March  2022, 27(3): 1765-1787. doi: 10.3934/dcdsb.2021109

A switching feedback control approach for persistence of managed resources

1. 

Departamento de Matemática Aplicada, E.T.S.I. Industriales, Universidad Nacional de Educación a Distancia (UNED), c/Juan del Rosal 12, 28040, Madrid, Spain

2. 

School of Engineering & the Built Environment, Edinburgh Napier University, Merchiston Campus, 10 Colinton Road, Edinburgh EH10 5DT, UK

3. 

Department of Mathematical Sciences, University of Bath, Claverton Down, Bath BA2 7AY, UK

4. 

Environment and Sustainability Institute, College of Engineering, Mathematics and Physical Sciences, University of Exeter, Penryn campus, Penryn, TR10 9FE, UK

* Corresponding author: Chris Guiver

Received  August 2020 Revised  March 2021 Published  March 2022 Early access  April 2021

An adaptive switching feedback control scheme is proposed for classes of discrete-time, positive difference equations, or systems of equations. In overview, the objective is to choose a control strategy which ensures persistence of the state, consequently avoiding zero which corresponds to absence or extinction. A robust feedback control solution is proposed as the effects of different management actions are assumed to be uncertain. Our motivating application is to the conservation of dynamic resources, such as populations, which are naturally positive quantities and where discrete and distinct courses of management actions, or control strategies, are available. The theory is illustrated with examples from population ecology.

Citation: Daniel Franco, Chris Guiver, Phoebe Smith, Stuart Townley. A switching feedback control approach for persistence of managed resources. Discrete and Continuous Dynamical Systems - B, 2022, 27 (3) : 1765-1787. doi: 10.3934/dcdsb.2021109
References:
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S. N. Amin and M. Zafar, Studies on age, growth and virtual population analysis of Coilia dussumieri from the neritic water of Bangladesh, J. Biol. Sci, 4 (2004), 342-344. 

[2]

K. J. Astrom, Adaptive feedback control, Proc. IEEE, 75 (1987), 185-217.  doi: 10.1109/PROC.1987.13721.

[3]

I. Barkana, Simple adaptive control—a stable direct model reference adaptive control methodology—brief survey, Internat. J. Adapt. Control Signal Process., 28 (2014), 567-603.  doi: 10.1002/acs.2411.

[4]

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

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

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B. L. Byrd, T. V. Cole, L. Engleby, L. P. Garrison, J. M. Hatch, A. Henry, S. C. Horstman, J. A. Litz, M. Lyssikatos, K. Mullin, et al., US Atlantic and Gulf of Mexico Marine Mammal Stock Assessments-2015, 2016.

[7]

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

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C. W. Clark, Mathematical Bioeconomics, 3$^{rd}$ edition, The Mathematics of Conservation, John Wiley & Sons, Inc., Hoboken, 2010.

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J. Cooke, Eubalaena glacialis, The IUCN Red List of Threatened Species, (2018).

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M. De Lara and L. Doyen, Sustainable Management of Natural Resources: Mathematical Models and Methods, Springer-Verlag, Berlin, 2008.

[11]

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.

[12]

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.

[13]

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

[14]

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

[15]

H. I. Freedman and J. W.-H. So, Persistence in discrete semidynamical systems, SIAM J. Math. Anal., 20 (1989), 930-938.  doi: 10.1137/0520062.

[16]

M. Fujiwara and H. Caswell, Demography of the endangered North Atlantic right whale, Nature, 414 (2001), 537-541.  doi: 10.1038/35107054.

[17]

T. A. GowanJ. G. Ortega-OrtizJ. A. HostetlerP. K. HamiltonA. R. KnowltonK. A. JacksonR. C. GeorgeC. R. Taylor and and P. J. Naessig, Temporal and demographic variation in partial migration of the north atlantic right whale, Sci. Rep., 9 (2019), 1-11.  doi: 10.1038/s41598-018-36723-3.

[18]

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.

[19]

E. Halfon, The systems identification problem and the development of ecosystem models, Simulation, 25 (1975), 149-152.  doi: 10.1177/003754977502500604.

[20] E. Halfon, Theoretical Systems Ecology, Academic Press, New York, 1979. 
[21]

M. P. Hassell, J. H. Lawton and R. May, Patterns of dynamical behaviour in single-species populations, J. Anim. Ecol., (1976), 471–486. doi: 10.2307/3886.

[22]

B. IngramC. BarlowJ. BurchmoreG. GooleyS. Rowland and and A. Sanger, Threatened native freshwater fishes in {A}ustralia–some case histories, J. Fish Biol., 37 (1990), 175-182.  doi: 10.1111/j.1095-8649.1990.tb05033.x.

[23]

J. Koehn, M. Lintermans, J. Lieschke and D. Gilligan, Maccullochella macquariensis, The IUCN Red List of Threatened Species 2019: e.T12574A123378211, 2019.

