JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE) ›› 2023, Vol. 58 ›› Issue (10): 54-66.doi: 10.6040/j.issn.1671-9352.0.2022.635

Previous Articles     Next Articles

Dynamics of a stochastic predation model with fear effect and Crowley-Martin functional response

Yuke ZHANG(),Xinzhu MENG*()   

  1. College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao 266590, Shandong, China
  • Received:2022-11-27 Online:2023-10-20 Published:2023-10-17
  • Contact: Xinzhu MENG E-mail:zhangyukekkk@163.com;mxz721106@sdust.edu.cn

RichHTML

3

PDF (PC)

93

Abstract:

A stochastic predator-prey model with fear effect and Crowley-Martin functional response is studied. We first give the existence and uniqueness of global positive solutions and the boundedness of solutions on the stochastic model. We explore sufficient conditions for the persistence in mean and the extinction of the populations by using stochastic qualitative analysis theory. The existence of unique ergodic stationary distribution is proved by establishing suitable Lyapunov functions. Finally, numerical simulations are conducted to reveal the effects of fear and white noise on the theoretical results of population dynamics.

Key words: Crowley-Martin functional response, global positive solution, stochastic qualitative analysis, ergodic stationary distribution

CLC Number: 

  • O175

Table 1

Biological meaning of each parameter"

参数 含义
r 内禀增长率
f 由捕食者引起的恐惧程度
δ 食饵的密度制约系数
β 捕食率
q 能量转换率
d 捕食者的死亡率
h 捕食者的密度制约系数

Table 2

Parameter values of Model (2)"

参数 r f β δ q d h a b
取值 1.5 0.2 0.5 0.01 0.9 0.042 0.2 2 2

Fig.1

Time series plots of Model (2) with σ1=1.9, σ2=1.2"

Fig.2

Persistence of Model (2) with σ1=0.1, σ2=0.1"

Fig.3

Probability density functions of x(t) and y(t) of Model (2)"

Fig.4

Time series plots of Model (2) with f=0, 0.2, 0.8"

Fig.5

Time series plots of Model (2) with f=0.8, σ1=0.8, σ2=0.8"

