Management and Analysis of Biological Populations by BEAN-SAN GOH (Eds.)

By BEAN-SAN GOH (Eds.)

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An interesting conclusion from this example is that a constant quota harvesting policy for a single-species population cannot be globally stable. We can show that this conclusion is not dependent on the assumption that the natural dynamics is described by a logistic equation. Note that a Liapunov need not be continuously differentiable everywhere in a region Ω . 2 2 2 X 2 2 Example 3 . 3 . 2 . L e t Ν χ and N be the densities of a prey and its predator respectively. Let the prey population be harvested at the constant rate of 1 .

4. SINGLE-SPECIES MODELS A common criticism of the logistic model of a single-species population is that it assumes that the per capita birth rate of the population is a linear function of the density of the population. This criticism is not as serious as some ecologists believe it t o be. The crucial assumption in the logistic model is that a single species population can be described by a single nonlinear differential equation. We shall show that the logistic model of a single species has the same qualitative dynamical behavior as all proposed models of a single species, and much more.

We can show that Ν* is locally asymptotically stable if the real parts of all the eigenvalues of (Ν*α ) are negative. A better approximation of model ( 3 . 1 . 1 ) is /; ϋ m ^ = 2 NflvW - # * ) , i = 1, 2, . . , m. 4) y= ι This approximation is better because it contains the second order terms in the Taylor's expansion of ( 3 . 1 . 1 ) . Moreover, model ( 3 . 1 . 4 ) is valid in a larger neighbourhood than that for ( 3 . 1 . 3 ) . Model ( 3 . 1 . 4 ) is none other than a Lotka—Volterra model. We conclude that a study of a Lotka—Volterra model in the neighbourhood of an equilibrium can provide more information than an eigenvalue analysis.

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