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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  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
References:
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D. Angeli, A Lyapunov approach to incremental stability properties, IEEE Trans. Automat. Control, 47 (2002), 410-421.  doi: 10.1109/9.989067.  Google Scholar

[2]

M. Arcak and P. Kokotovic, Observer-based control of systems with slope-restricted nonlinearities, IEEE Trans. Automat. Control, 46 (2001), 1146-1150.  doi: 10.1109/9.935073.  Google Scholar

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S. N. DashkovskiyD. V. Efimov and E. D. Sontag., Input-to-state stability and allied system properties, Autom. Remote Control, 72 (2011), 1579-1614.  doi: 10.1134/S0005117911080017.  Google Scholar

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M. R. EasterlingS. P. Ellner and P. M. Dixon, Size-specific sensitivity: Applying a new structured population model, Ecology, 81 (2000), 694-708.   Google Scholar

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M. E. Gilmore, C. Guiver and H. Logemann, Stability and convergence properties of forced infinite-dimensional discrete-time Lur'e systems, Int. J. Control, (2019), 1–40. Google Scholar

[24]

J. L. GouzéA. Rapaport and M. Z. Hadj-Sadok, Interval observers for uncertain biological systems, Ecol. Modelling, 133 (2000), 45-56.  doi: 10.1016/S0304-3800(00)00279-9.  Google Scholar

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C. GuiverD. Hodgson and S. Townley, Positive state controllability of positive linear systems, Systems Control Lett., 65 (2014), 23-29.  doi: 10.1016/j.sysconle.2013.12.002.  Google Scholar

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C. Guiver, H. Logemann and B. Rüffer, Small-gain stability theorems for positive Lur'e inclusions, Positivity, 23 (2019), 249–289. doi: 10.1007/s11117-018-0605-2.  Google Scholar

[29]

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M. Z. Hadj-Sadok and J. L. Gouzé, Estimation of uncertain models of activated sludge processes with interval observers, J. Process Control, 11 (2001), 299-310.  doi: 10.1016/S0959-1524(99)00074-8.  Google Scholar

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E. Halfon (ed), Theoretical Systems Ecology, Academic Press, New York, 1979. Google Scholar

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show all references

References:
[1]

D. Angeli, A Lyapunov approach to incremental stability properties, IEEE Trans. Automat. Control, 47 (2002), 410-421.  doi: 10.1109/9.989067.  Google Scholar

[2]

M. Arcak and P. Kokotovic, Observer-based control of systems with slope-restricted nonlinearities, IEEE Trans. Automat. Control, 46 (2001), 1146-1150.  doi: 10.1109/9.935073.  Google Scholar

[3]

A. Berman, M. Neumann and R. J. Stern, Nonnegative Matrices in Dynamic Systems, John Wiley & Sons Inc., New York, 1989.  Google Scholar

[4]

A. Berman and R. J. Plemmons, Nonnegative Matrices in the Mathematical Sciences, SIAM, Philadelphia, 1994. doi: 10.1137/1.9781611971262.  Google Scholar

[5]

G. E. P. Box, G. M. Jenkins, G. C. Reinsel and G. M. Ljung, Time Series Analysis, fifth edn., John Wiley & Sons, Inc., Hoboken, 2016.  Google Scholar

[6]

H. Caswell, Matrix Population Models : Construction, Analysis and Interpretation, Sinauer, Massachusetts, 2001. Google Scholar

[7]

I. ChadèsE. McDonald-MaddenM. A. McCarthyB. WintleM. Linkie and H. P. Possingham, When to stop managing or surveying cryptic threatened species, Proc. Nat. Acad. Sci., 105 (2008), 13936-13940.   Google Scholar

[8]

T. Chen and B. Francis, Optimal Sampled-Data Control Systems, Communications and Control Engineering Series, Springer-Verlag London, Ltd., London, 1996.  Google Scholar

[9]

C. k. Chui and G. Chen, Kalman Filtering, with Real-Time Applications, Springer-Verlag, Berlin, 1991. doi: 10.1007/978-3-662-02666-3.  Google Scholar

[10]

K. R. CrooksM. A. Sanjayan and D. F. Doak, New insights on cheetah conservation through demographic modeling, Conservation Biology, 12 (1998), 889-895.   Google Scholar

[11]

J. M. Cushing, An Introduction to Structured Population Dynamics, SIAM, Philadelphia, 1998. doi: 10.1137/1.9781611970005.  Google Scholar

[12]

