A country is to establish a zone in the sea adjacent to its coast in which no fishing is to be allowed. This is to be done to protect a species of fish which, in the absence of fishing, would grow exponentially due to births which are proportional to the density of the fish. Outside the zone, which extends from the shore to a distance L out to sea, all the fish are harvested by deep-sea trawlers. The fish move about at random in the direction from the shore to the zone's boundary and so effectively diffuse in this direction. If L is too small, the effectively immediate destruction of the fish which wander out of the protection zone will cause a loss of fish that may exceed their ability to reproduce; this would lead to eventual extinction. 1. Formulate a model of the dynamics based on the conservation of fish, their breeding and their movement.

A country is to establish a zone in the sea adjacent to its coast in which no fishing is to be allowed. This is to be done to protect a species of fish which, in the absence of fishing, would grow exponentially due to births which are proportional to the density of the fish. Outside the zone, which extends from the shore to a distance L out to sea, all the fish are harvested by deep-sea trawlers. The fish move about at random in the direction from the shore to the zone's boundary and so effectively diffuse in this direction. If L is too small, the effectively immediate destruction of the fish which wander out of the protection zone will cause a loss of fish that may exceed their ability to reproduce; this would lead to eventual extinction. 1. Formulate a model of the dynamics based on the conservation of fish, their breeding and their movement.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

A country is to establish a zone in the sea adjacent to its coast in which no
fishing is to be allowed. This is to be done to protect a species of fish which,
in the absence of fishing, would grow exponentially due to births which are
proportional to the density of the fish. Outside the zone, which extends from
the shore to a distance L out to sea, all the fish are harvested by deep-sea
trawlers. The fish move about at random in the direction from the shore to the
zone's boundary and so effectively diffuse in this direction. If L is too small, the
effectively immediate destruction of the fish which wander out of the protection
zone will cause a loss of fish that may exceed their ability to reproduce; this
would lead to eventual extinction.
1. Formulate a model of the dynamics based on the conservation of fish, their
breeding and their movement.

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Follow-up Question

Find the minimum zone width L that prevents the extinction of the fish
by the trawlers. Note that a boundary condition is obtained from realising
that the fish cannot swin on-shore (i.e. there is no flux of fish past the
shoreline). Another boundary condition is obtained by realising that there
are effectively no fish outside the protection zone.

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