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:
[1]

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

[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.

[3]

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

[4]

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

[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.

[6]

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

[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. 

[8]

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

[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.

[10]

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

[11]

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

[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.

[13]

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

[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.

[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.

[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.

[17]

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

[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.

[19]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[29]

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

[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.

[31]

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

[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.

[33]

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

[34]

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

[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.

[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.

[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.

[38]

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

[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.

[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.

[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.

[42]

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

[43]

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

[44]

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

[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.

[46]

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

[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.

[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.

[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.

[50]

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

[51]

M. A. Rami and F. Tadeo, Positive observation problem for linear discrete positive systems, in Proceedings of the 45th IEEE Conference on Decision and Control, IEEE, (2006). doi: 10.1109/CDC.2006.377749.

[52]

P. Reichert and M. Omlin, On the usefulness of overparameterized ecological models, Ecol. Mod., 95 (1997), 289-299.  doi: 10.1016/S0304-3800(96)00043-9.

[53]

K. E. RoseS. M. Louda and M. Rees, Demographic and evolutionary impacts of native and invasive insect herbivores on cirsium canescens, Ecology, 86 (2005), 453-465.  doi: 10.1890/03-0697.

[54]

M. de la Sen, Non-periodic and adaptive sampling. A tutorial review, Informatica, 7 (1996), 175-228. 

[55]

H. L. Smith and H. R. Thieme, Persistence and global stability for a class of discrete time structured population models, Discrete Contin. Dyn. Syst., 33 (2013), 4627-4646.  doi: 10.3934/dcds.2013.33.4627.

[56]

T. Söderström and P. Stoica, System Identification, Prentice-Hall, Inc., London, 1989.

[57]

E. D. Sontag, Smooth stabilization implies coprime factorization, IEEE Trans. Automat. Control, 34 (1989), 435-443.  doi: 10.1109/9.28018.

[58]

E. D. Sontag, Input-to-state stability: Basic concepts and results, in Nonlinear and Optimal Control Theory, 163–220, Lecture Notes in Math., 1932, Springer, Berlin, 2008. doi: 10.1007/978-3-540-77653-6_3.

[59]

E. D. Sontag, Mathematical Control Theory, second ed., Springer-Verlag, New York, 1998. doi: 10.1007/978-1-4612-0577-7.

[60]

M. Soroush, Nonlinear state-observer design with application to reactors, Chemical Engineering Science, 52 (1997), 387-404.  doi: 10.1016/S0009-2509(96)00391-0.

[61]

I. StottS. TownleyD. Carslake and D. J. Hodgson, On reducibility and ergodicity of population projection matrix models, Methods Ecol. Evol., 1 (2010), 242-252.  doi: 10.1111/j.2041-210X.2010.00032.x.

[62]

F. Tadeo and M. Rami, Selection of time-after-injection in bone scanning using compartmental observers, in Proceeding of World Engineering Congress, WCE (2010).

[63]

B. TenhumbergS. LoudaJ. Eckberg and M. Takahashi, Monte carlo analysys of parameter uncertainy in matrix models for the weed, J. Appl. Ecol., 45 (2008), 439-447. 

[64]

S. TownleyR. Rebarber and B. Tenhumberg, Feedback control systems analysis of density dependent population dynamics, Systems Control Lett., 61 (2012), 309-315.  doi: 10.1016/j.sysconle.2011.11.014.

[65]

H. L. Trentelman, A. A. Stoorvogel and M. Hautus, Control Theory for Linear Systems, Communications and Control Engineering Series, Springer-Verlag London, Ltd., London, 2001. doi: 10.1007/978-1-4471-0339-4.

[66]

J. M. Van Den Hof, Positive linear observers for linear compartmental systems, SIAM J. Control Optim., 36 (1998), 590-608.  doi: 10.1137/S036301299630611X.

[67]

X.-H. Xia and W.-B. Gao., Nonlinear observer design by observer error linearization, SIAM J. Control Optim., 27 (1989), 199-216.  doi: 10.1137/0327011.

[68]

J. I. Yuz and G. C. Goodwin, Sampled-Data Models for Linear and Nonlinear Systems, Communications and Control Engineering Series, Springer, London, 2014. doi: 10.1007/978-1-4471-5562-1.

[69]

J. ZhangX. ZhaoR. Zhang and Y. Chen, Improved controller design for uncertain positive systems and its extension to uncertain positive switched systems, Asian J. Control, 20 (2018), 159-173.  doi: 10.1002/asjc.1553.

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.

[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.

[3]

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

[4]

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

[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.