[24]

S. D. Kraus, Rates and potential causes of mortality in North Atlantic right whales (Eubalaena glacialis), Mar. Mammal Sci., 6 (1990), 278-291. 

[25]

E. L. Meyer-Gutbrod and C. H. Greene, Uncertain recovery of the North Atlantic right whale in a changing ocean, Glob. Change Biol., 24 (2018), 455-464.  doi: 10.1111/gcb.13929.

[26]

R. M. Pace IIIP. J. Corkeron and S. D. Kraus, State–space mark–recapture estimates reveal a recent decline in abundance of North Atlantic right whales, Ecol. Evol., 7 (2017), 8730-8741. 

[27]

R. RebarberB. Tenhumberg and S. Townley, Global asymptotic stability of density dependent integral population projection models, Theo. Popul. Bio., 81 (2012), 81-87.  doi: 10.1016/j.tpb.2011.11.002.

[28]

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.

[29]

A. O. Shelton and M. Mangel, Fluctuations of fish populations and the magnifying effects of fishing, Proc. Natl. Acad. Sci. U.S.A, 108 (2011), 7075-7080.  doi: 10.1073/pnas.1100334108.

[30]

H. L. Smith and H. R. Thieme, Dynamical Systems and Population Persistence, American Mathematical Society, Providence, RI, 2011. doi: 10.1090/gsm/118.

[31]

H. L. Smith and H. R. Thieme, Persistence and global stability for a class of discrete time structured population models, Dynamical Systems, 33 (2013), 4627-4645.  doi: 10.3934/dcds.2013.33.4627.

[32]

P. Smith, A Control Theoretic Approach to the Conservation of Migratory Species, Ph.D thesis, University of Bath, in preparation.

[33]

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.

[34]

C. R. ToddS. J. Nicol and J. D. Koehn, Density-dependence uncertainty in population models for the conservation management of trout cod, Maccullochella macquariensis, Ecol. Mod., 171 (2004), 359-380. 

[35]

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.

[36]

B. K. Williams, Adaptive management of natural resources — framework and issues, J. Environ. Manage., 92 (2011), 1346-1353.  doi: 10.1016/j.jenvman.2010.10.041.

show all references

References:
[1]

S. N. Amin and M. Zafar, Studies on age, growth and virtual population analysis of Coilia dussumieri from the neritic water of Bangladesh, J. Biol. Sci, 4 (2004), 342-344. 

[2]

K. J. Astrom, Adaptive feedback control, Proc. IEEE, 75 (1987), 185-217.  doi: 10.1109/PROC.1987.13721.

[3]

I. Barkana, Simple adaptive control—a stable direct model reference adaptive control methodology—brief survey, Internat. J. Adapt. Control Signal Process., 28 (2014), 567-603.  doi: 10.1002/acs.2411.

[4]

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

[5]

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

[6]

B. L. Byrd, T. V. Cole, L. Engleby, L. P. Garrison, J. M. Hatch, A. Henry, S. C. Horstman, J. A. Litz, M. Lyssikatos, K. Mullin, et al., US Atlantic and Gulf of Mexico Marine Mammal Stock Assessments-2015, 2016.

[7]

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

[8]

C. W. Clark, Mathematical Bioeconomics, 3$^{rd}$ edition, The Mathematics of Conservation, John Wiley & Sons, Inc., Hoboken, 2010.

[9]

J. Cooke, Eubalaena glacialis, The IUCN Red List of Threatened Species, (2018).

[10]

M. De Lara and L. Doyen, Sustainable Management of Natural Resources: Mathematical Models and Methods, Springer-Verlag, Berlin, 2008.

[11]

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.

[12]

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.

[13]

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

[14]

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

[15]

H. I. Freedman and J. W.-H. So, Persistence in discrete semidynamical systems, SIAM J. Math. Anal., 20 (1989), 930-938.  doi: 10.1137/0520062.

[16]

M. Fujiwara and H. Caswell, Demography of the endangered North Atlantic right whale, Nature, 414 (2001), 537-541.  doi: 10.1038/35107054.

[17]

T. A. GowanJ. G. Ortega-OrtizJ. A. HostetlerP. K. HamiltonA. R. KnowltonK. A. JacksonR. C. GeorgeC. R. Taylor and and P. J. Naessig, Temporal and demographic variation in partial migration of the north atlantic right whale, Sci. Rep., 9 (2019), 1-11.  doi: 10.1038/s41598-018-36723-3.

[18]

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.

[19]

E. Halfon, The systems identification problem and the development of ecosystem models, Simulation, 25 (1975), 149-152.  doi: 10.1177/003754977502500604.

[20] E. Halfon, Theoretical Systems Ecology, Academic Press, New York, 1979. 
[21]

M. P. Hassell, J. H. Lawton and R. May, Patterns of dynamical behaviour in single-species populations, J. Anim. Ecol., (1976), 471–486. doi: 10.2307/3886.

