## Dynamical complexities in the Leslie-Gower predator-prey model as consequences of the Allee effect on prey.(English)Zbl 1202.34079

Appl. Math. Modelling 35, No. 1, 366-381 (2011); erratum ibid. 36, No. 2, 860-862 (2012).
Summary: This work deals with the analysis of a predator-prey model derived from the Leslie-Gower type model, where the most common mathematical form to express the Allee effect in the prey growth function is considered.
It is well-known that the Leslie-Gower model has a unique globally asymptotically stable equilibrium point. However, it is shown here the Allee effect significantly modifies the original system dynamics, as the studied model involves many non-topological equivalent behaviors.
None, one or two equilibrium points can exist at the interior of the first quadrant of the modified Leslie-Gower model with strong Allee effect on prey. However, a collapse may be seen when two positive equilibrium points exist.
Moreover, we proved the existence of parameter subsets for which the system can have: a cusp point (Bogdanov-Takens bifurcation), homoclinic curves (homoclinic bifurcation), Hopf bifurcation and the existence of two limit cycles, the innermost stable and the outermost unstable, in inverse stability as they usually appear in the Gause-type predator-prey models.
In contrast, the system modelling an special of weak Allee effect, may include none or just one positive equilibrium point and no homoclinic curve; the latter implies a significant difference between the mathematical properties of these forms of the phenomenon, although both systems show some rich and interesting dynamics.

### MSC:

 34C23 Bifurcation theory for ordinary differential equations 92D25 Population dynamics (general)

Mathematica
Full Text:

