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September  2017, 22(7): 2651-2667. doi: 10.3934/dcdsb.2017129

Extinction in stochastic predator-prey population model with Allee effect on prey

University of Niš, Faculty of Sciences and Mathematics, Višegradska 33,18000 Niš, Serbia

* Corresponding author: Miljana Jovanović

Received  July 2016 Revised  December 2016 Published  April 2017

Fund Project: The authors were supported by the Grant No 174007 of MNTRS

This paper presents the analysis of the conditions which lead the stochastic predator-prey model with Allee effect on prey population to extinction. In order to find these conditions we first prove the existence and uniqueness of global positive solution of considered model using the comparison theorem for stochastic differential equations. Then, we establish the conditions under which extinction of predator and prey populations occur. We also find the conditions for parameters of the model under which the solution of the system is globally attractive in mean. Finally, the numerical illustration with real life example is carried out to confirm our theoretical results.

Citation: Miljana JovanoviĆ, Marija KrstiĆ. Extinction in stochastic predator-prey population model with Allee effect on prey. Discrete & Continuous Dynamical Systems - B, 2017, 22 (7) : 2651-2667. doi: 10.3934/dcdsb.2017129
References:
[1]

A. AcklehL. Allen and J. Carter, Establishing a beachhead: A stochastic population model with an Allee effect applied to species invasion, Theor. Popul. Biol., 71 (2007), 290-300. doi: 10.1016/j.tpb.2006.12.006.

[2] W. C. Allee, Animal Aggregations, a Study in General Sociology, University of Chicago Press, Chicago, 1931.
[3]

I. Barbalat, Systems d'equations differential d'oscillations nonlinearies, Revue Roumaine de Mathematiques Pures et Appliquees, 4 (1959), 267-270.

[4]

F. CourchampT. Clutton-Brock and B. Grenfell, Inverse density dependence and the Allee effect, TREE, 14 (1999), 405-410. doi: 10.1016/S0169-5347(99)01683-3.

[5]

O. Duman and H. Merdan, Stability analysis of continuous population model involving predation and Allee effect, Chaos, Sol. & Frac., 41 (2009), 1218-1222. doi: 10.1016/j.chaos.2008.05.008.

[6]

J. D. Flores and E. Gonzales-Olivares, Dynamics of predator-prey model with Allee effect on prey and ratio-dependent functional response, Ecol. Compl., 18 (2014), 59-66. doi: 10.1016/j.ecocom.2014.02.005.

[7]

I. I. Gikhman and A. V. Skorokhod, Stochastic differential equations, The Theory of Stochastic Processes Ⅲ: Part of the series Classics in Mathematics, (1968), 113-219. doi: 10.1007/978-3-540-49941-1_2.

[8]

C. JiD. Jiang and X. Li, Qualitative analysis of a stochastic ratio-dependent predator-prey system, J. Comput. Appl. Math., 235 (2011), 1326-1341. doi: 10.1016/j.cam.2010.08.021.

[9]

C. JiD. Jiang and N. Shi, Analysis of a predator-prey model with modified Leslie-Gower and Holling-type Ⅱ schemes with stochastic perturbation, J. Math. Anal. Appl., 359 (2009), 482-498. doi: 10.1016/j.jmaa.2009.05.039.

[10]

M. Jovanović and M. Krstić, The influence of time-dependent delay on behavior of stochastic population model with the Allee effect, Appl. Math. Model., 39 (2015), 733-746. doi: 10.1016/j.apm.2014.06.019.

[11] I. Karatzas and S. E. Shreve, Brownian Motion and Stochastic Calculus, 2 edition, Springer-Verlag, Berlin, 1991. doi: 10.1007/978-1-4612-0949-2.
[12]

A. KentC. P. Doncaster and T. Sluckin, Consequences for predators of rescue and Allee effects on prey, Ecol. Model., 162 (2003), 233-245. doi: 10.1016/S0304-3800(02)00343-5.

[13] P. E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations, Springer, Berlin, 1992. doi: 10.1007/978-3-662-12616-5.
[14]

M. Krstić and M. Jovanović, On stochastic population model with the Allee effect, Math. Comput. Model., 52 (2010), 370-379. doi: 10.1016/j.mcm.2010.02.051.

[15]

R. LessardS. MartellC. WaltersT. Essington and J. Kitchell, Should ecosystem management involve active control of species abundances?, Ecology and Society, 10 (2005), p1. doi: 10.5751/ES-01313-100201.

[16]

A. Liebhold and J. Bascompte, The Allee effect, stochastic dynamics and the eradiction of alien species, Ecol. Lett., 6 (2003), 133-140.

