Department of Mathematics, University of Southern California, Los Angeles, CA 90089-2532, USA.
J Biol Dyn. 2007 Oct;1(4):437-53. doi: 10.1080/17513750701605572.
In a difference or differential equation one is usually interested in finding solutions having certain properties, either intrinsic properties (e.g. bounded, periodic, almost periodic) or extrinsic properties (e.g. stable, asymptotically stable, globally asymptotically stable). In certain instances it may happen that the dependence of these equations on the state variable is such that one may (1) alter that dependency by replacing part of the state variable by a function from a class having some of the above properties and (2) solve the 'reduced' equation for a solution having the remaining properties and lying in the same class. This then sets up a mapping Τ of the class into itself, thus reducing the original problem to one of finding a fixed point of the mapping. The procedure is applied to obtain a globally asymptotically stable periodic solution for a system of difference equations modeling the interaction of wild and genetically altered mosquitoes in an environment yielding periodic parameters. It is also shown that certain coupled periodic systems of difference equations may be completely decoupled so that the mapping Τ is established by solving a set of scalar equations. Periodic difference equations of extended Ricker type and also rational difference equations with a finite number of delays are also considered by reducing them to equations without delays but with a larger period. Conditions are given guaranteeing the existence and global asymptotic stability of periodic solutions.
在差分方程或微分方程中，人们通常感兴趣的是找到具有某些性质的解，这些性质可以是内在性质（例如有界的、周期的、几乎周期的），也可以是外在性质（例如稳定的、渐近稳定的、全局渐近稳定的）。在某些情况下，这些方程对状态变量的依赖关系可能是这样的，人们可以（1）通过用具有上述某些性质的类中的函数来替换状态变量的一部分来改变这种依赖性，（2）为具有剩余性质且位于同一类中的解求解“简化”方程。这就建立了一个将类映射到自身的映射 Τ，从而将原始问题简化为找到映射的不动点的问题。该方法应用于一个差分方程系统，该系统用于模拟在产生周期参数的环境中野生和遗传改变的蚊子的相互作用，以获得全局渐近稳定的周期解。还表明，某些耦合的周期差分方程系统可以完全解耦，从而通过求解一组标量方程来建立映射 Τ。还通过将扩展的 Ricker 型周期差分方程和具有有限个时滞的有理差分方程简化为具有较大周期但没有时滞的方程来考虑它们。给出了保证周期解存在和全局渐近稳定性的条件。
