffusion and Voterra-Lotka type of interaction. They are harvested for economic profit, leading to reduction of growth rates; and the problem is to control the spatial distributions of harvests so as to optimize the return. Precise conditions are found so that the optimal control can be rigorously characterized as the solution of an optimality system of nonlinear elliptic partial differential equations. Moreover, a constructive approximation scheme for optimal control is given.

A general monotone scheme for elliptic systems with applications to ecological models

We consider weakly-coupled elliptic systems of the type with each f i being either an increasing or a decreasing function of each u j . Assuming the existence of coupled super- and subsolutions, we prove the existence of solutions, and provide a constructive iteration scheme to approximate the solutions. We then apply our results to study the steady-states of two-species interaction in the Volterra–Lotka model with diffusion.

Limiting behaviour for a prey-predator model with diffusion and crowding effects

where a~, ~2, a, b, c, p , q, r are posit ive constants related to diffusion, growth, interaction and death rates of the populations ux, u2. The unknown populat ion functions Ul, u2 are defined for time t > /0 and space x = (xx . . . . . x~) e ~ (f2 is a given bounded open connected subset of R n, n >/ 1). When u~ are functions of t alone, (1.1) reduces to the generalized Volterra-Lotka model for prey-predator interactions, with b, q expressing the crowding effect on the growth of prey ul and predator u2 respectively. The parameters a, r, c , p represent growth, death and interaction rates. (See e.g. Poole [5] or Rescigno and Richardson [7]). With the inclusion of diffusion terms a~Aul, a2Au2, (A _= ~ = 1 ~2/~x2), Williams and Chow [8] studied Eq. (1.1) with no crowding effect (i.e. b = q = 0). Since the effects of crowding tend to occur in realistic biological systems, this article studies Eq. (1.1) with b > 0, q > 0, extending the results in [8].

Existence of positive solutions for elliptic systems―Degenerate and nondegenerate ecological models

with homogeneous Dirichlet conditions w,=O on dD. The function

Conditions for Global Stability Concerning a Prey-Predator Model with Delay Effects

(s) satisfies the conditions rl/ E C’[O, oo),

Nonlinear systems of partial differential equations : applications to life and physical sciences

(Q) =O, and

Periodic optimal control for competing parabolic Volterra-Lotka-type systems

‘(s)>O for s>O. Problems of this nature are of interest in reaction-diffusion processes in biology and chemistry. For example, the case for

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

(u) = urn, m > 1, or m E (0, 1) .for single parabolic equations (i.e., U, = du” +f(x, u)) has been studied recently for porous medium analysis and population dynamics (cf. [2, 7, 15, 18, 191). As t -+ cc these solutions tend to a solution of the corresponding elliptic scalar equation. Studies have also been carried out in these and other papers (e.g., [3, 4, 16, 171) with urn replaced by

Asymptotically stable invariant manifold for coupled nonlinear parabolic-hyperbolic partial differential equations

(u) satisfying the conditions described above. In this article, however, the hypotheses on f are quite different from those in the papers mentioned above. For example, f is not necessarily Lipschitz in u and may depend discontinuously on x. Thus even for the scalar case, the results here are not covered by the other papers, although there are some overlaps. Moreover, our emphasis here is on degenerate systems (Section 3) and their applications (Section 4). These results are quite different and generalize the papers mentioned above to practical interesting cases. Other relevant results for the nondegenerate cases can be found in [S, 9, 111. In Section 2, we discuss some existence and uniqueness theorems for