Periodic environments and periodic harvesting. (English) Zbl 1067.92056

Brauer, Fred; Sánchez, David A.

Nat. Resour. Model. 16, No. 3, 233-244 (2003); errata ibid. 16, No. 4, 505 (2003).

In this nicely written paper, the authors discuss the effect of harvesting in several well-known mathematical models of the growth of a population, namely, logistic, Gompertz, Nisbet-Gurney and spruce budworm. A general result on the existence of periodic solutions is formulated for an autonomous system \(y^{\prime}=f(t,y)\), and then specified for the differential equation (1) \(y^{\prime}=g(y)-H(t)\), which models population growth in the presence of harvesting \(H(t).\) In the remaining part of the paper, particular cases of (1) are studied.
It has been shown that under periodic harvesting there is either extinction of the population or an asymptotically stable periodic solution. Furthermore, the examples considered in the paper suggest that a seasonal harvesting may produce a larger average yield compared to a constant harvesting, and there could be situations in which moderate seasonal harvesting may be useful for suppressing the population to a lower density, the latter useful, for instance, to fight harmful bugs or insects.

Reviewer: Svitlana P. Rogovchenko (Famagusta)

MSC:

92D40	Ecology
34C25	Periodic solutions to ordinary differential equations
34D20	Stability of solutions to ordinary differential equations
91B76	Environmental economics (natural resource models, harvesting, pollution, etc.)

Keywords:

population dynamics; periodic environment; harvesting; periodic solutions; asymptotic stability

References:

[1]	Brauer, Periodic Solutions of Some Ecological Models, J. Theoret. Biol. 69 pp 143– (1977)
[2]	Brauer, Constant Rate Population Harvesting: Equilibrium and Stability, Theoret. Pop. Biol. 8 pp 12– (1975) · Zbl 0313.92012
[3]	Coleman, Nonautonomous Logistic Equations as Models of the Adjustment of Populations to Environmental Change, Math. Biosci. 45 pp 159– (1979) · Zbl 0425.92013
[4]	Coleman, On the Optimal Choice of r for a Population in a Periodic Environment, Math. Biosci. 46 pp 71– (1979) · Zbl 0429.92022
[5]	Friedrichs, Advanced Ordinary Differential Equations (1966)
[6]	Lazer, Periodic Equilibria under Periodic Harvesting, Math. Mag. 57 pp 156– (1984) · Zbl 0539.92026
[7]	Ludwig, Qualitative Analysis of Insect Outbreak Systems: The Spruce Budworm and the Forest, J. Animal Ecol. 47 pp 315– (1978)
[8]	Nisbet, Modeling Fluctuating Populations (1982)
[9]	Pliss, Nonlocal Problems of the Theory of Oscillations (1966)
[10]	Sànchez, Nonlinear Phenomena in Mathematical Sciences pp 883– (1982)
[11]	Sánchez, Ordinary Differential Equations: A Brief Eclectic Tour (2002) · Zbl 1004.34001