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June  2022, 27(6): 3155-3175. doi: 10.3934/dcdsb.2021177

Dynamical behavior of a stochastic predator-prey model with general functional response and nonlinear jump-diffusion

Xinhong Zhang , and  Qing Yang

College of Science, China University of Petroleum, Qingdao 266580, Shandong Province, China

* Corresponding author: Xinhong Zhang

Received  March 2021 Revised  May 2021 Published  June 2022 Early access  July 2021

Fund Project: The first author is supported by National Natural Science Foundation of China Grant 11801566 and the Fundamental Research Funds for the Central Universities of China grant 19CX02059A

In this paper, we consider a stochastic predator-prey model with general functional response, which is perturbed by nonlinear Lévy jumps. Firstly, We show that this model has a unique global positive solution with uniform boundedness of $ \theta\in(0,1] $-th moment. Secondly, we obtain the threshold for extinction and exponential ergodicity of the one-dimensional Logistic system with nonlinear perturbations. Then based on the results of Logistic system, we introduce a new technique to study the ergodic stationary distribution for the stochastic predator-prey model with general functional response and nonlinear jump-diffusion, and derive the sufficient and almost necessary condition for extinction and ergodicity.

Keywords: Predator-prey model, Lévy jumps, functional response, ergodicity, extinction.
Mathematics Subject Classification: Primary: 92B05, 60G51; Secondary: 60H10, 60G57.
Citation: Xinhong Zhang, Qing Yang. Dynamical behavior of a stochastic predator-prey model with general functional response and nonlinear jump-diffusion. Discrete and Continuous Dynamical Systems - B, 2022, 27 (6) : 3155-3175. doi: 10.3934/dcdsb.2021177
References:
[1] D. Applebaum, Lévy Process and Stochastic Calculus, Cambridge University Press, New York, 2009.  doi: 10.1017/CBO9780511809781.
[2]

J. Bao and C. Yuan, Comparison theorem for stochastic differential delay equations with jumps, Acta Appl. Math., 116 (2011), 119-132.  doi: 10.1007/s10440-011-9633-7.

[3]

J. Bao, X. Mao, G. Yin and C. Yuan, Competitive Lotka-Volterra population dynamics with jumps, Nonlinear Anal.-Throe., 74 (2011), 6601-6616.  doi: 10.1016/j.na.2011.06.043.

[4]

J. R. Beddington, Mutual interference between parasites or predators and its effect on searching efficiency, J. Anim. Ecol., 44 (1975), 331-340.  doi: 10.2307/3866.

[5]

Y. Cai and X. Mao, Stochastic prey-predator system with foraging arena scheme, Appl. Math. Model., 64 (2018), 357-371.  doi: 10.1016/j.apm.2018.07.034.

[6]

R. S. Cantrell and C. Cosner, On the dynamics of predator-prey models with the Beddington-Deangelis functional response, J. Math. Anal. Appl., 257 (2001), 206-222.  doi: 10.1006/jmaa.2000.7343.

[7]

S. Chen, J. Wei and J. Yu, Stationary patterns of a diffusive predator-prey model with Crowley-Martin functional response, Nonlinear Anal.-Real World Appl., 39 (2018), 33-57.  doi: 10.1016/j.nonrwa.2017.05.005.

[8]

Z. Cowan, M. S. Pratchett, V. Messmer and S. Ling, Known predators of crown-of-thorns starfish (Acanthaster spp.) and their role in mitigating, if not preventing, population outbreaks, Diversity-Basel, 9 (2017), 7. doi: 10.3390/d9010007.

[9]

P. H. Crowley and E. K. Martin, Functional responses and interference within and between year classes of a dragonfly population, J. North Am. Benthol. Soc., 8 (1989), 211-221.  doi: 10.2307/1467324.

[10]

A. Das and G. P. Samanta, Stochastic prey-predator model with additional food for predator, Physica A, 512 (2018), 121-141.  doi: 10.1016/j.physa.2018.08.138.

[11]

N. H. Du, N. H. Dang and G. Yin, Conditions for permanence and ergodicity of certain stochastic predator-prey models, J. Appl. Probab., 53 (2016), 187-202.  doi: 10.1017/jpr.2015.18.

[12]

N. H. Du, D. H. Dang and G. Yin, Conditions for permanence and ergodicity of certain stochastic predator-prey models, J. Appl. Probab., 53 (2016), 187-202.  doi: 10.1017/jpr.2015.18.

[13]

R. Has'miniskii, Ergodic properties of recurrent diffusion processes and stabilization of the Cauchy problem for parabolic equations, Theory Probab. Appl., 5 (1960), 179-196. 

