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The Lévy Flight Foraging Hypothesis in Bounded Regions: Subordinate Brownian Motions and High-risk/High-gain Strategies
 
Serena Dipierro The University of Western Australia, Australia
Giovanni Giacomin The University of Western Australia, Australia
Enrico Valdinoci The University of Western Australia, Australia
A publication of European Mathematical Society
Softcover ISBN:  978-3-98547-068-6
Product Code:  EMSMEM/10
List Price: $75.00
AMS Member Price: $60.00
Not yet published - Preorder Now!
Expected availability date: May 06, 2024
Please note AMS points can not be used for this product
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The Lévy Flight Foraging Hypothesis in Bounded Regions: Subordinate Brownian Motions and High-risk/High-gain Strategies
Serena Dipierro The University of Western Australia, Australia
Giovanni Giacomin The University of Western Australia, Australia
Enrico Valdinoci The University of Western Australia, Australia
A publication of European Mathematical Society
Softcover ISBN:  978-3-98547-068-6
Product Code:  EMSMEM/10
List Price: $75.00
AMS Member Price: $60.00
Not yet published - Preorder Now!
Expected availability date: May 06, 2024
Please note AMS points can not be used for this product
  • Book Details
     
     
    Memoirs of the European Mathematical Society
    Volume: 102024; 99 pp
    MSC: Primary 92; Secondary 35; 60

    The authors investigate the problem of the Lévy flight foraging hypothesis in an ecological niche described by a bounded region of space, with either absorbing or reflecting boundary conditions. To this end, they consider a forager diffusing according to a fractional heat equation in a bounded domain, and they define several efficiency functionals whose optimality is discussed in relation to the fractional exponent \(s \in (0, 1)\) of the diffusive equation. Such an equation is taken to be the spectral fractional heat equation (with Dirichlet or Neumann boundary conditions).

    The authors analyze the biological scenarios in which a target is close to the forager or far from it. In particular, for all the efficiency functionals considered here, they show that if the target is close enough to the forager, then the most rewarding search strategy will be in a small neighborhood of \(s = 0\). Interestingly, we show that \(s = 0\) is a global pessimizer for some of the efficiency functionals. From this, together with the aforementioned optimality results, the authors deduce that the most rewarding strategy can be unsafe or unreliable in practice, given its proximity with the pessimizing exponent; thus, the forager may opt for a less performant, but safer, hunting method.

    However, the biological literature has already collected several pieces of evidence of foragers diffusing with very low Lévy exponents, often in relation with a high energetic content of the prey. It is thereby suggestive to relate these patterns, which are induced by distributions with a very fat tail, with a high-risk/high-gain strategy, in which the forager adopts a potentially very profitable, but also potentially completely unrewarding, strategy due to the high value of the possible outcome.

    A publication of the European Mathematical Society (EMS). Distributed within the Americas by the American Mathematical Society.

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    Review Copy – for publishers of book reviews
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Volume: 102024; 99 pp
MSC: Primary 92; Secondary 35; 60

The authors investigate the problem of the Lévy flight foraging hypothesis in an ecological niche described by a bounded region of space, with either absorbing or reflecting boundary conditions. To this end, they consider a forager diffusing according to a fractional heat equation in a bounded domain, and they define several efficiency functionals whose optimality is discussed in relation to the fractional exponent \(s \in (0, 1)\) of the diffusive equation. Such an equation is taken to be the spectral fractional heat equation (with Dirichlet or Neumann boundary conditions).

The authors analyze the biological scenarios in which a target is close to the forager or far from it. In particular, for all the efficiency functionals considered here, they show that if the target is close enough to the forager, then the most rewarding search strategy will be in a small neighborhood of \(s = 0\). Interestingly, we show that \(s = 0\) is a global pessimizer for some of the efficiency functionals. From this, together with the aforementioned optimality results, the authors deduce that the most rewarding strategy can be unsafe or unreliable in practice, given its proximity with the pessimizing exponent; thus, the forager may opt for a less performant, but safer, hunting method.

However, the biological literature has already collected several pieces of evidence of foragers diffusing with very low Lévy exponents, often in relation with a high energetic content of the prey. It is thereby suggestive to relate these patterns, which are induced by distributions with a very fat tail, with a high-risk/high-gain strategy, in which the forager adopts a potentially very profitable, but also potentially completely unrewarding, strategy due to the high value of the possible outcome.

A publication of the European Mathematical Society (EMS). Distributed within the Americas by the American Mathematical Society.

Review Copy – for publishers of book reviews
Accessibility – to request an alternate format of an AMS title
Please select which format for which you are requesting permissions.