A stochastic mutualism model with saturation effect and impulsive toxicant input in a polluted environment

Abstract

In this paper, we use a mean-reverting Ornstein–Uhlenbeck process to model the stochastic perturbations in the environment, and then a stochastic mutualism model with saturation effect and pulse toxicant input in a polluted environment is proposed. A set of sufficient conditions including exponential extinction, persistence in the mean, permanence in time average and stochastic permanence are derived. Numerical simulations are worked out to support the analysis results. Also, we look at the effects of the volatile intensity, reversion rate, impulsive period and impulsive toxicant input amount on the survival of two species.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7

References

  1. 1.

    Nelson, S.A.: The problem of oil population of the sea. Adv. Mar. Biol. 8, 215–306 (1971)

  2. 2.

    Jensen, A.L., Marshall, J.S.: Application of a surplus production model to assess environmental impacts on exploited populations of Daphnia pluex in the laboratory. Environ. Pollut. Ser. A 28, 273–280 (1982)

    Google Scholar 

  3. 3.

    Shukla, J.B., Freedman, H.I., Pal, V.M., Misra, O.P., Agarwal, M., Shukla, A.: Degradation and subsequent regeneration of a forestry resource: a mathematical model. Ecol. Model. 44, 219–229 (1989)

    Google Scholar 

  4. 4.

    Hallam, T.G., Clark, C.E., Lassiter, R.R.: Effects of toxicant on population: a qualitative approach I. Equilibrium environmental exposure. Ecol. Model. 18, 291–304 (1983)

    MATH  Google Scholar 

  5. 5.

    Hallam, T.G., Ma, Z.E.: Persistence in population models with demographic fluctuations. J. Math. Biol. 24, 327–339 (1986)

    MathSciNet  MATH  Google Scholar 

  6. 6.

    Ma, Z.E., Song, B.J., Hallam, T.G.: The threshold of survival for systems in a fluctuating environment. Bull. Math. Biol. 51, 311–323 (1989)

    MATH  Google Scholar 

  7. 7.

    Freedman, H.I., Shukla, J.B.: Models for the effect of toxicant in single-species and predator–prey systems. J. Math. Biol. 30, 15–30 (1991)

    MathSciNet  MATH  Google Scholar 

  8. 8.

    Thomas, D.M., Snell, T.W., Jaffer, S.M.: A control problem in a polluted environment. Math. Biosci. 133, 139–163 (1996)

    MATH  Google Scholar 

  9. 9.

    Chattopadhyay, J.: Effect of toxic substances on a two-species competitive system. Ecol. Model. 84, 287–289 (1996)

    Google Scholar 

  10. 10.

    He, J.W., Wang, K.: The survival analysis for a popuation in a polluted environment. Nonlinear Anal. RWA 10, 1555–1571 (2009)

    MATH  Google Scholar 

  11. 11.

    Dubey, B., Hussain, J.: Modelling the interaction of two biological species in a polluted environment. J. Math. Anal. Appl. 246, 58–79 (2000)

    MathSciNet  MATH  Google Scholar 

  12. 12.

    Johnston, E.L., Keough, M.J.: Field assessment of effects of timing and frequency of copper pulses on settlement of sessile marine invertebrates. Mar. Biol. 137, 1017–1029 (2000)

    Google Scholar 

  13. 13.

    Liu, B., Chen, L.S., Zhang, Y.J.: The effects of impulsive toxicant input on a population in a polluted environment. J. Biol. Syst. 11, 265–274 (2003)

    MATH  Google Scholar 

  14. 14.

    Yang, X.F., Jin, Z., Xue, Y.K.: Weak average persistence and extinction of a predator–prey system in a polluted environment with impulsive toxicant input. Chaos Solitons Fractals 31, 726–735 (2007)

    MathSciNet  MATH  Google Scholar 

  15. 15.

    Liu, B., Zhang, L.: Dynamics of a two-species Lotka–Volterra competition system in a polluted environment with pulse toxicant input. Appl. Math. Comput. 214, 155–162 (2009)

    MathSciNet  MATH  Google Scholar 

  16. 16.

    Jiao, J.J., Long, W., Chen, L.S.: A single stage-structured population model with mature individuals in a polluted environment and pulse input of environmental toxin. Nonlinear Anal. Real World Appl. 10, 3073–3081 (2009)

    MathSciNet  MATH  Google Scholar 

  17. 17.

    Wang, X.H., Jia, J.W.: Dynamic of a delayed predator–prey model with birth pulse and impulsive harvesting in a polluted environment. Phys. A 422, 1–15 (2015)

    MathSciNet  MATH  Google Scholar 

  18. 18.

    Li, D.M., Guo, T., Xu, Y.J.: The effects of impulsive toxicant input on a single-species population in a small polluted environment. Math. Biosci. Eng. 16, 8179–8194 (2019)

    MathSciNet  Google Scholar 

  19. 19.

    May, R.M.: Stability and Complexity in Model Ecosystems. Princeton University Press, Princeton (2001)

    Google Scholar 

  20. 20.