[14]

C. S. Holling, The components of predation as revealed by a study of small-mammal predation of the european pine sawfly, Can. Entomol., 91 (1959), 293-320.  doi: 10.4039/Ent91293-5.

[15]

C. S. Holling, Some characteristics of simple types of predation and parasitism, Can. Entomol., 91 (1959), 385-398.  doi: 10.4039/Ent91385-7.

[16]

C. S. Holling, The functional response of predators to prey density and its role in mimicry and population regulation, Memoirs of the Entomological Society of Canada, 97 (1965), 5-60.  doi: 10.4039/entm9745fv.

[17]

K. Ichihara and H. Kunita, A classification of the second order degenerate elliptic operators and its probabilistic characerization, Z. Wahrsch. Verw. Gebiete, 39 (1977), 81-84.  doi: 10.1007/BF01844875.

[18]

S. Li, J. Wu and Y. Dong, Effects of a degeneracy in a diffusive predato-prey model with Holling Ⅱ functional response, Nonlinear Anal.-Real World Appl., 43 (2018), 78-95.  doi: 10.1016/j.nonrwa.2018.02.003.

[19]

Q. Liu and D. Jiang, Influence of the fear factor on the dynamics of a stochastic predator-prey model, Appl. Math. Lett., 112 (2021), 106756. doi: 10.1016/j.aml.2020.106756.

[20]

M. Liu and P. S. Mandal, Dynamical behavior of a one-prey two-prey model with random perturbations, Commun. Nonlinear Sci. Numer. Simulat., 28 (2015), 123-137.  doi: 10.1016/j.cnsns.2015.04.010.

[21]

M. Liu and Y. Zhu, Stationary distribution and ergodicity of a stochastic hybrid competition model with Levy jumps, Nonlinear Anal-Hybrid Syst., 30 (2018), 225-239.  doi: 10.1016/j.nahs.2018.05.002.

[22]

Q. Liu, D. Jiang, T. Hayat, A. Alsaedi and B. Ahmad, A stochastic SIRS epidemic model with logistic growth and general nonlinear incidence rate, Physica A, 551 (2020), 124152. doi: 10.1016/j.physa.2020.124152.

[23]

Q. Liu, D. Jiang, T. Hayat and B. Ahmad, Stationary distribution and extinction of a stochastic predator-prey model with additional food and nonlinear perturbation, Appl. Math. Comput., 320 (2018), 226-239.  doi: 10.1016/j.amc.2017.09.030.

[24]

A. J. Lotka, Elements of Physical Biology, William and Wilkins, Baltimore, 1925.

[25]

X. Mao, G. Marion and E. Renshaw, Environmental Brownian noise suppresses explosions in population dynamics, Stoch. Proc. Appl., 97 (2002), 95-110.  doi: 10.1016/S0304-4149(01)00126-0.

[26] X. Mao and C. Yuan, Stochastic Differential Equations With Markovian Switching, Imperial College Press, London, 2006.  doi: 10.1142/p473.
[27]

X. Mao, G. Marion and E. Renshaw, Environmental Brownian noise suppresses explosions in population dynamics, Stochastic Process. Appl., 97 (2002), 95-110.  doi: 10.1016/S0304-4149(01)00126-0.

[28]

R. M. May, Stability and Complexity in Model Ecosystems, Princeton University, 1973.

[29]

S. P. Meyn and R. L. Tweedie, Stability of Markovian processes Ⅲ: Foster-Lyapunov criteria for continuous-time processes, Dav. Appl. Prob., 25 (1993), 518-548.  doi: 10.2307/1427522.

[30]

H. M. Safuan, H. S. Sidhu, Z. Jovanoski and I. N. Towers, Impacts of biotic resource enrichment on a predator-prey population, Bull. Math. Biol., 75 (2013), 1798-1812.  doi: 10.1007/s11538-013-9869-7.

[31]

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

[32]

L. Wang and D. Jiang, Ergodic property of the chemostat: A stochastic model under regime switching and with general response function, Nonlinear Anal-Hybrid Syst., 27 (2018), 341-352.  doi: 10.1016/j.nahs.2017.10.001.

[33]

F. Xi, Asymptotic properties of jump-diffusion processes with state-dependent switching, Stoch. Proc. Appl., 119 (2009), 2198-2221.  doi: 10.1016/j.spa.2008.11.001.

[34]

X. Yu and S. Yuan, Asymptotic properties of a stochastic chemostat model with two distributed delays and nonlinear perturbation, Discrete Continuous Dynam. Systems - B, 25 (2020), 2373-2390.  doi: 10.3934/dcdsb.2020014.

[35]

X. Zhang, Y. Li and D. Jiang, Dynamics of a stochastic Holliing type Ⅱ predator-prey model with hyperboic mortality, Nonlinear Dyn., 87 (2017), 2011-2020. 

