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August  2018, 23(6): 2487-2498. doi: 10.3934/dcdsb.2018066

A new flexible discrete-time model for stable populations

Eduardo Liz

Departamento de Matemática Aplicada Ⅱ, Universidade de Vigo, 36310 Vigo, Spain

Received  May 2017 Published  August 2018 Early access  February 2018

Fund Project: This research has been supported by the Spanish Government and FEDER, under grant MTM2013-43404-P.

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.

Keywords: Discrete population model, global stability, density dependence, Allee effects, bifurcations of fixed points, extinction.
Mathematics Subject Classification: 39A10, 92D25.
Citation: Eduardo Liz. A new flexible discrete-time model for stable populations. Discrete and Continuous Dynamical Systems - B, 2018, 23 (6) : 2487-2498. doi: 10.3934/dcdsb.2018066
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.

Figure 1.  Different graphs of the map $f$ defined in (1.1). (a): $f$ is unimodal for $\gamma<1$; (b): $f$ is increasing for $\gamma = 1$, with a unique positive fixed point if $\beta>1$; (c) and (d): $f$ is increasing for $1<\gamma<2$, and can have 0, 1, or $2$ positive fixed points; (e): $f$ is increasing and convex for $\gamma = 2$, with linear growth at infinity; it has a unique positive fixed point if $\beta>\delta$ and no positive fixed points if $\beta\leq\delta$; (f): $f$ is increasing and convex, with superlinear growth at infinity, if $\gamma>2$. In all cases, the red dashed line represents the graph of $y = x$
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Figure 2.  Graph of the map $\beta = F_{\delta}(\gamma)$ showing the survival/extinction switches for (1.1), which only occur if $\beta<1+\delta$
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Figure 3.  Relative position of the graphs of $f_1(x) = \beta x^{\gamma-1}$ (red color) and $f_2(x) = 1+\delta x$ (blue color) when equation $f_1(x) = f_2(x)$ has two positive solutions
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Figure 4.  Bifurcation diagrams for equation (1.1), using $\gamma$ as the bifurcation parameter. Red dashed lines correspond to unstable equilibria, which, in case of bistability, establish the boundary between the basins of attraction of the extinction equilibrium 0 and the nontrivial attractor $p$. (a): $\beta = 3, \delta = 1$; (b): $\beta = 2, \delta = 1$; (c): $\beta = 2, \delta = 1.5$; (b): $\beta = 2, \delta = 2.5$. Each case is an example of the corresponding case in Theorem 5.1
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Figure 5.  Bifurcation diagrams for equation (1.1), using $\gamma$ as the bifurcation parameter. Red dashed lines correspond to unstable equilibria. (a): $\beta = 0.9, \delta = 1.5$; (b): $\beta = 0.9, \delta = 0.5$
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