Abstract
We consider persistence and stability properties for a class of forced discretetime difference equations with three defining properties: the solution is constrained to evolve in the nonnegative orthant, the forcing acts multiplicatively, and the dynamics are described by socalled Lur’e systems, containing both linear and nonlinear terms. Many discretetime biological models encountered in the literature may be expressed in the form of a Lur’e system and, in this context, the multiplicative forcing may correspond to
harvesting, culling or timevarying (such as seasonal) vital rates or environmental conditions. Drawing upon techniques from systems and control theory, and assuming that the forcing is bounded, we provide conditions under which persistence occurs and, further, that a unique nonzero equilibrium is stable with respect to the forcing in a sense which is reminiscent of inputtostate stability, a concept wellknown in nonlinear control theory. The theoretical results are illustrated with several examples. In particular, we discuss how our results relate to previous literature on stabilization of chaotic systems by socalled proportional feedback control.
harvesting, culling or timevarying (such as seasonal) vital rates or environmental conditions. Drawing upon techniques from systems and control theory, and assuming that the forcing is bounded, we provide conditions under which persistence occurs and, further, that a unique nonzero equilibrium is stable with respect to the forcing in a sense which is reminiscent of inputtostate stability, a concept wellknown in nonlinear control theory. The theoretical results are illustrated with several examples. In particular, we discuss how our results relate to previous literature on stabilization of chaotic systems by socalled proportional feedback control.
Original language  English 

Pages (fromto)  4661 
Journal  Physica D: Nonlinear Phenomena 
Volume  360 
Early online date  31 Aug 2017 
DOIs  
Publication status  Published  1 Dec 2017 
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Hartmut Logemann
 Centre for Mathematical Biology
 EPSRC Centre for Doctoral Training in Statistical Applied Mathematics (SAMBa)
 Department of Mathematical Sciences  Professor Emeritus
Person: Honorary / Visiting Staff