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On the stock estimation for some fishery systems

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Abstract

In this work we address the stock estimation problem for two fishery models. We show that a tool from nonlinear control theory called “observer” can be helpful to deal with the resource stock estimation in the field of renewable resource management. It is often difficult or expensive to measure all the state variables characterising the evolution of a given population system, therefore the question arises whether from the observation of certain indicators of the considered system, the whole state of the population system can be recovered or at least estimated. The goal of this paper is to show how some techniques of control theory can be applied for the approximate estimation of the unmeasurable state variables using only the observed data together with the dynamical model describing the evolution of the system. More precisely we shall consider two fishery models and we shall show how to built for each model an auxiliary dynamical system (the observer) that uses the available data (the total of caught fish) and which produces a dynamical estimation \(\hat x(t)\) of the unmeasurable stock state x(t). Moreover the convergence speed of \(\hat x(t)\) towards x(t) can be chosen.

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References

  • Alamir M, Calvillo-Corona L (2002) Further results on nonlinear receding-horizon observers. IEEE Trans Autom Control 47(7):1184–1188

    Article  Google Scholar 

  • Alcaraz-Gonzalez V, Salazar-Pena R, Gonzalez-Alvarez V, Gouze J-L, Steyer J-P (2005) A tunable multivariable nonlinear robust observer for biological systems. Comptes Rendus Biol 328(4):317–325

    Article  Google Scholar 

  • Aniţa S (1998) Optimal harvesting for a nonlinear age-dependent population dynamics. J Math Anal Appl 226(1):6–22

    Article  Google Scholar 

  • Atassi A, Khalil H (2000) Separation results for the stabilization of nonlinear systems using different high-gain observer designs. Syst Control Lett 39(3):183–191

    Article  Google Scholar 

  • Bernard O, Sallet G, Sciandra A (1998) Nonlinear observers for a class of biological systems: application to validation of a phytoplanktonic growth model. IEEE Trans Autom Control 43:1056–1065

    Article  Google Scholar 

  • Bestle D, Zeitz M (1983) Canonical form observer design for non-linear time-variable systems. Int J Control 38(2):419–431

    Article  Google Scholar 

  • Bornard G, Hammouri H (1991) A high gain observer for a class of uniformly observable systems. Proceedings of the 30th IEEE conference on decision and control, vol 2, pp 1494–1496, December 1991

  • Clark CW (1990) Mathematical bioeconomics. The optimal management of renewable resources, 2nd edn. Wiley-Interscience Publication, New York

    Google Scholar 

  • Dubey B, Chandra P, Sinha P (2003) A model for fishery resource with reserve area. Nonlinear Anal Real World Appl 4(4):625–637

    Article  Google Scholar 

  • Farza M, Busawon K, Hammouri H (1998) Simple nonlinear observers for on-line estimation of kinetic rates in bioreactors. Automatica 34:301–318

    Article  Google Scholar 

  • Gao S, Chen L, Sun L (2005) Optimal pulse fishing policy in stage-structured models with birth pulses. Chaos, Solitons & Fractals 25(5):1209–1219

    Article  Google Scholar 

  • Gauthier JP, Kupka I (1994) Observability and observers for nonlinear systems. SIAM J Control Optim 32(4):975–994

    Article  Google Scholar 

  • Gauthier JP, Kupka I (2001) Deterministic observation theory and applications. Cambridge University Press, Cambridge

  • Gauthier JP, Hammouri H, Othman S (1992) A simple observer for nonlinear systems applications to bioreactors. IEEE Trans Autom Control 37:875–880

    Article  Google Scholar 

  • Guiro A, Iggidr A, Ngom D, Touré H (2008) A non linear observer for a fishery model. In: Proceedings of 17th Triennial IFAC World Congress, Seoul, Korea, July 6–11, 2008

  • Gulland JA (1983) Fish stock assessment, a manual of basic methods. Wiley, Chichester

    Google Scholar 

  • Horwood JW, Whittle P (1986) The optimal harvest from a multicohort stock. IMA J Math Appl Med Biol 3(2):143–155

