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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.  Google Scholar

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

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar
[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.  Google Scholar

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

[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.  Google Scholar

[16]

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

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

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

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

[21]

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

[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.   Google Scholar

[23]

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

[24]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

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

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar
[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.  Google Scholar

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

[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.  Google Scholar

[16]

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

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

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

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

[21]

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

[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.   Google Scholar

[23]

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

[24]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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