UNIVERSITY THE OF SYDNEY Differential Equations & Biomathematics Semester 1 Assignment 1 2008
This assignment is due at or before the beginning of the Thursday lecture on April 10th. It can be handed to me in room 526 or at the beginning of the lecture.
Harvesting Model with Capacity Constraints Suppose we want to model the harvesting of a resource population, where the capacity to exploit the resource is limited, not only by the eort that can be put into harvesting, but by capacity constraints which limit the ability to process the harvest. For example the infrastructure available to store and process it. We also assume that the eort available to expend on harvesting increases with the capacity to process the harvest. Suppose we want investigate the the behaviour of this model as the capacity constraints change. We make the following plausible assumption about the harvest rate. (1) The rate is an increasing function of population. To model the conditions on limited available eort and processing capacity we make the following further assumptions. (2) When the population is small the harvest rate is proportional to population size. (3) There is a limit to the rate at which the resource can harvested imposed by the limited capacity to deal with the harvest. (4) The eort expended on harvesting is proportional to the ability to process a large catch. Let P be population as a function of time t. Let h(P ) be the harvest rate. Assume that the underlying population dynamics are logistic with intrinsic relative birth rate r, and environmental carry capacity K. So the harvested model is dP = rP (1 P/K) h(P ). dt The conditions (2), (1), (3) and (4) on the harvesting rate above are satised if h = h(P ) is dened and dierentiable on [0, ) and satises the corresponding conditions (1), (2), (3) and (4) below. (1) The derivative h (P ) > 0, on [0, ). (2) The relative harvest rate h(P )/P tends to positive limit as t 0. Equivalently h(0) = 0 and h (0) > 0. (3) As P , h(P ) tends to a limit C 0. Call C the processing capacity. (4) h (0) is proportional to C. Note C = 0 corresponds to having no capacity to process the harvest. Increasing C corresponds to easing the capacity constraints. Answer the following questions about an explicit model with control parameter C.
1. Show that
CP , P +B B > 0 a positive parameter satises the conditions imposed above on h. h(P ) =
2. Write down the dimensions of the model parameters r, K, C, B of the corresponding harvesting model CP dP = rP (1 P/K) , dt P +B in terms of [P ] and [t]. 3. Show that this model with r, K and B xed and control parameter...