S. N. DashkovskiyD. V. Efimov and E. D. Sontag., Input-to-state stability and allied system properties, Autom. Remote Control, 72 (2011), 1579-1614.  doi: 10.1134/S0005117911080017.  Google Scholar

[13]

N. Dautrebande and G. Bastin, Positive linear observers for positive linear systems, Proceedings of the European Control Conference, 1999, 1092–1095. Google Scholar

[14]

E. A. Eager, Modelling and analysis of population dynamics using Lur'e systems accounting for competition from adult conspecifics, Lett. Biomath., 3 (2016), 41-58.  doi: 10.30707/LiB3.1Eager.  Google Scholar

[15]

E. A. Eager and R. Rebarber, Sensitivity and elasticity analysis of a Lur'e system used to model a population subject to density-dependent reproduction, Math. Biosci., 282 (2016), 34-45.  doi: 10.1016/j.mbs.2016.09.016.  Google Scholar

[16]

E. A. EagerR. Rebarber and B. Tenhumberg, Global asymptotic stability of plant-seed bank models, J. Math. Biology, 69 (2014), 1-37.  doi: 10.1007/s00285-013-0689-z.  Google Scholar

[17]

M. R. EasterlingS. P. Ellner and P. M. Dixon, Size-specific sensitivity: Applying a new structured population model, Ecology, 81 (2000), 694-708.   Google Scholar

[18]

X. Fan and M. Arcak, Observer design for systems with multivariable monotone nonlinearities, Systems Control Lett., 50 (2003), 319-330.  doi: 10.1016/S0167-6911(03)00170-1.  Google Scholar

[19]

L. Farina and S.Rinaldi, Positive Linear Systems: Theory and Applications, Wiley-Interscience, New York, 2000. doi: 10.1002/9781118033029.  Google Scholar

[20]

D. FrancoC. GuiverH. Logemann and J. Perán, Semi-global persistence and stability for a class of forced discrete-time population models, Phys. D, 360 (2017), 46-61.  doi: 10.1016/j.physd.2017.08.001.  Google Scholar

[21]

D. FrancoC. GuiverH. Logemann and J. Perán., Boundedness, persistence and stability for classes of forced difference equations arising in population ecology, J. Math. Biol., 79 (2019), 1029-1076.  doi: 10.1007/s00285-019-01388-7.  Google Scholar

[22]

D. FrancoH. Logemann and J. Perán, Global stability of an age-structured population model, Systems Control Lett., 65 (2014), 30-36.  doi: 10.1016/j.sysconle.2013.11.012.  Google Scholar

[23]

M. E. Gilmore, C. Guiver and H. Logemann, Stability and convergence properties of forced infinite-dimensional discrete-time Lur'e systems, Int. J. Control, (2019), 1–40. Google Scholar

[24]

J. L. GouzéA. Rapaport and M. Z. Hadj-Sadok, Interval observers for uncertain biological systems, Ecol. Modelling, 133 (2000), 45-56.  doi: 10.1016/S0304-3800(00)00279-9.  Google Scholar

[25]

C. GuiverD. Hodgson and S. Townley, Positive state controllability of positive linear systems, Systems Control Lett., 65 (2014), 23-29.  doi: 10.1016/j.sysconle.2013.12.002.  Google Scholar

[26]

C. GuiverC. EdholmY. JinM. MuellerJ. PowellR. RebarberB. Tenhumberg and S. Townley, Simple adaptive control for positive linear systems with applications to pest management, SIAM J. Appl. Math, 76 (2016), 238-275.  doi: 10.1137/140996926.  Google Scholar

[27]

C. GuiverH. LogemannR. RebarberA. BillB. TenhumbergD. Hodgson and S. Townley, Integral control for population management, J. Math. Biol., 70 (2005), 1015-1063.  doi: 10.1007/s00285-014-0789-4.  Google Scholar

[28]

C. Guiver, H. Logemann and B. Rüffer, Small-gain stability theorems for positive Lur'e inclusions, Positivity, 23 (2019), 249–289. doi: 10.1007/s11117-018-0605-2.  Google Scholar

[29]

W. M. Haddad, V. Chellaboina and Q. Hui, Nonnegative and Compartmental Dynamical Systems, Princeton University Press, Princeton, 2010. doi: 10.1515/9781400832248.  Google Scholar

[30]

M. Z. Hadj-Sadok and J. L. Gouzé, Estimation of uncertain models of activated sludge processes with interval observers, J. Process Control, 11 (2001), 299-310.  doi: 10.1016/S0959-1524(99)00074-8.  Google Scholar