    Article  Google Scholar 

  • Iggidr A (2004) Controllability, observability and stability of mathematical models, in Mathematical Models. In: Filar JA (ed) Encyclopedia of life support systems (EOLSS). Developed under the auspices of the UNESCO, Eolss Publishers, Oxford, UK. http://www.eolss.net. Retrieved 31 Jan 2006

  • Iggidr A, Sallet G (1993) Exponential stabilization of nonlinear systems by an estimated state feedback. In: Proceedings of the 2nd European control conference ECC’93. Groningen, Pays-Bas

  • Isidori A (1995) Nonlinear control systems, 3rd edn. Communications and Control Engineering Series. Springer, Berlin

    Google Scholar 

  • Jacobs OLR, Ballance DJ, Horwood JW (1991) Fishery management as a problem in feedback-control. Automatica 27(4):627–639

    Article  Google Scholar 

  • Kang W (2006) Moving horizon numerical observers of nonlinear control systems. IEEE Trans Automat Control 51(2):344–350

    Article  Google Scholar 

  • Kar TK (2004) Management of a fishery based on continuous fishing effort. Nonlinear Anal: Real World Appl 5(4):629–644

    Article  Google Scholar 

  • Kreisselmeier G, Engel R (2003) Nonlinear observers for autonomous Lipschitz continuous systems. IEEE Trans Automat Control 48:451–464

    Article  Google Scholar 

  • Krener AJ, Respondek W (1985) Nonlinear observers with linearizable error dynamics. SIAM J Control Optim 23:197–216

    Article  Google Scholar 

  • Krener AJ, Xiao M (2002) Nonlinear observer design in the Siegel domain. SIAM J Control Optim 41:932–953

    Article  Google Scholar 

  • López I, Gámez M, Garay J, Varga Z (2007) Monitoring in a lotka-volterra model. Biosystems 87(1):68–74

    Article  PubMed  Google Scholar 

  • Luenberger DG (1971) An introduction to observers. IEEE Trans Automat Control 16:596–602

    Article  Google Scholar 

  • Marutani T (2008) On the optimal path in the dynamic pool model for a fishery. Rev Fish Biol Fish 18(2):133–141

    Article  Google Scholar 

  • Mchich R, Charouki N, Auger P, Raissi N, Ettahiri O (2006) Optimal spatial distribution of the fishing effort in a multi fishing zone model. Ecol Model 197(3-4):274–280

    Article  Google Scholar 

  • Michalska H, Mayne D (1995) Moving horizon observers and observer-based control. IEEE Trans Autom Control 40(6):995–1006

    Article  Google Scholar 

  • Moraal P, Grizzle J (1995) Observer design for nonlinear systems with discrete-time measurements. IEEE Trans Autom Control 40(3):395–404

    Article  Google Scholar 

  • Ngom D, Iggidr A, Guiro A, Ouahbi A (2008) An observer for a nonlinear age-structured model of a harvested fish population. Math Biosci Eng 5(2):337–354

    PubMed  Google Scholar 

  • Ouahbi A, Iggidr A, El Bagdouri M (2003) Stabilization of an exploited fish population. Syst Anal Model Simulation 43:513–524

    Article  Google Scholar 

  • Rapaport A, Maloum A (2004) Design of exponential observers for nonlinear systems by embedding. Int J Robust Nonlinear Control 14(3):273–288

    Article  Google Scholar 

  • Sontag ED (1998) Mathematical control theory. Deterministic finite-dimensional systems, volume 6 of Texts in applied mathematics. Springer-Verlag, New York

    Google Scholar 

  • Tornambe A (1989) Use of asymptotic observers having-high-gains in the state and parameter estimation. Proceedings of the 28th IEEE conference on decision and control, vol 2, pp 1791–1794, December 1989

  • Torres LA, Ibarra-Junquera V, Escalante-Minakata P, Rosu HC (2007) High-gain nonlinear observer for simple genetic regulation process. Phys A Stat Mech Appl 380:235–240

    Article  CAS  Google Scholar 

  • Touzeau S (1997) Modèles de contrôle en gestion des pêches. Thesis, University of Nice-Sophia Antipolis, France

  • Touzeau S, Gouzé J-L (1998) On the stock–recruitment relationships in fish population models. Environ Model Assess 3:87–93

    Article  Google Scholar 

  • White C, Kendall BE (2007) A reassessment of equivalence in yield from marine reserves and traditional fisheries managament. Oikos 116(12):2039–2043

    Article  Google Scholar 

  • Zeitz M (1987) The extended Luenberger observer for nonlinear systems. Syst Control Lett 9:149–156

    Article  Google Scholar 

  • Zimmer G (1994) State observation by on-line minimization. Int J Control 60:595–606

    Article  Google Scholar 

Download references

Acknowledgments

We thank the anonymous referees for their valuable comments and suggestions that have helped us to improve the presentation of this article.