### References:

 [1] Li, Y.; Xiao, D., Bifurcations of a predator-prey system of Holling and Leslie types, Chaos Solitons Fract., 34, 8606-8620 (2007) [2] Turchin, P., Complex Population Dynamics. A Theoretical/Empirical Synthesis. Complex Population Dynamics. A Theoretical/Empirical Synthesis, Monographs in Population Biology, vol. 35 (2003), Princeton University Press · Zbl 1062.92077 [3] Berec, L.; Angulo, E.; Courchamp, F., Multiple Allee effects and population management, Trends Ecol. Evol., 22, 185-191 (2007) [4] Courchamp, F.; Clutton-Brock, T.; Grenfell, B., Inverse density dependence and the Allee effect, Trends Ecol. Evol., 14, 10, 405-410 (1999) [5] Seo, G.; Kot, M., A comparison of two predator-prey models with Holling’s type I functional response, Math. Biosci., 212, 161-179 (2008) · Zbl 1138.92033 [6] Aziz-Alaoui, M. A.; Daher Okiye, M., Boundedness and global stability for a predator-prey model with modified Leslie-Gower and Holling-type II schemes, Appl. Math. Lett., 16, 1069-1075 (2003) · Zbl 1063.34044 [7] Cheng, K. S., Uniqueness of a limit cycle for a predator-prey system, SIAM J. Math. Anal., 12, 541-548 (1981) · Zbl 0471.92021 [8] Chicone, C., Ordinary Differential Equations with Applications. Ordinary Differential Equations with Applications, Texts in Applied Mathematics, vol. 34 (2006), Springer · Zbl 1120.34001 [9] González-Olivares, E.; González-Yañez, B.; S áez, E.; Szántó, I., On the number of limit cycles in a predator-prey model with non-monotonic functional response, Discrete Contin. Dynam. Syst. Ser. B, 6, 525-534 (2006) · Zbl 1092.92045 [10] Korobeinikov, A., A Lyapunov function for Leslie-Gower predator-prey models, Appl. Math. Lett., 14, 697-699 (2001) · Zbl 0999.92036 [11] Hanski, I.; Hentonnen, H.; Korpimaki, E.; Oksanen, L.; Turchin, P., Small-rodent dynamics and predation, Ecology, 82, 1505-1520 (2001) [12] Collings, J. B., The effect of the functional response on the bifurcation behavior of a mite predator-prey interaction model, J. Math. Biol., 36, 149-168 (1997) · Zbl 0890.92021 [13] Stephens, P. A.; Sutherland, W. J., Consequences of the Allee effect for behaviour, ecology and conservation, Trends Ecol. Evol., 14, 10, 401-405 (1999) [14] Liermann, M.; Hilborn, R., Depensation: evidence, models and implications, Fish Fisheries, 2, 33-58 (2001) [15] Clark, C. W., Mathematical Bioeconomics: The Optimal Management of Renewable Resources (1990), John Wiley and Sons · Zbl 0712.90018 [16] Clark, C. W., The Worldwide Crisis in Fisheries: Economic Model and Human Behavior (2007), Cambridge University Press [17] Dennis, B., Allee effects: population growth, critical density, and the chance of extinction, Nat. Resour. Model., 3, 4, 481-538 (1989) · Zbl 0850.92062 [18] Stephens, P. A.; Sutherland, W. J.; Freckleton, R. P., What is the Allee effect?, Oikos, 87, 185-190 (1999) [19] Courchamp, F.; Berec, L.; Gascoigne, J., Allee Effects in Ecology and Conservation (2008), Oxford University Press [20] Conway, E. D.; Smoller, J. A., Global analysis of a system of predator-prey equations, SIAM J. Appl. Math., 46, 4, 630-642 (1986) · Zbl 0608.92016 [21] Bazykin, A. D.; Berezovskaya, F. S.; Isaev, A. S.; Khlebopros, R. G., Dynamics of forest insect density: bifurcation approach, J. Theoret. Biol., 186, 267-278 (1997) [22] Freedman, H. I., Deterministic Mathematical Models in Population Ecology (1980), Marcel Dekker · Zbl 0448.92023 [23] Bazykin, A. D., Nonlinear Dynamics of Interacting Populations (1998), World Scientific · Zbl 0605.92015 [24] van Voorn, G. A.K.; Hemerik, L.; Boer, M. P.; Kooi, B. W., Heteroclinic orbits indicate overexploitation in predator-prey systems with a strong Allee, Math. Biosci., 209, 451-469 (2007) · Zbl 1126.92062 [25] Boukal, D. S.; Berec, L., Single-species models and the Allee effect: extinction boundaries, sex ratios and mate encounters, J. Theoret. Biol., 218, 375-394 (2002) [26] Boukal, D. S.; Sabelis, M. W.; Berec, L., How predator functional responses and Allee effects in prey affect the paradox of enrichment and population collapses, Theoret. Popul. Biol., 72, 136-147 (2007) · Zbl 1123.92034 [27] Wang, G.; Liang, X.-G.; Wang, F.