# American Institute of Mathematical Sciences

## 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

Received  June 2019 Revised  May 2020 Published  August 2020

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.

Citation: Chris Guiver, Nathan Poppelreiter, Richard Rebarber, Brigitte Tenhumberg, Stuart Townley. Dynamic observers for unknown populations. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2020232
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Illustration of a dynamic observer. The dashed lines in the right-hand figure at computed using the measured variable $y(t)$
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
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
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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