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Correspondence to A. Iggidr.

Appendices

Appendix A: Positive invariance of D w

Let N = x 1 + x 2.

$$\dot N = -q E x_1+r_1 \left(1-\frac{x_1} {K_1}\right)x_1 +(N-x_1) \left(1-\frac{N-x_1} {K_2}\right) r_2$$

Let w be a positive real number, for N = w, we have

$$\dot N = -q E x_1+r_1 \left(1-\frac{x_1} {K_1}\right)x_1 +(w-x_1) \left(1-\frac{w-x_1} {K_2}\right) r_2 = g(x_1)$$

The function g is defined for 0 ≤ x 1 ≤ w.

\(g(0)= w \left(1-\frac{w} {K_2}\right) r_2\)

\(g(w)= -q u w+w \left(1-\frac{w} {K_1}\right) r_1\)

\(g'(x_1) = r_1-r_2 -q u+\frac{2 w r_2} {K_2}-2\left(\frac{ r_1} {K_1}+\frac{ r_2} {K_2}\right) x_1\)

\(g'(x_1) = 0 \Leftrightarrow x_1=\bar x_1=\frac{K_1 \left(K_2 r_1-K_2 r_2-q u K_2+2 w r_2\right)} {2 \left(K_2 r_1+K_1 r_2\right)}\)

The maximum value of the function g is then given by the expression

$$\frac{K_1 K_2 \left(q u-r_1+r_2\right){}^2+\left(4 K_2 r_1 r_2+K_1 \left(-4 q u r_2+4 r_1 r_2\right)\right) w-4 \left(r_1 r_2\right) w^2} {4 \left(K_2 r_1+K_1 r_2\right)}$$

It is therefore clear that this maximum is non positive if w ≥ w 0 with

$$ w_0= \frac{ r_1 r_2 (K_1+K_2)-q u K_1 r_2+\sqrt{r_2 \left(K_2 r_1+K_1 r_2\right) \left(K_1 \left(-q u+r_1\right){}^2+K_2 r_1 r_2\right)}} {2 r_1 r_2} $$
(20)

This shows that for any real number w ≥ w 0, the compact set

$$ D_w= \{(x_1,x_2) \in {\mathbb{R}}^{2}_{+}\;:\; x_1 +x_2 \leq w \} $$

is positively invariant for system (17).

Appendix B: Construction of the Lipschitz extension of \(\varphi\)

The function \(\varphi\) is Lipschitz on the compact set D = [a 0,b 0] × [a 1,b 1] × [a 2,b 2]. Our aim is to extend it to a function \(\tilde\varphi\) which is Lipschitz with the same Lipschitz coefficient in the whole \({\mathbb{R}}^3.\)

Let a(a 0, a 1, a 2), (respectively b(b 0, b 1, b 2)), the lower corner, (respectively the upper corner) of the domain D and x(x 0, x 1, x 2) an unspecified point of \({\mathbb{R}}^3.\)

The problem of the extension is set for point \(x\notin\) D; in this situation we have 26 possibilities according to the situation of x. The different situations correspond to x i  ≤ a i , a i  ≤ x i  ≤ b i , or x i  ≥ b i .

The principle of this prolongation is to compose the function \(\varphi\) with the function π (the projection function of the point x on the domain D).

The extension of function \(\varphi\) is described by the following algorithm:

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Guiro, A., Iggidr, A., Ngom, D. et al. On the stock estimation for some fishery systems. Rev Fish Biol Fisheries 19, 313–327 (2009). https://doi.org/10.1007/s11160-009-9104-7

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