### Mathematical Biosciences and Engineering

2015, Issue 1: 83-97. doi: 10.3934/mbe.2015.12.83

# Delayed population models with Allee effects and exploitation

• Received: 01 April 2014 Accepted: 29 June 2018 Published: 01 December 2014
• MSC : Primary: 34K18, 34K20, 92D25; Secondary: 37E05.

• Allee effects make populations more vulnerable to extinction, especially under severe harvesting or predation. Using a delay-differential equation modeling the evolution of a single-species population subject to constant effort harvesting, we show that the interplay between harvest strength and Allee effects leads not only to collapses due to overexploitation; large delays can interact with Allee effects to produce extinction at population densities that would survive for smaller time delays.In case of bistability, our estimations on the basins of attraction of the two coexisting attractors improve some recent results in this direction. Moreover, we show that the persistent attractor can exhibit bubbling: a stable equilibrium loses its stability as harvesting effort increases, giving rise to sustained oscillations, but higher mortality rates stabilize the equilibrium again.

Citation: Eduardo Liz, Alfonso Ruiz-Herrera. Delayed population models with Allee effects and exploitation[J]. Mathematical Biosciences and Engineering, 2015, 12(1): 83-97. doi: 10.3934/mbe.2015.12.83

### Related Papers:

• Allee effects make populations more vulnerable to extinction, especially under severe harvesting or predation. Using a delay-differential equation modeling the evolution of a single-species population subject to constant effort harvesting, we show that the interplay between harvest strength and Allee effects leads not only to collapses due to overexploitation; large delays can interact with Allee effects to produce extinction at population densities that would survive for smaller time delays.In case of bistability, our estimations on the basins of attraction of the two coexisting attractors improve some recent results in this direction. Moreover, we show that the persistent attractor can exhibit bubbling: a stable equilibrium loses its stability as harvesting effort increases, giving rise to sustained oscillations, but higher mortality rates stabilize the equilibrium again.

 [1] J. Theoret. Biol., 218 (2002), 375-394. [2] Math. Biosci., 248 (2014), 78-87. [3] $2^{nd}$ edition, John Wiley & Sons, Hoboken, New Jersey, 1990. [4] Oxford University Press, New York, 2008. [5] Proc. Natl. Acad. Sci. USA, 99 (2002), 12907-12912. [6] Springer-Verlag, New York, 1995. [7] J. Biol. Dyn., 4 (2010), 397-408. [8] J. Math. Biol., 65 (2012), 1411-1415. [9] Electron. J. Qual. Theory Differ. Equ., 11 (2008), 1-18. [10] J. Differential Equations, 256 (2014), 2101-2114. [11] Differential Equations Dynam. Systems, 11 (2003), 33-54. [12] Dynam. Report. Expositions Dynam. Systems (N.S.), 1 (1992), 164-224. [13] Theor. Ecol., 7 (2014), 335-349. [14] Period. Math. Hungar., 56 (2008), 83-95. [15] Qual. Theory Dyn. Syst., 10 (2011), 169-196. [16] Academic Press, Boston, 1993. [17] Theor. Ecol., 3 (2010), 209-221. [18] Quart. Appl. Math., 63 (2005), 56-70. [19] Discrete Contin. Dyn. Syst., 24 (2009), 1215-1224. [20] J. Differential Equations, 255 (2013), 4244-4266. [21] SIAM J. Math. Anal., 35 (2003), 596-622. [22] Ann. Mat. Pura Appl., 145 (1986), 33-128. [23] Appl. Math. Comput., 190 (2007), 846-850. [24] Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 463 (2007), 2655-2669. [25] in Delay differential equations and applications, NATO Sci. Ser. II Math. Phys. Chem., Springer, Dordrecht, 205 (2006), 477-517. [26] J. Math. Biol., 42 (2001), 239-260. [27] Theor. Popul. Biol., 64 (2003), 201-209. [28] Kluwer Academic Publishers, Dordrecht, 1997. [29] Kluwer Academic Publishers, Dordrecht, 1993. [30] SIAM J. Appl. Math., 35 (1978), 260-267. [31] Math. Biosci., 232 (2011), 66-77. [32] Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 466 (2010), 2955-2973.
###### 通讯作者: 陈斌, bchen63@163.com
• 1.

沈阳化工大学材料科学与工程学院 沈阳 110142

1.285 1.3