[17] A. Lotka, Elements of Physical Biology, Williams and Wilkins, Baltimore, Md, 1924.
[18]

J. Lv and K. Wang, Analysis of stochastic predator-prey model with modified Leslie-Gower response Abstr. Appl. Anal. (2011), Art. ID 518719, 16 pp. doi: 10.1155/2011/518719.

[19] X. Mao, Stochastic Differential Equations and Applications, 2 edition, Horvood, Chichester, UK, 2008. doi: 10.1533/9780857099402.
[20]

X. Mao, Stochastic version of the Lassalle theorem, J. Differential Equations, 153 (1999), 175-195. doi: 10.1006/jdeq.1998.3552.

[21]

V. Volterra, Variazioni e fluttuazioni del numero d'individui in specie d'animali conviventi, Mem. Acad. Lincei, 2 (1926), 31-113.

[22]

J. A. Vucetich and R. O. Peterson, The influence of prey consumption and demographic stochasticity on population growth rate of Isle Royale wolves Canis lupus, Oikos, 107 (2004), 309-320.

[23]

J. A. Vucetich and R. O. Peterson, Ecological Studies of Wolves on Isle Royal 2015. Available from: www.isleroyalewolf.org.

[24]

Inbred wolf population on Isle Royale collapses 2015. Available from: www.sciencemag.org/news/2015/04/inbred-wolf-population-isle-royale-collapses

[25]

Q. Yang and D. Jiang, A note on asymptotic behaviors of stochastic population model with Allee effect, Appl. Math. Model., 35 (2011), 4611-4619. doi: 10.1016/j.apm.2011.03.034.

[26]

B. ZimmermannH. SandP. WabakkenO. Liberg and H. P. Andreassen, Predator-dependent functional response in wolves: From food limitation to surplus killing, J. Anim. Ecol., 84 (2015), 102-112. doi: 10.1111/1365-2656.12280.

[27]

S. R. ZhouY. F. Liu and G. Wang, The stability of predator-prey systems subject to the Allee effects, Theor. Popul. Biol., 67 (2005), 23-31. doi: 10.1016/j.tpb.2004.06.007.

[28]

J. Zu and M. Mimura, The impact of Allee effect on a predator-prey system with Holling type Ⅱ functional response, Appl. Math. Comput., 217 (2010), 3542-3556. doi: 10.1016/j.amc.2010.09.029.

show all references

References:
[1]

A. AcklehL. Allen and J. Carter, Establishing a beachhead: A stochastic population model with an Allee effect applied to species invasion, Theor. Popul. Biol., 71 (2007), 290-300. doi: 10.1016/j.tpb.2006.12.006.

[2] W. C. Allee, Animal Aggregations, a Study in General Sociology, University of Chicago Press, Chicago, 1931.
[3]

I. Barbalat, Systems d'equations differential d'oscillations nonlinearies, Revue Roumaine de Mathematiques Pures et Appliquees, 4 (1959), 267-270.

[4]

F. CourchampT. Clutton-Brock and B. Grenfell, Inverse density dependence and the Allee effect, TREE, 14 (1999), 405-410. doi: 10.1016/S0169-5347(99)01683-3.

[5]

O. Duman and H. Merdan, Stability analysis of continuous population model involving predation and Allee effect, Chaos, Sol. & Frac., 41 (2009), 1218-1222. doi: 10.1016/j.chaos.2008.05.008.

[6]

J. D. Flores and E. Gonzales-Olivares, Dynamics of predator-prey model with Allee effect on prey and ratio-dependent functional response, Ecol. Compl., 18 (2014), 59-66. doi: 10.1016/j.ecocom.2014.02.005.

[7]

I. I. Gikhman and A. V. Skorokhod, Stochastic differential equations, The Theory of Stochastic Processes Ⅲ: Part of the series Classics in Mathematics, (1968), 113-219. doi: 10.1007/978-3-540-49941-1_2.

[8]

C. JiD. Jiang and X. Li, Qualitative analysis of a stochastic ratio-dependent predator-prey system, J. Comput. Appl. Math., 235 (2011), 1326-1341. doi: 10.1016/j.cam.2010.08.021.

[9]

C. JiD. Jiang and N. Shi, Analysis of a predator-prey model with modified Leslie-Gower and Holling-type Ⅱ schemes with stochastic perturbation, J. Math. Anal. Appl., 359 (2009), 482-498. doi: 10.1016/j.jmaa.2009.05.039.

[10]

M. Jovanović and M. Krstić, The influence of time-dependent delay on behavior of stochastic population model with the Allee effect, Appl. Math. Model., 39 (2015), 733-746. doi: 10.1016/j.apm.2014.06.019.