1 LOTKA A J . Elements of physical biology[M]. Baltimore: Williams & Wilkins Company, 1925: 460.
2 VOLTERRA V . Fluctuations in the abundance of a species considered mathematically[J]. Nature, 1926, 118 (2972): 558- 560.
doi: 10.1038/118558a0
3 HOLLING C S . Some characteristics of simple types of predation and parasitism[J]. The Canadian Entomologist, 1959, 91 (7): 385- 398.
doi: 10.4039/Ent91385-7
4 AGIZA H N , ELABBASY E M , EL-METWALLY H , et al. Chaotic dynamics of a discrete prey-predator model with Holling type Ⅱ[J]. Nonlinear Analysis: Real World Applications, 2009, 10 (1): 116- 129.
doi: 10.1016/j.nonrwa.2007.08.029
5 HUANG Yunjin , CHEN Fengde , ZHONG Li . Stability analysis of a prey-predator model with Holling type Ⅲ response function incorporating a prey refuge[J]. Applied Mathematics and Computation, 2006, 182 (1): 672- 683.
doi: 10.1016/j.amc.2006.04.030
6 LIU Xianqing , ZHONG Shouming , TIAN Baodan , et al. Asymptotic properties of a stochastic predator-prey model with Crowley-Martin functional response[J]. Journal of Applied Mathematics and Computing, 2013, 43 (1): 479- 490.
7 LU Chun . Dynamical analysis and numerical simulations on a Crowley-Martin predator-prey model in stochastic environment[J]. Applied Mathematics and Computation, 2022, 413, 126641.
doi: 10.1016/j.amc.2021.126641
8 BEDDINGTON J R . Mutual interference between parasites or predators and its effect on searching efficiency[J]. The Journal of Animal Ecology, 1975, 44, 331.
doi: 10.2307/3866
9 DEANGELIS D L , GOLDSTEIN R A , O'NEILL R V . A model for tropic interaction[J]. Ecology, 1975, 56 (4): 881- 892.
doi: 10.2307/1936298
10 LESLIE P H , GOWER J C . The properties of a stochastic model for the predator-prey type of interaction between two species[J]. Biometrika, 1960, 47 (3/4): 219- 234.
doi: 10.2307/2333294
11 ZANETTE L Y , WHITE A F , ALLEN M C , et al. Perceived predation risk reduces the number of offspring songbirds produce per year[J]. Science, 2011, 334 (6061): 1398- 1401.
doi: 10.1126/science.1210908
12 CREEL S , CHRISTIANSON D , LILEY S , et al. Predation risk affects reproductive physiology and demography of elk[J]. Science, 2007, 315 (5814): 960.
doi: 10.1126/science.1135918
13 CREEL S , CHRISTIANSON D . Relationships between direct predation and risk effects[J]. Trends in Ecology & Evolution, 2008, 23 (4): 194- 201.
14 HUA F Y , SIEVING K E , FLETCHER R J , et al. Increased perception of predation risk to adults and offspring alters avian reproductive strategy and performance[J]. Behavioral Ecology, 2014, 25 (3): 509- 519.
doi: 10.1093/beheco/aru017
15 CAI Yongmei , CAI Siyang , MAO Xuerong . Stochastic delay foraging arena predator-prey system with Markov switching[J]. Stochastic Analysis and Applications, 2020, 38 (2): 191- 212.
doi: 10.1080/07362994.2019.1679645
16 LIU Qun , JIANG Daqing . Influence of the fear factor on the dynamics of a stochastic predator-prey model[J]. Applied Mathematics Letters, 2021, 112, 106756.
doi: 10.1016/j.aml.2020.106756
17 ROY J , BARMAN D , ALAM S . Role of fear in a predator-prey system with ratio-dependent functional response in deterministic and stochastic environment[J]. Biosystems, 2020, 197, 104176.
doi: 10.1016/j.biosystems.2020.104176
18 QI H K , MENG X Z , HAYAT T , et al. Stationary distribution of a stochastic predator-prey model with hunting cooperation[J]. Applied Mathematics Letters, 2022, 124, 107662.
doi: 10.1016/j.aml.2021.107662
19 QI Haokun , MENG Xinzhu . Threshold behavior of a stochastic predator-prey system with prey refuge and fear effect[J]. Applied Mathematics Letters, 2021, 113, 106846.
doi: 10.1016/j.aml.2020.106846
20 LIU Guodong , QI Haokun , CHANG Zhengbo , et al. Asymptotic stability of a stochastic May mutualism system[J]. Computers & Mathematics with Applications, 2020, 79 (3): 735- 745.
21 PENG Hao , ZHANG Xinhong . The dynamics of stochastic predator-prey models with non-constant mortality rate and general nonlinear functional response[J]. Journal of Nonlinear Modeling and Analysis, 2020, 2, 495- 511.
22 WEI Ning , LI Mei . Stochastically permanent analysis of a non-autonomous HollingⅡ predator-prey model with a complex type of noises[J]. Journal of Applied Analysis & Computation, 2022, 12 (2): 479- 496.
23 CAO Boqiang , SHAN Meijing , ZHANG Qimin , et al. A stochastic SIS epidemic model with vaccination[J]. Physica A: Statistical Mechanics and Its Applications, 2017, 486, 127- 143.
doi: 10.1016/j.physa.2017.05.083
24 ZHANG Shengqiang , YUAN Sanling , ZHANG Tonghua . A predator-prey model with different response functions to juvenile and adult prey in deterministic and stochastic environments[J]. Applied Mathematics and Computation, 2022, 413, 126598.
doi: 10.1016/j.amc.2021.126598
25 CHENG S R . Stochastic population systems[J]. Stochastic Analysis and Applications, 2009, 27 (4): 854- 874.
doi: 10.1080/07362990902844348
26 ROY J , ALAM S . Fear factor in a prey-predator system in deterministic and stochastic environment[J]. Physica A: Statistical Mechanics and Its Applications, 2020, 541, 123359.
doi: 10.1016/j.physa.2019.123359
27 KHASMINSKII R . Stochastic stability of differential equations[M]. 2nd ed. Michigan: Springer Science & Business Media, 2011.
[1] Qian CAO,Yanling LI,Weihua SHAN. Dynamics of a reaction-diffusion predator-prey model incorporating prey refuge and fear effect [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2023, 58(10): 43-53.
[2] Gaihui GUO,Jingjing WANG,Wangrui LI. Hopf bifurcation of a vegetation-water reaction-diffusion model with time delay [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2023, 58(10): 32-42, 53.
[3] Yadi WANG,Hailong YUAN. Hopf bifurcation analysis in the Lengyel-Epstein reaction diffusion system with time delay [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2023, 58(8): 92-103.
[4] Yun NI,Xiping LIU. Existence and Ulam stability for positive solutions of conformable fractional coupled systems [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2023, 58(8): 82-91.
[5] Zipeng HE,Yaying DONG. Steady-state solutions of a Holling type Ⅱ competition model in heterogeneous environment [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2023, 58(8): 73-81.
[6] Zhongbo CAI,Jihong ZHAO. Global existence of large solutions for a class of chemotaxis-fluid model [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2023, 58(6): 84-91.
[7] Pan SUN,Xuping ZHANG. Continuous dependence of solution for impulsive evolution equations with infinite delay [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2023, 58(6): 77-83, 91.
[8] Congcong KANG. Existence and multiplicity of positive solutions for a class of second-order periodic boundary value problems [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2023, 58(6): 68-76.
[9] Li LI,He YANG. Existence of mild solutions for the nonlocal problem of second-order impulsive evolution equations [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2023, 58(6): 57-67.
[10] Yanmin GUO,Song ZHAO,Yuxia TONG. Global BMO estimations for obstacle problems of a class of non-uniformly elliptic equations [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2023, 58(6): 46-56.
[11] HAN Qing-xiu, LIU Hong-xia, WU Yun. Bifurcation phenomena and nonlinear wave solutions of BKK equation [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2022, 57(8): 88-94.
[12] LI Chun-ping, SANG Yan-bin. Multiple solutions of fractional p-q-Laplacian system with sign-changing weight functions [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2022, 57(8): 95-102.
[13] PANG Yu-ting, ZHAO Dong-xia, BAO Fang-xia. Stability of the bidirectional ring networks with multiple time delays and multiple parameters [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2022, 57(8): 103-110.
[14] SU Xiao-xiao, ZHANG Ya-li. Existence of positive solutions for periodic boundary value problems of secondorder damped difference equations [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2021, 56(2): 56-63.
[15] WU Ruo-fei. Existence of solutions for singular fourth-order m-point boundary value problems [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2021, 56(2): 75-83.
Viewed
Full text