[22]

B. IngramC. BarlowJ. BurchmoreG. GooleyS. Rowland and and A. Sanger, Threatened native freshwater fishes in {A}ustralia–some case histories, J. Fish Biol., 37 (1990), 175-182.  doi: 10.1111/j.1095-8649.1990.tb05033.x.

[23]

J. Koehn, M. Lintermans, J. Lieschke and D. Gilligan, Maccullochella macquariensis, The IUCN Red List of Threatened Species 2019: e.T12574A123378211, 2019.

[24]

S. D. Kraus, Rates and potential causes of mortality in North Atlantic right whales (Eubalaena glacialis), Mar. Mammal Sci., 6 (1990), 278-291. 

[25]

E. L. Meyer-Gutbrod and C. H. Greene, Uncertain recovery of the North Atlantic right whale in a changing ocean, Glob. Change Biol., 24 (2018), 455-464.  doi: 10.1111/gcb.13929.

[26]

R. M. Pace IIIP. J. Corkeron and S. D. Kraus, State–space mark–recapture estimates reveal a recent decline in abundance of North Atlantic right whales, Ecol. Evol., 7 (2017), 8730-8741. 

[27]

R. RebarberB. Tenhumberg and S. Townley, Global asymptotic stability of density dependent integral population projection models, Theo. Popul. Bio., 81 (2012), 81-87.  doi: 10.1016/j.tpb.2011.11.002.

[28]

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.

[29]

A. O. Shelton and M. Mangel, Fluctuations of fish populations and the magnifying effects of fishing, Proc. Natl. Acad. Sci. U.S.A, 108 (2011), 7075-7080.  doi: 10.1073/pnas.1100334108.

[30]

H. L. Smith and H. R. Thieme, Dynamical Systems and Population Persistence, American Mathematical Society, Providence, RI, 2011. doi: 10.1090/gsm/118.

[31]

H. L. Smith and H. R. Thieme, Persistence and global stability for a class of discrete time structured population models, Dynamical Systems, 33 (2013), 4627-4645.  doi: 10.3934/dcds.2013.33.4627.

[32]

P. Smith, A Control Theoretic Approach to the Conservation of Migratory Species, Ph.D thesis, University of Bath, in preparation.

[33]

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.

[34]

C. R. ToddS. J. Nicol and J. D. Koehn, Density-dependence uncertainty in population models for the conservation management of trout cod, Maccullochella macquariensis, Ecol. Mod., 171 (2004), 359-380. 

[35]

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.

[36]

B. K. Williams, Adaptive management of natural resources — framework and issues, J. Environ. Manage., 92 (2011), 1346-1353.  doi: 10.1016/j.jenvman.2010.10.041.

Figure 2.1.  Illustration of the conditions (NL3)(a) and (NL3)(b) in panels (A) and (B), respectively. The dashed straight lines have gradient $ p_h >0 $
Figure 3.1.  Numerical simulations of the adaptive switching feedback control scheme (2.4) for the North Atlantic right whale model described in Example 3.1
Figure 3.2.  Functions $ g_h $, panel (a), with parameters, panel (b), from Example 3.2
Figure 3.3.  Numerical simulations of the adaptive switching feedback control scheme (2.4) for the trout cod model from Example 3.2
Figure 3.4.  Numerical simulations of the adaptive switching feedback control scheme (2.4) for the trout cod model from Example 3.2 with 100 random initial conditions $ x_0 $
Figure 3.5.  Functions $ g_h $, panel (a), with parameters, panel (b), from Example 3.3
Figure 3.6.  Numerical simulations of the adaptive switching feedback control scheme (2.4) for the Gold-spotted grenadier anchovy model from Example 3.3
Figure 3.7.  Numerical simulations of the adaptive switching feedback control scheme (2.4) for the trout cod model discussed in Section 3.1
Table 3.1.  Vital rates used in the population projection matrices $ A_h $ in (3.1)
Strategy ($ h $) Vital rates
$ s_{2, 1} $ $ s_{2, 2} $ $ s_{3, 2} $ $ s_{3, 3} $ $ s_{3, 4} $ $ s_{4, 2} $ $ s_{4, 3} $ $ f_{1, 2} $ $ f_{1, 3} $
1 0.85 0.85 0.08 0.8 0.64 0.02 0.19 0.0080 0.0760
2 0.92 0.86 0.08 0.8 0.83 0.02 0.19 0.0091 0.0865
Strategy ($ h $) Vital rates
$ s_{2, 1} $ $ s_{2, 2} $ $ s_{3, 2} $ $ s_{3, 3} $ $ s_{3, 4} $ $ s_{4, 2} $ $ s_{4, 3} $ $ f_{1, 2} $ $ f_{1, 3} $
1 0.85 0.85 0.08 0.8 0.64 0.02 0.19 0.0080 0.0760
2 0.92 0.86 0.08 0.8 0.83 0.02 0.19 0.0091 0.0865
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