Applications of mathematical systems theory in population biology
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APPLICATIONS OF MATHEMATICAL SYSTEMS THEORY IN POPULATION BIOLOGY ´n Varga1 Zolta
Dedicated to the memory of Professor Mikl´ os Farkas
Institute of Mathematics and Informatics, Szent Istv´ an University P´ ater K. u. 1., H-2103 G¨ od¨ oll˝ o, Hungary E-mail: [email protected] (Received August 6, 2007; Accepted September 23, 2007)
Abstract
This paper is a review of recent developments of a research line proposed on the turn of the decades, 1980s to 1990s. The main results concern basic qualitative properties of nonlinear models of population biology, such as controllability and observability. The methods applied are different for the density-dependent models of population ecology and for the frequency-dependent models of population genetics and evolutionary theory. While in the first case the classical theorems of nonlinear systems theory can be used, in the second one an extension of classical results to systems with invariant manifold is necessary.
1. Introduction In engineering practice, it is a typical situation that an object (e.g. a machine) is controlled by a human intervention to influence the state of the object, or observing a transform of the state the task is to recover the state process of the object. The corresponding concepts of controllability, observability and the related state space model played important role in the development of Mathematical Systems Theory (MST). A first comprehensive monograph on this discipline was Kalman et al. [4]. In biology, particular tools of MST play an important role in Mathematics subject classification numbers: 93B05, 93C10, 93B07, 92D10, 92D40. Key words and phrases: nonlinear systems, controllability, observability, population biology. 1 Supported by the Hungarian NFSR (OTKA K 62000, K 68187). 0031-5303/2008/$20.00
c Akad´emiai Kiad´o, Budapest
Akad´ emiai Kiad´ o, Budapest Springer, Dordrecht
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Z. VARGA
mathematical modelling at infra-individual level. This paper is aimed at surveying a research line concerning the application of MST in supra-individual biology, dealing with problems of controllability and observability of continuous-time systems modelling populations. In Section 2, following a brief summary of necessary concepts and theorems concerning linear and nonlinear control systems, a controllability problem of population ecology is considered. Section 3 is devoted to a similar treatment of the observability and observer design for the monitoring of population systems. In these sections the application of classical results of Lee and Markus [5], and the general methods of observer design are illustrated. In Sections 4 and 5, based on general sufficient conditions for controllability and observability of nonlinear systems with invariant manifold proved in Varga [18] and Varga [19], particular controllability and observability problems of population genetics will be presented. Finally, in Section 6 a short survey of the application of the above methodology will be given.
2. Controllability problems of population ecology Controllability analysi
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