×

Stability results for delayed-recruitment models in population dynamics. (English) Zbl 0533.92017

This paper deals with discrete models of population dynamics (for populations of whales). Dividing the population into a number of classes leads to a recurrence system: \(\bar N(t+1)=F(\bar N(t))\), which can be reduced to a scalar delayed difference equation: \(N(t+1)=sN(t)+(1- s)g(N(t-k)) (0<s<1).\)
The interest of the paper is that the questions of stability and global stability are considered via a Lyapunov function associated to the system: \(V:{\mathbb{R}}^ n\to {\mathbb{R}}^+\), such that: V(F(x))\(\leq V(x)\), \(x\in {\mathbb{R}}^ n\). V is in fact obtained from a function v having the same properties with respect to the scalar recurrence equation \(x_{n+1}=g(x_ n)\). The main result is that if v is convex and \(v(x)\to +\infty\), \(x\to +\infty\), then the non-trivial equilibrium is globally asymptotically stable (we observe that the convexity here is certainly meant in the non-degenerate sense: there is at most one point \(x_ 0\) with \(0\in \partial v(x_ 0))\). This result is then localized \(({\mathbb{R}}^ n\) replaced by a subset G), and illustrated with several ”wellknown baleen whale models”.
Reviewer: O.Arino

MSC:

92D25 Population dynamics (general)
39A11 Stability of difference equations (MSC2000)
34D20 Stability of solutions to ordinary differential equations
39A10 Additive difference equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Allen, K. R.: Analysis of stock-recruitment relations in Antarctic fin whales. Cons. Int. pour l’Explor. Mer-Rapp. Proc.-Verb. 164, 132-137 (1963)
[2] Beddington, J. R.: On the dynamics of Sei whales under exploitation. Rep. Int. Whal. Commn. 28, 169-172 (1978a)
[3] Beddington, J. R.: On the risks associated with different harvesting strategies in the harvesting of baleen whales. Rep. Int. Whal. Commn. 28, 165-167 (1978b)
[4] Beddington, J. R., May, R. M.: Harvesting natural populations in a randomly fluctuating environment. Science 197, 463-465 (1977) · doi:10.1126/science.197.4302.463
[5] Beverton, R. J. H., Holt, S. J.: On the dynamics of exploited fish populations. Fish. Invest. London 19, 1-533 (1957)
[6] Clark, C. W.: A delayed-recruitment model of population dynamics, with an application to baleen whale populations. J. Math. Biol. 3, 381 -391 (1976) · Zbl 0337.92011
[7] Goh, B. S.: Management and analysis of biological populations. Amsterdam: Elsevier 1980
[8] Horwood, J. W., Knights, B. J., Overy, R. W.: Harvesting of whale populations subject to stochastic variability. Rep. Int. Whal. Commn. 29, 219-224 (1979)
[9] International Whaling Commission: Twenty-Sixth Annual Report of the International Commission on Whaling. London (1976)
[10] International Whaling Commission: Twenty-Eighth Annual Report of the International Commission on Whaling. London (1978)
[11] La Salle, J. P.: The stability of dynamical systems. SIAM, Philadelphia (1976)
[12] Levin, S. A., May, R. M.: A note on difference-delay equations. Theoret. Population Biol. 9, 178-187 (1976) · Zbl 0338.92021 · doi:10.1016/0040-5809(76)90043-5
[13] May, R. M.: Biological populations with nonoverlapping generations: Stable points, stable cycles and chaos. Science 186, 645-647 (1974) · doi:10.1126/science.186.4164.645
[14] Ricker, W. E.: Stock and recruitment. J. Fish. Res. Bd. Can. 11, 559-623 (1954)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.