-Z., The competitive dynamics of populations subject to an Allee effect, Ecol. Model., 124, 183-192 (1999) [28] E. González-Olivares, B. González-Yañez, J. Mena-Lorca, R. Ramos-Jiliberto, Modelling the Allee effect: Are the different mathematical forms proposed equivalents? in: R. Mondaini (Ed.), Proceedings of the International Symposium on Mathematical and Computational Biology BIOMAT 2006, E-papers Serviços Editoriais Ltda., Rio de Janeiro, 2007, pp. 53-71.; E. González-Olivares, B. González-Yañez, J. Mena-Lorca, R. Ramos-Jiliberto, Modelling the Allee effect: Are the different mathematical forms proposed equivalents? in: R. Mondaini (Ed.), Proceedings of the International Symposium on Mathematical and Computational Biology BIOMAT 2006, E-papers Serviços Editoriais Ltda., Rio de Janeiro, 2007, pp. 53-71. [29] J.D. Flores, J. Mena-Lorca, B. González-Yañez, E. González-Olivares, Consequences of depensation in a Smith’s bioeconomic model for open-access fishery, in: R. Mondaini (Ed.), Proceedings of the International Symposium on Mathematical and Computational Biology BIOMAT 2006, E-papers Serviços Editoriais Ltda., 2007, pp. 219-232.; J.D. Flores, J. Mena-Lorca, B. González-Yañez, E. González-Olivares, Consequences of depensation in a Smith’s bioeconomic model for open-access fishery, in: R. Mondaini (Ed.), Proceedings of the International Symposium on Mathematical and Computational Biology BIOMAT 2006, E-papers Serviços Editoriais Ltda., 2007, pp. 219-232. [30] Aguirre, P.; González-Olivares, E.; Sáez, E., Two limit cycles in a Leslie-Gower predator-prey model with additive Allee effect, Nonlinear Anal.: Real World Appl., 10, 3, 1401-1416 (2009) · Zbl 1160.92038 [31] Aguirre, P.; González-Olivares, E.; Sáez, E., Three limit cycles in a Leslie-Gower predator-prey model with additive Allee effect, SIAM J. Appl. Math., 69, 5, 1244-1262 (2009) · Zbl 1184.92046 [32] Taylor, R. J., Predation (1984), Chapman and Hall [33] L.M. Gallego-Berrı´o, Consecuencias del efecto Allee en el modelo de depredación de May-Holling-Tanner, Master Thesis on Biomatemathics, Universidad del Quindı´o, Colombia, 2004.; L.M. Gallego-Berrı´o, Consecuencias del efecto Allee en el modelo de depredación de May-Holling-Tanner, Master Thesis on Biomatemathics, Universidad del Quindı´o, Colombia, 2004. [34] González-Yañez, B.; González-Olivares, E.; Mena-Lorca, J., Multistability on a Leslie-Gower type predator-prey model with nonmonotonic functional response, (Mondaini, R.; Dilao, R., BIOMAT 2006 - International Symposium on Mathematical and Computational Biology (2007), World Scientific Co. Pvt. Ltd.), 359-384 [35] Sáez, E.; González-Olivares, E., Dynamics on a predator-prey model, SIAM J. Appl. Math., 59, 5, 1867-1878 (1999) · Zbl 0934.92027 [36] J. Mena-Lorca, E. González-Olivares, B. González-Ya ñez, The Leslie-Gower predator-prey model with Allee effect on prey: a simple model with a rich and interesting dynamics, in: R. Mondaini (Ed.), Proceedings of the International Symposium on Mathematical and Computational Biology BIOMAT 2006, E-papers Serviços Editoriais Ltda., R ı´o de Janeiro, 2007, pp. 105-132.; J. Mena-Lorca, E. González-Olivares, B. González-Ya ñez, The Leslie-Gower predator-prey model with Allee effect on prey: a simple model with a rich and interesting dynamics, in: R. Mondaini (Ed.), Proceedings of the International Symposium on Mathematical and Computational Biology BIOMAT 2006, E-papers Serviços Editoriais Ltda., R ı´o de Janeiro, 2007, pp. 105-132. [37] González-Olivares, E.; Mena-Lorca, J.; Meneses-Alcay, H.; González-Yañez, B.; Flores, J. D., Allee effect, emigration and immigration in a class of predator-prey models, Biophys. Rev. Lett. (BRL), 3, 1/2, 195-215 (2008) [38] H. Meneses-Alcay, E. González-Olivares, Consequences of the Allee effect on Rosenzweig-McArthur predator-prey model, in: R. Mondaini (Ed.), Proceedings of the Third Brazilian Symposium on Mathematical and Computational Biology BIOMAT 2003, E-papers Serviços Editoriais Ltda., Rı´o de Janeiro, vol. 2, 2004, pp. 264-277.; H. Meneses-Alcay, E. González-Olivares, Consequences of the Allee effect on Rosenzweig-McArthur predator-prey model, in: R. Mondaini (Ed.), Proceedings of the Third Brazilian Symposium on Mathematical and Computational Biology BIOMAT 2003, E-papers Serviços Editoriais Ltda., Rı´o de Janeiro, vol. 2, 2004, pp. 264-277. [39] Dumortier, F.; Llibre, J.; Artés, J. C., Qualitative Theory of Planar Differential Systems (2006), Springer · Zbl 1110.34002 [40] Perko, L., Differential Equations and Dynamical Systems (1991), Springer-Verlag · Zbl 0717.34001 [41] Arrowsmith, D. K.; Place, C. M., Dynamical Systems, Differential Equations, Maps and Chaotic Behaviour (1992), Chapman and Hall · Zbl 0754.34001 [42] Wolfram Research, Mathematica: A System for Doing Mathematics by Computer, Champaign, IL, 1988.; Wolfram Research, Mathematica: A System for Doing Mathematics by Computer, Champaign, IL, 1988. · Zbl 0671.65002
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