The effect of seasonal harvesting on a single-species discrete population model with stage structure and birth pulses. (English) Zbl 1061.92059

Summary: We propose an exploited single-species discrete model with stage structure for the dynamics in a fish population for which births occur in a single pulse once per time period. Using the stroboscopic map, we obtain an exact cycle of the system, and obtain the threshold conditions for its stability. Bifurcation diagrams are constructed with the birth rate as the bifurcation parameter, and are observed to display complex dynamic behaviors, including chaotic bands with period windows, pitch-fork and tangent bifurcation. This suggests that birth pulse provides a natural period or cyclicity that makes the dynamical behavior more complex. Moreover, we show that the timing of harvesting has a strong impact on the persistence of the fish population, on the volume of mature fish stock and on the maximum annual-sustainable yield. An interesting result is obtained that, after the birth pulse, the earlier culling the mature fish, the larger harvest can tolerate.


92D40 Ecology
37N25 Dynamical systems in biology
39A10 Additive difference equations
92D25 Population dynamics (general)
91B76 Environmental economics (natural resource models, harvesting, pollution, etc.)
39A12 Discrete version of topics in analysis
Full Text: DOI


[1] Aiello, W.G.; Freedman, H.I., A time-delay model of single-species growth with stage structure, Math. biosci., 101, 139-153, (1990) · Zbl 0719.92017
[2] Aiello, W.G.; Freedman, H.I.; Wu, J., Analysis of a model representing stage-structured population growth with state-dependent time delay, SIAM J. appl. math., 3, 855-869, (1992) · Zbl 0760.92018
[3] Hastings, A., Delay in recruitment at different trophic levels: effects on stability, J. math. biol., 21, 35-44, (1984) · Zbl 0547.92014
[4] Cai, Y.; Fan, J.; Gard, T.C., The effects of a stage-structured population growth model, Nonlin. anal. th. mech. appl., 16, 95-105, (1992) · Zbl 0777.92014
[5] Freedman, H.I.; Wu, J., Persistence and global asymptotic stability of single species dispersal models with stage structure, Quart. appl. math., 2, 351-371, (1991) · Zbl 0732.92021
[6] Gurney, W.S.C.; Nisbet, R.M.; Lawton, J.H., The systematic formulation of tractable single species population models incorporating age structure, J. animal ecol., 52, 479-485, (1983)
[7] Bainov, D.D.; Simeonov, P.S., System with impulsive effect: stability, theory and applications, (1989), John Wiley & Sons New York · Zbl 0676.34035
[8] Hastings, A.; Higgins, K., Persistence of transients in spatially structured ecological models, Science, 263, 1133-1137, (1994)
[9] Freedman, H.I., Deterministic mathematical models in population ecology, (1980), Marcel Dekker New York · Zbl 0448.92023
[10] Hinggins, K.; Hastings, A.; Botsford, L., Density dependence and age structure: nonlinear dynamics and population behavior, Am. nat., 149, 247-269, (1997)
[11] Gao, S.; Chen, L., Dynamic complexities in a single-species discrete population model with stage structure and birth pulses, Chaos, solitons & fractals, 23, 519-527, (2005) · Zbl 1066.92041
[12] Tang, S.; Chen, L., Density-dependent birth rate, birth pulses and their population dynamic consequences, J. math. biol., 64, 169-184, (2002)
[13] Robert, M.G.; Kao, P.R., The dynamics of an infectious disease in a population with birth pulses, Math. biosci., 149, 23-36, (1998) · Zbl 0928.92027
[14] Tang, S.; Chen, L., The effect of seasonal harvesting on stage-structured population models, J. math. biol., 48, 357-374, (2003) · Zbl 1058.92051
[15] Jury, E.I., Inners and stability of dynamic system, (1974), Wiley New York · Zbl 0307.93025
[16] May, R.M., Biological population with nonoverlapping generations: stable points, stable cycles, and chaos, Science, 186, 645-647, (1974)
[17] May, R.M.; Oster, G.F., Bifurcations and dynamics complexity in simple ecological models, Am. nat., 110, 573-599, (1976)
[18] Eckmann, J.P., Routes to chaos with special emphasis on period doubling, () · Zbl 0616.58032
[19] Collet, P.; Eckmann, J.P., Iterated maps of the interval as dynamical systems, (1980), Birkhauser Boston · Zbl 0456.58016
[20] Hauser, M.J.B.; Olsen, L.F.; Bronnikova, T.V.; Schaffer, W.M., Routs to chaos in the peroxidase-oxidase reaction: period-doubling and period-adding, J. phys. chem. B, 101, 5075-5083, (1997)
[21] Kaneko, K., On the period-adding phenomena at the frequency locking in a one-dimensional mapping, Prog. theor. phys., 69, 403-414, (1983)
[22] Guckenheimer, J.; Oster, G.; Ipaktchi, A., The dynamics of density dependent population models, J. math. biol., 4, 101-147, (1977) · Zbl 0379.92016
[23] Ellison, L.N., Shooting and compensatory mortality in tetarconid, Ornis scand., 22, 229-240, (1991)
[24] Smith, G.W.; Reynolds, R.E., Hunting and mallard survival, 1979-88, J. wildl. manage., 56, 306-316, (1992)
[25] Smith, G.W.; Reynolds, R.E., Hunting and mallard survival, a reply, J. wildl. manage., 58, 578-581, (1994)
[26] Small, R.J.; Holzwart, J.C.; Rusch, D.H., Predation and hunting mortality of ruffed grouse in central wisconsin, J. wildl. manage., 55, 512-520, (1991)
[27] Anderson, D.R.; Burnham, K.P., Population ecology of the mallard VI, the effect of exploitation on survival, US fish wildl. serv. resour. publ., 128, 1-66, (1976)
[28] Roseberry, J.L., Bobwhite population responses to exploitation: real and simulated, J. wildl. manage., 43, 285-305, (1979)
[29] Clark, C.W., Mathematical bioeconomics: the optimal management of renewal resources, (1976), Wiley New York · Zbl 0364.90002
[30] Nicholson, N.J., An outline of the dynamics of animal populations, Aust. J. zool., 2, 9-65, (1954)
[31] Gurney, W.S.C.; Nisbet, R.M.; Lawton, J.L., The systematic formulation of tractable single-species population models incorporating age-structure, J. anim. ecol., 52, 479-495, (1983)
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