[6]

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

[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. 

[8]

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

[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.

[10]

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

[11]

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

[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.

[13]

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

[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.

[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.

[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.

[17]

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

[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.

[19]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[29]

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

[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.

[31]

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

[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.

[33]

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

[34]

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

[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.

[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.

[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.

[38]

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

[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.

[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.

[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.

[42]

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

[43]

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

[44]

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

[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.

[46]

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

[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.

[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.

[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.

[50]

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

[51]

M. A. Rami and F. Tadeo, Positive observation problem for linear discrete positive systems, in Proceedings of the 45th IEEE Conference on Decision and Control, IEEE, (2006). doi: 10.1109/CDC.2006.377749.

[52]

P. Reichert and M. Omlin, On the usefulness of overparameterized ecological models, Ecol. Mod., 95 (1997), 289-299.  doi: 10.1016/S0304-3800(96)00043-9.

[53]

K. E. RoseS. M. Louda and M. Rees, Demographic and evolutionary impacts of native and invasive insect herbivores on cirsium canescens, Ecology, 86 (2005), 453-465.  doi: 10.1890/03-0697.

[54]

M. de la Sen, Non-periodic and adaptive sampling. A tutorial review, Informatica, 7 (1996), 175-228. 

[55]

H. L. Smith and H. R. Thieme, Persistence and global stability for a class of discrete time structured population models, Discrete Contin. Dyn. Syst., 33 (2013), 4627-4646.  doi: 10.3934/dcds.2013.33.4627.

[56]

T. Söderström and P. Stoica, System Identification, Prentice-Hall, Inc., London, 1989.

[57]

E. D. Sontag, Smooth stabilization implies coprime factorization, IEEE Trans. Automat. Control, 34 (1989), 435-443.  doi: 10.1109/9.28018.

[58]

E. D. Sontag, Input-to-state stability: Basic concepts and results, in Nonlinear and Optimal Control Theory, 163–220, Lecture Notes in Math., 1932, Springer, Berlin, 2008. doi: 10.1007/978-3-540-77653-6_3.

[59]

E. D. Sontag, Mathematical Control Theory, second ed., Springer-Verlag, New York, 1998. doi: 10.1007/978-1-4612-0577-7.

[60]

M. Soroush, Nonlinear state-observer design with application to reactors, Chemical Engineering Science, 52 (1997), 387-404.  doi: 10.1016/S0009-2509(96)00391-0.

[61]

I. StottS. TownleyD. Carslake and D. J. Hodgson, On reducibility and ergodicity of population projection matrix models, Methods Ecol. Evol., 1 (2010), 242-252.  doi: 10.1111/j.2041-210X.2010.00032.x.

[62]

F. Tadeo and M. Rami, Selection of time-after-injection in bone scanning using compartmental observers, in Proceeding of World Engineering Congress, WCE (2010).

[63]

B. TenhumbergS. LoudaJ. Eckberg and M. Takahashi, Monte carlo analysys of parameter uncertainy in matrix models for the weed, J. Appl. Ecol., 45 (2008), 439-447. 

[64]

S. TownleyR. Rebarber and B. Tenhumberg, Feedback control systems analysis of density dependent population dynamics, Systems Control Lett., 61 (2012), 309-315.  doi: 10.1016/j.sysconle.2011.11.014.

[65]

H. L. Trentelman, A. A. Stoorvogel and M. Hautus, Control Theory for Linear Systems, Communications and Control Engineering Series, Springer-Verlag London, Ltd., London, 2001. doi: 10.1007/978-1-4471-0339-4.

[66]

J. M. Van Den Hof, Positive linear observers for linear compartmental systems, SIAM J. Control Optim., 36 (1998), 590-608.  doi: 10.1137/S036301299630611X.

[67]

X.-H. Xia and W.-B. Gao., Nonlinear observer design by observer error linearization, SIAM J. Control Optim., 27 (1989), 199-216.  doi: 10.1137/0327011.

[68]

J. I. Yuz and G. C. Goodwin, Sampled-Data Models for Linear and Nonlinear Systems, Communications and Control Engineering Series, Springer, London, 2014. doi: 10.1007/978-1-4471-5562-1.

[69]

J. ZhangX. ZhaoR. Zhang and Y. Chen, Improved controller design for uncertain positive systems and its extension to uncertain positive switched systems, Asian J. Control, 20 (2018), 159-173.  doi: 10.1002/asjc.1553.

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
[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

Metrics

  • PDF downloads (268)
  • HTML views (334)
  • Cited by (0)

[Back to Top]