Dynamic observers for unknown populations

Chris Guiver; Nathan Poppelreiter; Richard Rebarber; Brigitte Tenhumberg; Stuart Townley

doi:10.3934/dcdsb.2020232

Article Contents

2021, Volume 26, Issue 6: 3279-3302. Doi: 10.3934/dcdsb.2020232

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Dynamic observers for unknown populations

1.
Department of Mathematical Sciences, University of Bath, Bath, BA2 7AY, UK
2.
Boston Fusion Corporation, Lexington, MA 02421
3.
Department of Mathematics, University of Nebraska-Lincoln, Lincoln, NE 68588-0130
4.
Department of Biological Sciences and Mathematics, University of Nebraska-Lincoln, Lincoln, NE 68588-0130
5.
Environment and Sustainability Institute, College of Engineering, Mathematics and Physical Sciences, University of Exeter, Penryn campus, Penryn, TR10 9FE, UK

^* Corresponding author
^* Corresponding author

Received: May 31, 2019

Revised: April 30, 2020

Early access: August 2020

Published: June 2021

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Abstract

Dynamic observers are considered in the context of structured-population modeling and management. Roughly, observers combine a known measured variable of some process with a model of that process to asymptotically reconstruct the unknown state variable of the model. We investigate the potential use of observers for reconstructing population distributions described by density-independent (linear) models and a class of density-dependent (nonlinear) models. In both the density-dependent and -independent cases, we show, in several ecologically reasonable circumstances, that there is a natural, optimal construction of these observers. Further, we describe the robustness these observers exhibit with respect to disturbances and uncertainty in measurement.

Keywords:

Mathematics Subject Classification: Primary: 39A30, 92D40, 93B51, 93B52, 93C05, 93C10, 93D09.

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Figure 1.1. Illustration of a dynamic observer. The dashed lines in the right-hand figure at computed using the measured variable $ y(t) $

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Figure 5.1. Numerical simulations from Example 5.1. Abundances from the model (LO) are plotted against time. The solid line and dotted lines are the abundance of the eighth stage-class $ x_8(t) $ of the (forced) cheetah population, and its corresponding observer state $ z_8(t) $, respectively. The dashed and dashed-dotted lines are $ \| x(t)\|_1 $ and $ \| z(t)\|_1 $, the total population and its observer estimate, respectively

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Figure 5.2. Numerical simulations from Example 5.2. In both panels, the solid, dashed, dashed-dotted and dotted lines denote the total population $ \| x(t)\|_1 $, the corresponding estimate $ \| z(t)\|_1 $, the error $ \| x(t) - z(t) \|_1 $ and the unforced equilibrium $ \| x^*\|_1 $, respectively, each plotted against time $ t $. In (a), no forcing terms are present, so $ d = 0 $ and $ v = 0 $, and the error converges to zero. In (b), the forcing and measurement error terms are nonzero, described in the main text

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Figure 5.3. Numerical simulations from Example 5.3. The solid line denotes $ \|x(t)\|_1 $, and the dashed, dashed-dotted, and dotted lines denote the errors $ \| x(t) - z(t)\|_1 $ for increasing measurement error. See the main text

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