    Liu, M., Wang, K., Wu, Q.: Survival analysis of stochastic competitive models in a polluted environment and stochastic competitive exclusion principle. Bull. Math. Biol. 73, 1969–2012 (2011)

    MathSciNet  MATH  Google Scholar 

  21. 21.

    Han, Q.X., Jiang, D.Q., Ji, C.Y.: Analysis of a delayed stochastic predator–prey model in a polluted environment. Appl. Math. Model. 38, 3067–3080 (2014)

    MathSciNet  MATH  Google Scholar 

  22. 22.

    Liu, M., Bai, C.Z.: Persistence and extinction of a stochastic cooperative model in a polluted environment with pulse toxicant input. Filomat 29, 1329–1342 (2015)

    MathSciNet  MATH  Google Scholar 

  23. 23.

    Zhang, S.W., Tan, D.J.: Dynamics of a stochastic predator–prey system in a polluted environment with pulse toxicant input and impulsive perturbations. Appl. Math. Model. 39, 6319–6331 (2015)

    MathSciNet  MATH  Google Scholar 

  24. 24.

    Liu, Q., Chen, Q.M.: Dynamics of stochastic delay Lotka–Volterra systems with impulsive toxicant input and Lévy noise in polluted environments. Appl. Math. Comput. 256, 52–67 (2015)

    MathSciNet  MATH  Google Scholar 

  25. 25.

    Zhao, Y., Yuan, S.L., Ma, J.L.: Survival and stationary distribution analysis of a stochastic competitive model of three species in a polluted environment. Bull. Math. Biol. 77, 1285–1326 (2015)

    MathSciNet  MATH  Google Scholar 

  26. 26.

    Zhao, W.C., Li, J., Zhang, T.Q., Meng, X.Z., Zhang, T.H.: Persistence and ergodicty of plant disease model with Markov conversion and impulsive toxicant input. Commun. Nonlinear Sci. Numer. Simul. 48, 70–84 (2017)

    MathSciNet  MATH  Google Scholar 

  27. 27.

    Wei, F.Y., Geritz, S.A.H., Cai, J.Y.: A stochastic single-species population model with partial pollution tolerance in a polluted environment. Appl. Math. Lett. 63, 130–136 (2017)

    MathSciNet  MATH  Google Scholar 

  28. 28.

    Wei, F.Y., Chen, L.H.: Psychological effect on single-species population models in a polluted environment. Math. Biosci. 290, 22–30 (2017)

    MathSciNet  MATH  Google Scholar 

  29. 29.

    Zhao, Y., Yuan, S.L.: Optimal harvesting policy of a stochastic two-species competitive model with Lévy noise in a polluted environment. Phys. A 477, 20–33 (2017)

    MathSciNet  Google Scholar 

  30. 30.

    Liu, M., Du, C.X., Deng, M.L.: Persistence and extinction of a modified Leslie–Gower Holling-type II stochastic predator–prey model with impulsive toxicant input in polluted environments. Nonlinear Anal. Hybrid Syst. 27, 177–190 (2018)

    MathSciNet  MATH  Google Scholar 

  31. 31.

    Chi, M.N., Zhao, W.C.: Dynamical analysis of multi-nutrient and single microorganism chemostat model in a polluted environment. Adv. Differ. Equ. 120, 16 (2018)

    MathSciNet  MATH  Google Scholar 

  32. 32.

    Lv, X.J., Meng, X.Z., Wang, X.Z.: Extinction and stationary distribution of an impulsive stochastic chemostat model with nonlinear perturbation. Chaos Solitons Fractals 110, 273–279 (2018)

    MathSciNet  MATH  Google Scholar 

  33. 33.

    Gao, Y.X., Tian, S.Q.: Dynamics of a stochastic predator–prey model with two competitive preys and one predator in a polluted environment. Jpn. J. Ind. Appl. Math. 35, 861–889 (2018)

    MathSciNet  MATH  Google Scholar 

  34. 34.

    Yu, X.W., Yuan, S.L., Zhang, T.H.: Survival and ergodicity of a stochastic phytoplankton–zooplankton model with toxin-producing phytoplankton in an impulsive polluted environment. Appl. Math. Comput. 347, 249–264 (2019)

    MathSciNet  MATH  Google Scholar 

  35. 35.

    Lan, G.J., Wei, C.J., Zhang, S.W.: Long time behaviors of single-species population models with psychological effect and impulsive toxicant in polluted environments. Phys. A 521, 828–842 (2019)

    MathSciNet  Google Scholar 

  36. 36.

    Liu, G.D., Meng, X.Z.: Optimal harvesting strategy for a stochastic mutualism system in a polluted environment with regime switching. Phys. A 536, 12893 (2019)

    MathSciNet  Google Scholar 

  37. 37.

    Wang, H., Pan, F.M., Liu, M.: Survival analysis of a stochastic service-resource mutualism model in a polluted environment with pulse toxicant input. Phys. A 521, 591–606 (2019)

    MathSciNet  Google Scholar 

  38. 38.

