Numerical Stability Of The Delay Logistic Equations With Piecewise Continuous Arguments

This paper deals with the numerical stability of the delay Logistic equations withpiecewise continuous arguments in Biology.A typical equations with piecewise continuous arguments contains argumentsthat are constants on certain intervals. The solution is defined as a continuous, sec-tionally smooth function that satisfies the equation within these intervals and the so-lution is determined by a finite set of initial data rather than by an initial function asin the case of general delay differential equations.In this paper, the Runge-Kutta method is applied to the Logistic equations withthe delays [t], [t-1]. The stability and convergence of the method are studied. The con-ditions under which the numerical solutions preserve the local asymptotical stabilityand global asymptotical stability of the analytic solutions are abtained.Firstly, applied the explicit Euler method and the Runge-Kutta method directlyto the Logistic equations with one delay [t] and two delays [t], [t-1], the conditionsunder which the methods admit spurious solutions are given. It is shown that theexplicit Euler method and some Runge-Kutta methods admit spurious solutions.Secondly, a numerical scheme for these equations which does not admit spurioussolutions is constructed. The convergence of this scheme is investigated. It is shownthat the numerical solutions preserve the local asymptotical stability of the analyticsolutions.Finally, the global asymptotical stability of the numerical solutions of thisscheme is investigated. It is proved that the numerical solutions of the Logistic equa-tions with one delay [t] and two delays [t], [t-1] are globally asymptotically stableunder certain conditions.