zbMATH — the first resource for mathematics

The effects of planting and harvesting on endangered species in discrete competitive systems. (English) Zbl 0820.92024
Summary: The influence of planting an endangered species in a discretely reproducing ecosystem is studied. By applying constant rate and variable rate planting strategies we obtain that coexistence of species that would otherwise exclude each other with no planting occurs. We also show that with a high planting rate the endagered species not only recovers from the brink of extinction but it excludes the other competing species. A similar reversal of exclusion principles occurs if the dominant species is harvested with a sufficiently high harvesting constant while the endangered species is left undisturbed. In addition, we obtain mutual exclusion of species with harvesting where there is exclusion of only one species without harvesting.

92D40 Ecology
39A10 Additive difference equations
39A11 Stability of difference equations (MSC2000)
37N99 Applications of dynamical systems
Full Text: DOI
[1] Brauer, F.; Sanchez, A.A., Constant rate population harvesting: equilibrium and stability, Theor. popl. biol., 8, 12-30, (1975) · Zbl 0313.92012
[2] Collet, P.; Eckmann, J.-P., Iterated maps of the interval as dynamical systems, (1980), Birkhäuser Boston · Zbl 0456.58016
[3] Comins, H.N.; Hassell, M.P., Predation in multiprey communities, J. theor. biol., 62, 93-114, (1976)
[4] Devaney, R.L., An introduction to chaotic dynamical systems, (1986), Benjamin/Cummings Menlo Park, CA · Zbl 0632.58005
[5] Fonda, A., Uniformly persistent semi-dynamical systems, (), 111-116 · Zbl 0667.34065
[6] Franke, J.E.; Yakubu, A.-A., Global attractors in competitive systems, Nonlinear anal. theor. methods appl., 16, 111-129, (1991) · Zbl 0724.92024
[7] Franke, J.E.; Yakubu, A.-A., Mutual exclusion versus coexistence for discrete competitive systems, J. math. biol., 30, 161-168, (1991) · Zbl 0735.92023
[8] J. E. Franke and A.-A. Yakubu, Pioneer exclusion in a one hump discrete pioneer-climax competitive system, J. Math. Biol., in press. · Zbl 0828.92029
[9] Freedman, H.I.; So, J.W.-H., Persistence in discrete models of a population which may be subjected to harvesting, Nat. res. modeling, 2, 135-145, (1987) · Zbl 0850.92074
[10] Freedman, H.I.; So, J.W.-H., Persistence in semi-dynamical systems, SIAM J. math. anal., 20, 930-938, (1989) · Zbl 0676.92011
[11] Hallam, T.; Levin, S., Mathematical ecology: an introduction, ()
[12] Hao, B.-L., Chaos, (1984), World Scientific Singapore · Zbl 0559.58012
[13] Hassell, M.P.; Comins, H.N., Discrete time models for two-species competition, Theor. pop. biol., 9, 202-221, (1976) · Zbl 0338.92020
[14] Hofbauer, J.; Hutson, V.; Jansen, W., Coexistence for systems governed by difference equations of Lotka-Volterra type, J. math. biol., 25, 553-570, (1987) · Zbl 0638.92019
[15] Jiang, H.; Rogers, T.D., The discrete dynamics of symmetric competition, J. math. biol., 15, 573-596, (1987) · Zbl 0668.92011
[16] V. L. Kocić and G. Ladas, Global behavior of nonlinear difference equations of higher order with applications, in press.
[17] Lasalle, J.P., The stability and control of discrete processes, () · Zbl 0606.93001
[18] Lewis, E.R., Network models in population biology, () · Zbl 0383.92011
[19] Li, T.; Yorke, J., Period—3 implies chaos, Amer. math. monthly, 82, 985-998, (1975) · Zbl 0351.92021
[20] May, R.M., Biological populations with nonoverlapping generations: stable points, stable cycles, and chaos, Science, 186, 645-647, (1974)
[21] May, R.M., Biological populations obeying difference equations: stable points, stable cycles, and chaos, J. theor. biol., 51, 511-524, (1975)
[22] May, R.M., Simple mathematical models with very complicated dynamics, Nature, 261, 459-467, (1976) · Zbl 1369.37088
[23] McCallum, H.I., Effects of immigration on chaotic population dynamics, J. theor. biol., 154, 277-284, (1992)
[24] Neubert, M.G.; Kot, M., The subcritical collapse of predator populations in discrete-time predator-prey models, Math. biosci., 110, 45-66, (1992) · Zbl 0747.92024
[25] J. F. Selgrade, Planting and harvesting for pioneer-climax models, Rocky Mountain J. Math, in press. · Zbl 0801.92024
[26] Stone, L., Period doubling reversals and chaos in simple ecological models, Nature, 365, 617-620, (1993)
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.