# American Institute of Mathematical Sciences

June  2021, 26(6): 3279-3302. doi: 10.3934/dcdsb.2020232

## 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  June 2021 Early access  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 and Continuous Dynamical Systems - B, 2021, 26 (6) : 3279-3302. doi: 10.3934/dcdsb.2020232
##### References:

show all references

##### References:
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
 [1] Ruofeng Rao, Shouming Zhong. Input-to-state stability and no-inputs stabilization of delayed feedback chaotic financial system involved in open and closed economy. Discrete and Continuous Dynamical Systems - S, 2021, 14 (4) : 1375-1393. doi: 10.3934/dcdss.2020280 [2] Andrii Mironchenko, Hiroshi Ito. Characterizations of integral input-to-state stability for bilinear systems in infinite dimensions. Mathematical Control and Related Fields, 2016, 6 (3) : 447-466. doi: 10.3934/mcrf.2016011 [3] Pengfei Wang, Mengyi Zhang, Huan Su. Input-to-state stability of infinite-dimensional stochastic nonlinear systems. Discrete and Continuous Dynamical Systems - B, 2022, 27 (2) : 821-836. doi: 10.3934/dcdsb.2021066 [4] Zaizheng Li, Zhitao Zhang. Uniqueness and nondegeneracy of positive solutions to an elliptic system in ecology. Electronic Research Archive, 2021, 29 (6) : 3761-3774. doi: 10.3934/era.2021060 [5] Huijuan Li, Junxia Wang. Input-to-state stability of continuous-time systems via finite-time Lyapunov functions. Discrete and Continuous Dynamical Systems - B, 2020, 25 (3) : 841-857. doi: 10.3934/dcdsb.2019192 [6] Hiroshi Ito. Input-to-state stability and Lyapunov functions with explicit domains for SIR model of infectious diseases. Discrete and Continuous Dynamical Systems - B, 2021, 26 (9) : 5171-5196. doi: 10.3934/dcdsb.2020338 [7] Jeongsim Kim, Bara Kim. Stability of a cyclic polling system with an adaptive mechanism. Journal of Industrial and Management Optimization, 2015, 11 (3) : 763-777. doi: 10.3934/jimo.2015.11.763 [8] Shihe Xu, Fangwei Zhang, Meng Bai. Stability of positive steady-state solutions to a time-delayed system with some applications. Discrete and Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021286 [9] Jing Liu, Xiaodong Liu, Sining Zheng, Yanping Lin. Positive steady state of a food chain system with diffusion. Conference Publications, 2007, 2007 (Special) : 667-676. doi: 10.3934/proc.2007.2007.667 [10] István Györi, Ferenc Hartung. Exponential stability of a state-dependent delay system. Discrete and Continuous Dynamical Systems, 2007, 18 (4) : 773-791. doi: 10.3934/dcds.2007.18.773 [11] Leonid Shaikhet. Stability of a positive equilibrium state for a stochastically perturbed mathematical model of glassy-winged sharpshooter population. Mathematical Biosciences & Engineering, 2014, 11 (5) : 1167-1174. doi: 10.3934/mbe.2014.11.1167 [12] Zhanping Liang, Yuanmin Song, Fuyi Li. Positive ground state solutions of a quadratically coupled schrödinger system. Communications on Pure and Applied Analysis, 2017, 16 (3) : 999-1012. doi: 10.3934/cpaa.2017048 [13] Guofeng Che, Haibo Chen, Tsung-fang Wu. Bound state positive solutions for a class of elliptic system with Hartree nonlinearity. Communications on Pure and Applied Analysis, 2020, 19 (7) : 3697-3722. doi: 10.3934/cpaa.2020163 [14] Andrew P. Sage. Risk in system of systems engineering and management. Journal of Industrial and Management Optimization, 2008, 4 (3) : 477-487. doi: 10.3934/jimo.2008.4.477 [15] Xiang-Ping Yan, Wan-Tong Li. Stability and Hopf bifurcations for a delayed diffusion system in population dynamics. Discrete and Continuous Dynamical Systems - B, 2012, 17 (1) : 367-399. doi: 10.3934/dcdsb.2012.17.367 [16] Luca Gerardo-Giorda, Pierre Magal, Shigui Ruan, Ousmane Seydi, Glenn Webb. Preface: Population dynamics in epidemiology and ecology. Discrete and Continuous Dynamical Systems - B, 2020, 25 (6) : i-ii. doi: 10.3934/dcdsb.2020125 [17] Haibo Jin, Long Hai, Xiaoliang Tang. An optimal maintenance strategy for multi-state systems based on a system linear integral equation and dynamic programming. Journal of Industrial and Management Optimization, 2020, 16 (2) : 965-990. doi: 10.3934/jimo.2018188 [18] Maoding Zhen, Jinchun He, Haoyuan Xu, Meihua Yang. Positive ground state solutions for fractional Laplacian system with one critical exponent and one subcritical exponent. Discrete and Continuous Dynamical Systems, 2019, 39 (11) : 6523-6539. doi: 10.3934/dcds.2019283 [19] Alejandro Cataldo, Juan-Carlos Ferrer, Pablo A. Rey, Antoine Sauré. Design of a single window system for e-government services: the chilean case. Journal of Industrial and Management Optimization, 2018, 14 (2) : 561-582. doi: 10.3934/jimo.2017060 [20] Zhanyou Ma, Wuyi Yue, Xiaoli Su. Performance analysis of a Geom/Geom/1 queueing system with variable input probability. Journal of Industrial and Management Optimization, 2011, 7 (3) : 641-653. doi: 10.3934/jimo.2011.7.641

2020 Impact Factor: 1.327