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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Appendices
Appendix A: Positive invariance of D w
Let N = x 1 + x 2.
Let w be a positive real number, for N = w, we have
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
It is therefore clear that this maximum is non positive if w ≥ w 0 with
This shows that for any real number w ≥ w 0, the compact set
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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DOI: https://doi.org/10.1007/s11160-009-9104-7