[31]

E. Halfon (ed), Theoretical Systems Ecology, Academic Press, New York, 1979. Google Scholar

[32]

H. R. Heinimann, A concept in adaptive ecosystem management — An engineering perspective, Forest Ecol. Manag., 259 (2010), 848-856.  doi: 10.1016/j.foreco.2009.09.032.  Google Scholar

[33]

D. Hinrichsen and A. J. Pritchard, Mathematical Systems Theory I, Springer–Verlag, Berlin, 2005. doi: 10.1007/b137541.  Google Scholar

[34]

D. Hinrichsen and N. K. Son, Stability radii of positive discrete-time systems, Internat. J. Robust Nonlinear Control, 8 (1995). Google Scholar

[35]

S. Ibrir, Circle-criterion approach to discrete-time nonlinear observer design, Automatica, 43 (2007), 1432-1441.  doi: 10.1016/j.automatica.2007.01.012.  Google Scholar

[36]

R. E. Kalman, A new approach to linear filtering and prediction problems, J. Basic Engineering (ASME), 82 60), 34–45. doi: 10.1115/1.3662552.  Google Scholar

[37]

I. Karafyllis and Z.-P. Jiang, Stability and Stabilization of Nonlinear Systems, Springer-Verlag, London, 2011. doi: 10.1007/978-0-85729-513-2.  Google Scholar

[38]

N. Keyfitz and H. Caswell, Applied Mathematical Demography, Springer Science+Business Media, Inc., 2005. Google Scholar

[39]

R KleinN. A. ChaturvediJ. ChristensenJ. AhmedR. Findeisen and A. Kojic, Electrochemical model based observer design for a lithium-ion battery, IEEE Trans. Control Syst. Technol., 21 (2013), 289-301.  doi: 10.1109/TCST.2011.2178604.  Google Scholar

[40]

A. J. Krener and A. Isidori, Linearization by output injection and nonlinear observers, Systems Control Lett., 3 (1983), 47-52.  doi: 10.1016/0167-6911(83)90037-3.  Google Scholar

[41]

T. Liao and N. Huang, An observer-based approach for chaotic synchronization with applications to secure communications, IEEE Trans. Circuits Syst. I, 46 (1999), 1144–1150. doi: 10.1109/81.788817.  Google Scholar

[42]

L. Ljung, System Identification: Theory for the User, Prentice Hall Information and System Sciences Series. Prentice Hall, Inc., Englewood Cliffs, NJ, 1987.  Google Scholar

[43]

D. Luenberger, An introduction to observers, IEEE Trans. Automat. Control, 16 (1971), 596-602.  doi: 10.1109/TAC.1971.1099826.  Google Scholar

[44]

D. Luenberger, Observing the state of a linear system, IEEE Trans. Mil. Electronics, 8 (1964), 74-80.  doi: 10.1109/TME.1964.4323124.  Google Scholar

[45]

C. MerowJ. P. DahlgrenC. J. E. MetcalfD. Z. ChildsM. E. EvansE. JongejansS. RecordM. ReesR. Salguero-Gómez and S. M. McMahon, Advancing population ecology with integral projection models: A practical guide, Methods Ecol. Evol., 5 (2014), 99-110.  doi: 10.1111/2041-210X.12146.  Google Scholar

[46]

W. F. Morris and D. F. Doak, Quantitative Conservation biology: Theory and Practice of Population Viability Analysis, Sinauer Associates Sunderland, Massachusetts, USA, 2002. Google Scholar

[47]

M. Müller and C. A. Sierra, Application of input to state stability to reservoir models, Theor. Ecol., 10 (2017), 451–475. doi: 10.1007/s12080-017-0342-3.  Google Scholar

[48]

R. A. Myers, Stock and recruitment: Generalizations about maximum reproductive rate, density dependence, and variability using meta-analytic approaches, ICES J. Marine Science, 58 (2001), 937-951.  doi: 10.1006/jmsc.2001.1109.  Google Scholar

[49]

I. M. Navon, Data assimilation for numerical weather prediction: A review, in Data Assimilation for Atmospheric, Oceanic and Hydrologic Applications, S.K. Park and L. Xu eds. Springer, Berlin, 2009. doi: 10.1007/978-3-540-71056-1_2.  Google Scholar

[50]

N. Poppelreiter, Dynamic Observers for Unknown Populations, Phd Thesis, University of Nebraska, Lincoln, 2019. Google Scholar

[51]

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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) $
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
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
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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