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Algebraic limit cycles for quadratic polynomial differential systems
A new flexible discrete-time model for stable populations
Departamento de Matemática Aplicada Ⅱ, Universidade de Vigo, 36310 Vigo, Spain |
We propose a new discrete dynamical system which provides a flexible model to fit population data. For different values of the three involved parameters, it can represent both globally persistent populations (compensatory or overcompensatory), and populations with Allee effects. In the most relevant cases of parameter values, there is a stable positive equilibrium, which is globally asymptotically stable in the persistent case. We study how population abundance depends on the parameters, and identify extinction windows between two saddle-node bifurcations.
References:
[1] |
L. Avilés,
Cooperation and non-linear dynamics: An ecological perspective on the evolution of sociality, Evol. Ecol. Res., 1 (1999), 459-477.
|
[2] |
R. Beverton and S. Holt,
On the dynamics of exploited fish populations, Fisheries Investigations, Ser 2, 19 (1957), 1-533.
doi: 10.1007/978-94-011-2106-4. |
[3] |
C. W. Clark, Mathematical Bioeconomics, 2nd edition, John Wiley & Sons, Hoboken, NJ, 2010. |
[4] |
F. Courchamp, L. Berec and J. Gascoigne, Allee Effects in Ecology and Conservation, Oxford University Press, New York, 2008.
doi: 10.1093/acprof:oso/9780198570301.001.0001.![]() ![]() |
[5] |
D. Cushing,
The dependence of recruitment on parent stock in different groups of fishes, J. Conseil, 33 (1971), 340-362.
doi: 10.1093/icesjms/33.3.340. |
[6] |
H. T. M. Eskola and K. Parvinen,
The Allee effect in mechanistic models based on inter-individual interaction processes, Bull. Math. Biol., 72 (2010), 184-207.
doi: 10.1007/s11538-009-9443-5. |
[7] |
F. M. Hilker, M. Paliaga and E. Venturino,
Diseased social predators, Bull. Math. Biol., 79 (2017), 2175-2196.
doi: 10.1007/s11538-017-0325-y. |
[8] |
E. Liz, A global picture of the gamma-Ricker map: A flexible discrete-time model with factors of positive and negative density dependence,
Bull. Math. Biol. , (2017), to appear. |
[9] |
T. Iles,
A review of stock-recruitment relationships with reference to flatfish populations, Neth. J. Sea Res., 32 (1994), 399-420.
doi: 10.1016/0077-7579(94)90017-5. |
[10] |
M. Kot, Elements of Mathematical Ecology, Cambridge University Press, New York, 2001.
![]() ![]() |
[11] |
R. M. May,
Simple mathematical models with very complicated dynamics, Nature, 261 (1976), 459-467.
|
[12] |
T. J. Quinn and R. B. Deriso, Quantitative Fish Dynamics, Oxford University Press, New York, 1999.
![]() |
[13] |
W. E. Ricker,
Stock and recruitment, J. Fish. Res. Board Canada, 11 (1954), 559-623.
doi: 10.1139/f54-039. |
[14] |
S. J. Schreiber,
Allee effects, extinctions, and chaotic transients in simple population models, Theor. Popul. Biol., 64 (2003), 201-209.
doi: 10.1016/S0040-5809(03)00072-8. |
[15] |
J. G. Shepherd,
A versatile new stock-recruitment relationship for fisheries, and the construction of sustainable yield resources, J. Conserv. Int. Explor. Mer., 40 (1982), 67-75.
|
[16] |
M. Teixeira Alves and F. M. Hilker,
Hunting cooperation and Allee effects in predators, J. Theoret. Biol., 419 (2017), 13-22.
doi: 10.1016/j.jtbi.2017.02.002. |
[17] |
H. R. Thieme, Mathematics in Population Biology, Princeton Series in Theoretical and Computational Biology. Princeton University Press, Princeton, NJ, 2003. |
[18] |
S. Wiggins,
Introduction to Applied Nonlinear Dynamical Systems and Chaos, 2nd edition, Texts in Applied Mathematics, vol. 2, Springer-Verlag, New York, 2003. |
show all references
References:
[1] |
L. Avilés,
Cooperation and non-linear dynamics: An ecological perspective on the evolution of sociality, Evol. Ecol. Res., 1 (1999), 459-477.
|
[2] |
R. Beverton and S. Holt,
On the dynamics of exploited fish populations, Fisheries Investigations, Ser 2, 19 (1957), 1-533.
doi: 10.1007/978-94-011-2106-4. |
[3] |
C. W. Clark, Mathematical Bioeconomics, 2nd edition, John Wiley & Sons, Hoboken, NJ, 2010. |
[4] |
F. Courchamp, L. Berec and J. Gascoigne, Allee Effects in Ecology and Conservation, Oxford University Press, New York, 2008.
doi: 10.1093/acprof:oso/9780198570301.001.0001.![]() ![]() |
[5] |
D. Cushing,
The dependence of recruitment on parent stock in different groups of fishes, J. Conseil, 33 (1971), 340-362.
doi: 10.1093/icesjms/33.3.340. |
[6] |
H. T. M. Eskola and K. Parvinen,
The Allee effect in mechanistic models based on inter-individual interaction processes, Bull. Math. Biol., 72 (2010), 184-207.
doi: 10.1007/s11538-009-9443-5. |
[7] |
F. M. Hilker, M. Paliaga and E. Venturino,
Diseased social predators, Bull. Math. Biol., 79 (2017), 2175-2196.
doi: 10.1007/s11538-017-0325-y. |
[8] |
E. Liz, A global picture of the gamma-Ricker map: A flexible discrete-time model with factors of positive and negative density dependence,
Bull. Math. Biol. , (2017), to appear. |
[9] |
T. Iles,
A review of stock-recruitment relationships with reference to flatfish populations, Neth. J. Sea Res., 32 (1994), 399-420.
doi: 10.1016/0077-7579(94)90017-5. |
[10] |
M. Kot, Elements of Mathematical Ecology, Cambridge University Press, New York, 2001.
![]() ![]() |
[11] |
R. M. May,
Simple mathematical models with very complicated dynamics, Nature, 261 (1976), 459-467.
|
[12] |
T. J. Quinn and R. B. Deriso, Quantitative Fish Dynamics, Oxford University Press, New York, 1999.
![]() |
[13] |
W. E. Ricker,
Stock and recruitment, J. Fish. Res. Board Canada, 11 (1954), 559-623.
doi: 10.1139/f54-039. |
[14] |
S. J. Schreiber,
Allee effects, extinctions, and chaotic transients in simple population models, Theor. Popul. Biol., 64 (2003), 201-209.
doi: 10.1016/S0040-5809(03)00072-8. |
[15] |
J. G. Shepherd,
A versatile new stock-recruitment relationship for fisheries, and the construction of sustainable yield resources, J. Conserv. Int. Explor. Mer., 40 (1982), 67-75.
|
[16] |
M. Teixeira Alves and F. M. Hilker,
Hunting cooperation and Allee effects in predators, J. Theoret. Biol., 419 (2017), 13-22.
doi: 10.1016/j.jtbi.2017.02.002. |
[17] |
H. R. Thieme, Mathematics in Population Biology, Princeton Series in Theoretical and Computational Biology. Princeton University Press, Princeton, NJ, 2003. |
[18] |
S. Wiggins,
Introduction to Applied Nonlinear Dynamical Systems and Chaos, 2nd edition, Texts in Applied Mathematics, vol. 2, Springer-Verlag, New York, 2003. |





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