| We study planar systems of difference equations and their applications to biological models of species populations. Central to the analysis of this study is the idea of folding- the method of transforming systems of difference equations into higher order scalar difference equations. For example, a planar system is transformed into a core second order difference equation and a passive non-dynamic equation. Two classes of second order equations are studied in detail: quadratic fractional and exponential. In the study of the quadratic fractional equation, we investigate the boundedness and persistence of solutions, the global stability of the positive fixed point and the occurrence of periodic solutions with non-negative parameters and initial values. These results are then applied to a class of linear/rational systems of difference equations that can be transformed into a quadratic fractional second order difference equation via folding. These results apply to systems with negative parameters, instances not commonly considered in previous studies. Using the idea of folding, we also identify ranges of parameter values that provide sufficient conditions on existence of chaotic, as well as multiple stable orbits of different periods for the planar system. We also study a second order exponential difference equation with time varying parameters. We obtain sufficient conditions for boundedness of solutions and global convergence to zero. For the special, autonomous case (with constant parameters), we show occurrence of multistable periodic and nonperiodic orbits. For the case where parameters are periodic, we show that the nature of the solutions differs significantly depending on whether the period of the parameters is even or odd. The above results are applied to biological models of populations. We investigate a broad class of planar systems that arise in the study of so-called stage-structured (adult-juvenile) single species populations, with and without time-varying parameters. In some cases, these systems are of the rational sort (e.g. the Beverton-Holt type), while in other cases the systems involve the exponential (or Ricker) function. In biological contexts, these results include conditions that imply extinction or survival of the species in some balanced form, as well as possible occurrence of complex and chaotic behavior. Special rational and exponential cases of the model are considered where we explore the role of inter-stage competition, restocking strategies, as well as seasonal fluctuations in the vital rates. |
