Global convergence in a reaction-diffusion equation with piecewise constant argument. (English) Zbl 0988.35079
The authors consider the reaction-diffusion equation with piecewise constant argument
\[
{{\partial u}\over {\partial t}}= r u(x,t) (1-u(x,t))- E u(x,[t])u(x,t)+D\nabla^2 u
\]
on a bounded domain with positive constants \(r,\) \(E\) and \(D.\) Assuming \(E<r(1-\exp(-r))\) and employing the method of sub- and super-solutions, it is proved that all solutions with positive initial data converge to the positive uniform state.
Reviewer: Dian K.Palagachev (Bari)
References:
| [1] | S.A. Gourley, H.M. Byrne and M.A.J. Chaplain, Asymptotic behaviour of solutions of a scalar differential equation with piecewise constant argument, Diff. & Int. Eqns. (to appear).; S.A. Gourley, H.M. Byrne and M.A.J. Chaplain, Asymptotic behaviour of solutions of a scalar differential equation with piecewise constant argument, Diff. & Int. Eqns. (to appear). |
| [2] | Gopalsamy, K.; Liu, P., Presistence and global stability in a population model, J. Math. Anal. & Applic., 224, 59-80 (1998) · Zbl 0912.34040 |
| [3] | Cooke, K. L.; Huang, W., A theorem of George Seifert and an equation with state-dependent delay, (Fink, A. M.; Miller, R. K.; Kliemann, W., Delay and Differential Equations (1992), World Scientific: World Scientific Singapore) · Zbl 0820.34042 |
| [4] | Gopalsamy, K., Stability and Oscillations in Delay Differential Equations of Population Dynamics (1992), Kluwer: Kluwer Dordrecht · Zbl 0752.34039 |
| [5] | Györi, I.; Ladas, G., Linearized oscillations for equations with piecewise constant arguments, Diff. & Int. Eqns., 2, 123-131 (1989) · Zbl 0723.34058 |
| [6] | Gourley, S. A.; Britton, N. F., On a modified Volterra population equation with diffusion, Nonl. Anal. TMA, 21, 389-395 (1993) · Zbl 0805.35062 |
| [7] | Britton, N. F., Reaction-Diffusion Equations and Their Applications to Biology (1986), Academic Press: Academic Press London · Zbl 0602.92001 |
| [8] | Grindrod, P., Patterns and Waves: The Theory and Applications of Reaction-Diffusion equations (1991), Clarendon Press: Clarendon Press Oxford · Zbl 0743.35032 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