[11] I. Karatzas and S. E. Shreve, Brownian Motion and Stochastic Calculus, 2 edition, Springer-Verlag, Berlin, 1991. doi: 10.1007/978-1-4612-0949-2.
[12]

A. KentC. P. Doncaster and T. Sluckin, Consequences for predators of rescue and Allee effects on prey, Ecol. Model., 162 (2003), 233-245. doi: 10.1016/S0304-3800(02)00343-5.

[13] P. E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations, Springer, Berlin, 1992. doi: 10.1007/978-3-662-12616-5.
[14]

M. Krstić and M. Jovanović, On stochastic population model with the Allee effect, Math. Comput. Model., 52 (2010), 370-379. doi: 10.1016/j.mcm.2010.02.051.

[15]

R. LessardS. MartellC. WaltersT. Essington and J. Kitchell, Should ecosystem management involve active control of species abundances?, Ecology and Society, 10 (2005), p1. doi: 10.5751/ES-01313-100201.

[16]

A. Liebhold and J. Bascompte, The Allee effect, stochastic dynamics and the eradiction of alien species, Ecol. Lett., 6 (2003), 133-140.

[17] A. Lotka, Elements of Physical Biology, Williams and Wilkins, Baltimore, Md, 1924.
[18]

J. Lv and K. Wang, Analysis of stochastic predator-prey model with modified Leslie-Gower response Abstr. Appl. Anal. (2011), Art. ID 518719, 16 pp. doi: 10.1155/2011/518719.

[19] X. Mao, Stochastic Differential Equations and Applications, 2 edition, Horvood, Chichester, UK, 2008. doi: 10.1533/9780857099402.
[20]

X. Mao, Stochastic version of the Lassalle theorem, J. Differential Equations, 153 (1999), 175-195. doi: 10.1006/jdeq.1998.3552.

[21]

V. Volterra, Variazioni e fluttuazioni del numero d'individui in specie d'animali conviventi, Mem. Acad. Lincei, 2 (1926), 31-113.

[22]

J. A. Vucetich and R. O. Peterson, The influence of prey consumption and demographic stochasticity on population growth rate of Isle Royale wolves Canis lupus, Oikos, 107 (2004), 309-320.

[23]

J. A. Vucetich and R. O. Peterson, Ecological Studies of Wolves on Isle Royal 2015. Available from: www.isleroyalewolf.org.

[24]

Inbred wolf population on Isle Royale collapses 2015. Available from: www.sciencemag.org/news/2015/04/inbred-wolf-population-isle-royale-collapses

[25]

Q. Yang and D. Jiang, A note on asymptotic behaviors of stochastic population model with Allee effect, Appl. Math. Model., 35 (2011), 4611-4619. doi: 10.1016/j.apm.2011.03.034.

[26]

B. ZimmermannH. SandP. WabakkenO. Liberg and H. P. Andreassen, Predator-dependent functional response in wolves: From food limitation to surplus killing, J. Anim. Ecol., 84 (2015), 102-112. doi: 10.1111/1365-2656.12280.

[27]

S. R. ZhouY. F. Liu and G. Wang, The stability of predator-prey systems subject to the Allee effects, Theor. Popul. Biol., 67 (2005), 23-31. doi: 10.1016/j.tpb.2004.06.007.

[28]

J. Zu and M. Mimura, The impact of Allee effect on a predator-prey system with Holling type Ⅱ functional response, Appl. Math. Comput., 217 (2010), 3542-3556. doi: 10.1016/j.amc.2010.09.029.

Figure 1.  Deterministic and stochastic trajectories of moose and wolf population described by (2) with parameters (12), $\alpha=0.0002$, $A_1=0.5$, $\sigma_1^2=\sigma_2^2=0.005$
Figure 2.  Left: Deterministic and stochastic trajectories of moose population described by (2) with parameters (12), $\alpha=0.0002$, $A_1=5$, $\sigma_1^2=\sigma_2^2=0.005$; Right: Stochastic trajectories of moose and wolf populations in which we can observe behavior of these populations in 140 years
Figure 3.  Deterministic and stochastic trajectories of moose population described by (2) with parameters (12), $\alpha=0.01$, $A_1=0.5$ and different intensities of noise
Figure 4.  Deterministic and stochastic trajectories of moose and wolf population described by (2) with parameters (12), $\alpha=0.01$, $A_1=5$, $\sigma_1^2=0.14$, $\sigma_2^2=0.005$
Figure 5.  Stochastic trajectories of moose and wolf population described by (2) with three different initial values
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