[36]

X. Zhang, The global dynamics of stochastic Holling type Ⅱ predator-prey models with non constant mortality rate, Filomat, 31 (2017), 5811-5825.  doi: 10.2298/FIL1718811Z.

[37]

X. Zou, Y. Zhen, L. Zhang and J. Lv, Survivability and stochastic bifurcations for a stochatic Holling type Ⅱ predator-prey model, Commun. Nonlinear Sci. Numer. Simulat., 83 (2020), 105136. doi: 10.1016/j.cnsns.2019.105136.

[38]

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

show all references

References:
[1] D. Applebaum, Lévy Process and Stochastic Calculus, Cambridge University Press, New York, 2009.  doi: 10.1017/CBO9780511809781.
[2]

J. Bao and C. Yuan, Comparison theorem for stochastic differential delay equations with jumps, Acta Appl. Math., 116 (2011), 119-132.  doi: 10.1007/s10440-011-9633-7.

[3]

J. Bao, X. Mao, G. Yin and C. Yuan, Competitive Lotka-Volterra population dynamics with jumps, Nonlinear Anal.-Throe., 74 (2011), 6601-6616.  doi: 10.1016/j.na.2011.06.043.

[4]

J. R. Beddington, Mutual interference between parasites or predators and its effect on searching efficiency, J. Anim. Ecol., 44 (1975), 331-340.  doi: 10.2307/3866.

[5]

Y. Cai and X. Mao, Stochastic prey-predator system with foraging arena scheme, Appl. Math. Model., 64 (2018), 357-371.  doi: 10.1016/j.apm.2018.07.034.

[6]

R. S. Cantrell and C. Cosner, On the dynamics of predator-prey models with the Beddington-Deangelis functional response, J. Math. Anal. Appl., 257 (2001), 206-222.  doi: 10.1006/jmaa.2000.7343.

[7]

S. Chen, J. Wei and J. Yu, Stationary patterns of a diffusive predator-prey model with Crowley-Martin functional response, Nonlinear Anal.-Real World Appl., 39 (2018), 33-57.  doi: 10.1016/j.nonrwa.2017.05.005.

[8]

Z. Cowan, M. S. Pratchett, V. Messmer and S. Ling, Known predators of crown-of-thorns starfish (Acanthaster spp.) and their role in mitigating, if not preventing, population outbreaks, Diversity-Basel, 9 (2017), 7. doi: 10.3390/d9010007.

[9]

P. H. Crowley and E. K. Martin, Functional responses and interference within and between year classes of a dragonfly population, J. North Am. Benthol. Soc., 8 (1989), 211-221.  doi: 10.2307/1467324.

[10]

A. Das and G. P. Samanta, Stochastic prey-predator model with additional food for predator, Physica A, 512 (2018), 121-141.  doi: 10.1016/j.physa.2018.08.138.

[11]

N. H. Du, N. H. Dang and G. Yin, Conditions for permanence and ergodicity of certain stochastic predator-prey models, J. Appl. Probab., 53 (2016), 187-202.  doi: 10.1017/jpr.2015.18.

[12]

N. H. Du, D. H. Dang and G. Yin, Conditions for permanence and ergodicity of certain stochastic predator-prey models, J. Appl. Probab., 53 (2016), 187-202.  doi: 10.1017/jpr.2015.18.

[13]

R. Has'miniskii, Ergodic properties of recurrent diffusion processes and stabilization of the Cauchy problem for parabolic equations, Theory Probab. Appl., 5 (1960), 179-196. 

[14]

C. S. Holling, The components of predation as revealed by a study of small-mammal predation of the european pine sawfly, Can. Entomol., 91 (1959), 293-320.  doi: 10.4039/Ent91293-5.

[15]

C. S. Holling, Some characteristics of simple types of predation and parasitism, Can. Entomol., 91 (1959), 385-398.  doi: 10.4039/Ent91385-7.

[16]

C. S. Holling, The functional response of predators to prey density and its role in mimicry and population regulation, Memoirs of the Entomological Society of Canada, 97 (1965), 5-60.  doi: 10.4039/entm9745fv.

[17]

K. Ichihara and H. Kunita, A classification of the second order degenerate elliptic operators and its probabilistic characerization, Z. Wahrsch. Verw. Gebiete, 39 (1977), 81-84.  doi: 10.1007/BF01844875.

[18]

S. Li, J. Wu and Y. Dong, Effects of a degeneracy in a diffusive predato-prey model with Holling Ⅱ functional response, Nonlinear Anal.-Real World Appl., 43 (2018), 78-95.  doi: 10.1016/j.nonrwa.2018.02.003.

