TY - JOUR
T1 - STOCHASTIC DELAY DIFFERENTIAL EQUATIONS OF THREE-SPECIES PREY-PREDATOR SYSTEM WITH COOPERATION AMONG PREY SPECIES
AU - Rihan, Fathalla A.
AU - Alsakaji, Hebatallah J.
N1 - Funding Information:
Acknowledgments. This research is generously supported by UAE University. The authors would like to thank the reviewers and editor for their constructive comments and suggestions which improved this manuscript.
Funding Information:
2020 Mathematics Subject Classification. 34D20, 34F05, 92B05. Key words and phrases. DDEs, Extinction; Stationary distribution and ergodicity; Stochastic prey-predator model; Stochastic DDEs; Time-delays. This work supported by UPAR-Project (Code # G00003440). ∗ Corresponding author: F.A. Rihan (frihan@uaeu.ac.ae).
Publisher Copyright:
© 2022 American Institute of Mathematical Sciences. All rights reserved.
PY - 2022/2
Y1 - 2022/2
N2 - Environmental factors and random variation have strong effects on the dynamics of biological and ecological systems. In this paper, we propose a stochastic delay differential model of two-prey, one-predator system with cooperation among prey species against predator. The model has a global positive solution. Sufficient conditions of existence and uniqueness of an ergodic stationary distribution of the positive solution are provided, by constructing suitable Lyapunov functionals. Sufficient conditions for possible extinction of the predator populations are also obtained. The conditions are expressed in terms of a threshold parameter Rs 0 that relies on the environmental noise. Illustrative examples and numerical simulations, using Milstein's scheme, are carried out to illustrate the theoretical results. A small scale of noise can promote survival of the species. While relative large noises can lead to possible extinction of the species in such an environment.
AB - Environmental factors and random variation have strong effects on the dynamics of biological and ecological systems. In this paper, we propose a stochastic delay differential model of two-prey, one-predator system with cooperation among prey species against predator. The model has a global positive solution. Sufficient conditions of existence and uniqueness of an ergodic stationary distribution of the positive solution are provided, by constructing suitable Lyapunov functionals. Sufficient conditions for possible extinction of the predator populations are also obtained. The conditions are expressed in terms of a threshold parameter Rs 0 that relies on the environmental noise. Illustrative examples and numerical simulations, using Milstein's scheme, are carried out to illustrate the theoretical results. A small scale of noise can promote survival of the species. While relative large noises can lead to possible extinction of the species in such an environment.
KW - DDEs, Extinction
KW - Stationary distribution and ergodicity
KW - Stochastic DDEs
KW - Stochastic prey-predator model
KW - Time-delays
UR - http://www.scopus.com/inward/record.url?scp=85122587834&partnerID=8YFLogxK
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U2 - 10.3934/dcdss.2020468
DO - 10.3934/dcdss.2020468
M3 - Article
AN - SCOPUS:85122587834
SN - 1937-1632
VL - 15
SP - 245
EP - 263
JO - Discrete and Continuous Dynamical Systems - Series S
JF - Discrete and Continuous Dynamical Systems - Series S
IS - 2
ER -