Abstract

Cited
  Shared   
  Discussed   
[1] MAO Ai-qin1,2, YANG Ming-jun2, 3, YU Hai-yun2, ZHANG Pin1, PAN Ren-ming1*. Study on thermal decomposition mechanism of  pentafluoroethane fire extinguishing agent[J]. J4, 2013, 48(1): 51 -55 .
[2] LI Yong-ming1, DING Li-wang2. The r-th moment consistency of estimators for a semi-parametric regression model for positively associated errors[J]. J4, 2013, 48(1): 83 -88 .
[3] DONG Li-hong1,2, GUO Shuang-jian1. The fundamental theorem for weak Hopf module in  Yetter-Drinfeld module categories[J]. J4, 2013, 48(2): 20 -22 .
[4] ZHAO Jun1, ZHAO Jing2, FAN Ting-jun1*, YUAN Wen-peng1,3, ZHANG Zheng1, CONG Ri-shan1. Purification and anti-tumor activity examination of water-soluble asterosaponin from Asterias rollestoni Bell[J]. J4, 2013, 48(1): 30 -35 .
[5] YANG Yong-wei1, 2, HE Peng-fei2, LI Yi-jun2,3. On strict filters of BL-algebras#br#[J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2014, 49(03): 63 -67 .
[6] LI Min1,2, LI Qi-qiang1. Observer-based sliding mode control of uncertain singular time-delay systems#br#[J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2014, 49(03): 37 -42 .
[7] YANG Jun. Characterization and structural control of metalbased nanomaterials[J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2013, 48(1): 1 -22 .
[8] DONG Wei-wei. A new method of DEA efficiency ranking for decision making units with independent subsystems[J]. J4, 2013, 48(1): 89 -92 .
[9] ZHANG Jing-you, ZHANG Pei-ai, ZHONG Hai-ping. The application of evolutionary graph theory in the design of knowledge-based enterprises’ organization strucure[J]. J4, 2013, 48(1): 107 -110 .
[10] TANG Feng-qin1, BAI Jian-ming2. The precise large deviations for a risk model with extended negatively upper orthant dependent claim  sizes[J]. J4, 2013, 48(1): 100 -106 .
Copyright 2018 © Editorial office of JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE)
Supported by Beijing Magtech Co.Ltd