[19]

Q. Liu and D. Jiang, Influence of the fear factor on the dynamics of a stochastic predator-prey model, Appl. Math. Lett., 112 (2021), 106756. doi: 10.1016/j.aml.2020.106756.

[20]

M. Liu and P. S. Mandal, Dynamical behavior of a one-prey two-prey model with random perturbations, Commun. Nonlinear Sci. Numer. Simulat., 28 (2015), 123-137.  doi: 10.1016/j.cnsns.2015.04.010.

[21]

M. Liu and Y. Zhu, Stationary distribution and ergodicity of a stochastic hybrid competition model with Levy jumps, Nonlinear Anal-Hybrid Syst., 30 (2018), 225-239.  doi: 10.1016/j.nahs.2018.05.002.

[22]

Q. Liu, D. Jiang, T. Hayat, A. Alsaedi and B. Ahmad, A stochastic SIRS epidemic model with logistic growth and general nonlinear incidence rate, Physica A, 551 (2020), 124152. doi: 10.1016/j.physa.2020.124152.

[23]

Q. Liu, D. Jiang, T. Hayat and B. Ahmad, Stationary distribution and extinction of a stochastic predator-prey model with additional food and nonlinear perturbation, Appl. Math. Comput., 320 (2018), 226-239.  doi: 10.1016/j.amc.2017.09.030.

[24]

A. J. Lotka, Elements of Physical Biology, William and Wilkins, Baltimore, 1925.

[25]

X. Mao, G. Marion and E. Renshaw, Environmental Brownian noise suppresses explosions in population dynamics, Stoch. Proc. Appl., 97 (2002), 95-110.  doi: 10.1016/S0304-4149(01)00126-0.

[26] X. Mao and C. Yuan, Stochastic Differential Equations With Markovian Switching, Imperial College Press, London, 2006.  doi: 10.1142/p473.
[27]

X. Mao, G. Marion and E. Renshaw, Environmental Brownian noise suppresses explosions in population dynamics, Stochastic Process. Appl., 97 (2002), 95-110.  doi: 10.1016/S0304-4149(01)00126-0.

[28]

R. M. May, Stability and Complexity in Model Ecosystems, Princeton University, 1973.

[29]

S. P. Meyn and R. L. Tweedie, Stability of Markovian processes Ⅲ: Foster-Lyapunov criteria for continuous-time processes, Dav. Appl. Prob., 25 (1993), 518-548.  doi: 10.2307/1427522.

[30]

H. M. Safuan, H. S. Sidhu, Z. Jovanoski and I. N. Towers, Impacts of biotic resource enrichment on a predator-prey population, Bull. Math. Biol., 75 (2013), 1798-1812.  doi: 10.1007/s11538-013-9869-7.

[31]

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

[32]

L. Wang and D. Jiang, Ergodic property of the chemostat: A stochastic model under regime switching and with general response function, Nonlinear Anal-Hybrid Syst., 27 (2018), 341-352.  doi: 10.1016/j.nahs.2017.10.001.

[33]

F. Xi, Asymptotic properties of jump-diffusion processes with state-dependent switching, Stoch. Proc. Appl., 119 (2009), 2198-2221.  doi: 10.1016/j.spa.2008.11.001.

[34]

X. Yu and S. Yuan, Asymptotic properties of a stochastic chemostat model with two distributed delays and nonlinear perturbation, Discrete Continuous Dynam. Systems - B, 25 (2020), 2373-2390.  doi: 10.3934/dcdsb.2020014.

[35]

X. Zhang, Y. Li and D. Jiang, Dynamics of a stochastic Holliing type Ⅱ predator-prey model with hyperboic mortality, Nonlinear Dyn., 87 (2017), 2011-2020. 

[36]

X. Zhang, The global dynamics of stochastic Holling type Ⅱ predator-prey models with non constant mortality rate, Filomat, 31 (2017), 5811-5825.  doi: 10.2298/FIL1718811Z.

[37]

X. Zou, Y. Zhen, L. Zhang and J. Lv, Survivability and stochastic bifurcations for a stochatic Holling type Ⅱ predator-prey model, Commun. Nonlinear Sci. Numer. Simulat., 83 (2020), 105136. doi: 10.1016/j.cnsns.2019.105136.

[38]

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

Figure 1.  The left column shows the numbers of $ x(t),y(t) $ in system (26) with $ \alpha_{11} = \alpha_{12} = 0.2 $ and $ \alpha_{21} = \alpha_{22} = 0.1 $. The right column represents the histogram of the probability density functions of $ x,y $ individuals
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Figure 2.  Simulations of the solution in stochastic system (26) with initial value $ \alpha_{11} = \alpha_{12} = 0.2 $ and $ \alpha_{21} = \alpha_{22} = 1.5 $
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