We consider a difference equation which is equivalent to the following logistic equation with piecewise constant arguments:

x'(t) = r(t)x(t){1 - ax(t) - a₀x([t]) - a₁x([t-1]) - ... - a_mx([t-m])}, t 0,

where [ ] denotes the greatest integer function, r : [0, )(0, ) is continuous and a, a_j [0, ), j = 0, 1, 2, ..., m with a₀+ ... + a_m > 0.

For an arbitrary solution x(t) satisfying the initial conditions:

x(0) = x₀ > 0 and x(-j) = x_-j 0, j = 1, 2, ..., m,

we study sufficient conditions which ensure the convergence of x(t) to the positive equilibrium x* = 1/(a + a₀ + ... + a_m) as t .

A global stability theorem in n-dimensional predator-prey systems with delay arguments are also considered.

References

1. K. Gopalsamy, M.R.S. Kulenovic and G. Ladas, On a logistic equation with piecewise constant arguments, Differential and Integral Equations 4 (1991), 215-223.
2. K. Gopalsamy and P. Liu, Persistence and global stability in a population model, J. Math. Anal. Appl. 224 (1998), 59-80.
3. Z. Lu and W. Wang, Global stability for two-species Lotka-Volterra systems with delay, J. Math. Anal. Appl. 208 (1997), 277-280.
4. J.W.-H. So and J.S.Yu, Global stability in a logistic equation with piecewise constant arguments, Hokkaido Math. J. 24 (1995), 269-286.
5. W. Wendi and M. Zhien, Harmless delays for uniform persistence, J. Math. Anal. Appl. 158 (1991), 256-268.

MINISIMPOSIUM SESSION: 3. Numerical methods for delay equations (2 sessions)