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A switching feedback control approach for persistence of managed resources

  • * Corresponding author: Chris Guiver

    * Corresponding author: Chris Guiver 
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  • 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.

    Mathematics Subject Classification: Primary: 39A22, 39A30, 39A60, 92D25, 93D21. Secondary: 15B48, 93D20.


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  • 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
     | Show Table
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