NON-STANDARD DISCRETIZATION METHODS FOR SOME BIOLOGICAL MODELS - Abbas, Shawki A M (Al-Nahrain Journal of Science, 2009)
It has been observed for some time that the standard (classical) discretization methods of differential equations often produce difference equations that do not share their dynamics Mickens[21]. An illustrative example is the logistic difference equations .Where x(t) represent the density of species A at time t,  is positive number, Euler’s discretization scheme produces the logistic difference equation x(n+1) = x (n) (1-x(n)), Which possesses a remarkably different dynamics such as period-doubling bifurcation route to chaos. A more popular discretization method is to modify the given differential equation to another with piecewise-constant arguments and then to integrate the modified equation. In some instance, this produces a different equation whose dynamics is closed to its original differential equation. However, oftentimes this is not the case. Nevertheless, many authors [1, 3, 7, 8, 9, 10] find it interesting to study the resulting difference equations. This is not a criticism of these author’s research, since the study of nonlinear difference equations is of paramount importance regardless of whether or not they have connections with differential equations. But what we are actually saying is that from the point of view of numerical analysis such study is of less importance. This paper itself with those numerical schemes that produce difference equations whose dynamics resembles that of their continuous counter-parts. The most fruitful methods are those of Mickens[14] (for asymptotically stable systems) and of Kahan[16] (for periodic systems).The paper is organized as follows. Section 2 establishes the basic stability results for Lotka-Volterra differential systems. Section 3 surveys some classical discretization methods that are widely used and show their shortcomings. Section 4 provides the reader with essential intgredients of Mickens nonstandard discretization scheme. In section 5, we discretize a periodic Lotka-Volterra differential system using Kahan’s scheme[16]. It is shown that the solutions of the resulting difference equation lie on closed curves surrounding the positive equilibrium point. In section 6, we consider a Kolmogrove continuous model of cooperative system[13]. This model was discretized in[7] using the method of piecewise-constant argument. Surprisingly, the resulting difference equation is dynamically consistent with its continuous counterpart.