    Allen, E.: Environmental variability and mean-reverting processes. Discrete Contin. Dyn. Syst. Ser. B 21, 2073–2089 (2016)

    MathSciNet  MATH  Google Scholar 

  39. 39.

    Higham, D.J.: An algorithmic introduction to numerical simulation of stochastic diffrential equations. SIAM Rev. 43, 525–546 (2001)

    MathSciNet  MATH  Google Scholar 

  40. 40.

    Liu, Q., Jiang, D.Q., Hayat, T., Alsaedi, A.: Asymptotic behavior of a food-limited Lotka–Volterra mutualism model with Markovian switching and Lévy jumps. Phys. A 505, 94–104 (2018)

    MathSciNet  Google Scholar 

  41. 41.

    Gao, H.J., Wang, Y.: Stochastic mutualism model under regime switching with Lévy jumps. Phys. A 515, 355–375 (2019)

    MathSciNet  Google Scholar 

  42. 42.

    Liu, C., Xun, X.Y., Zhang, Q.L., Li, Y.K.: Dynamical analysis and optimal control in a hybrid stochastic double delayed bioeconomic system with impulsive contaminants emission and Lévy jumps. Appl. Math. Comput. 352, 99–118 (2019)

    MathSciNet  MATH  Google Scholar 

  43. 43.

    Liu, G.D., Qi, H.K., Chang, Z.B., Meng, X.Z.: Asymptotic stability of a stochastic May mutualism system. Comput. Math. Appl. 79, 735–745 (2020)

    MathSciNet  MATH  Google Scholar 

  44. 44.

    Zhang, X.H., Jiang, D.Q.: Periodic solutions of a stochastic food-limited mutualism model. Mathodol. Comput. Appl. 22, 267–278 (2020)

    MathSciNet  MATH  Google Scholar 

  45. 45.

    Mao, X.R., Marion, G., Renshaw, E.: Environmental Brownian noise suppresses explosions in population dynamics. Stoch. Process. Appl. 97, 95–110 (2002)

    MathSciNet  MATH  Google Scholar 

  46. 46.

    Mao, X.R.: Stochastic Differential Equations and Applications. Horwood Publishing, Chichester (1997)

    Google Scholar 

Download references

Acknowledgements

The authors would like to thank the editor and referees for their valuable comments and suggestions, which greatly improve the presentation of this paper. The work is supported by the NNSF of China (Nos. 11871201, 11961023) and the NSF of Hubei Province, China (No. 2019CFB241).

Author information

Affiliations

Authors

Corresponding author

Correspondence to Zhijun Liu.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Appendices

Appendix A: The proof of Lemma 2.4

Consider the first two equations of model (1.6):

$$\begin{aligned} \left\{ \begin{array}{ll} du(t) =u(t)\left[ R_{1e}+(R_1(0)-R_{1e})e^{-\alpha _1t}-r_1C_0(t)\right. \\ \ \ \ \ \ \ \ \ \ \ -\left. a_1u(t)+\frac{b_1v(t)}{1+v(t)}\right] dt+\xi _1(t)u(t)dW_1(t),\\ dv(t)=v(t)\left[ R_{2e}+(R_2(0)-R_{2e})e^{-\alpha _2t}-r_2C_0(t)\right. \\ \ \ \ \ \ \ \ \ \ \ -\left. a_2v(t)+\frac{b_2u(t)}{1+u(t)}\right] dt+\xi _2(t)v(t)dW_2(t). \end{array}\right. \end{aligned}$$
(A.1)

By the theory of stochastic differential equation, (A.1) has a unique local solution (u(t),  v(t)) on \([0,~t_e)\), where \(t_e\) denotes the explosion time. If we can show that \(t_e=+\infty \) a.s., then \((u(t),~v(t))\in R_+^2\) a.s. for all \(t\ge 0\). To achieve this objective, let \(k_0>0\) be sufficiently large, and u(0) and v(0) lying in the interval \((1/{k_0},~k_0)\). Then define a stopping time as follows

$$\begin{aligned} t_k:=\inf \big \{t\in [0,t_e):u(t)\notin (\frac{1}{k_0},k_0),~v(t)\notin (\frac{1}{k_0},k_0)\big \}, \end{aligned}$$

where we set \(\inf \emptyset =+\infty \) (\(\emptyset \) denotes the empty set). Let \(t_{+\infty }=\lim _{k\rightarrow +\infty }t_k\), obviously, \(t_{+\infty }\le t_e\). If one can show that \(t_{+\infty }=+\infty \) a.s., then \(t_e=+\infty \) a.s. and \((u(t),~v(t))\in R_+^2\) a.s. for all \(t\ge 0\). To complete the proof, it is sufficient to show that \(t_{+\infty }=+\infty \) a.s.

Consider a \(C^2\)-function \(V:R_+^2\rightarrow R_+\) with the form \(V(u,v)=u-1-\ln u+v-1-\ln v\). An application of Itô’s formula can show that

$$\begin{aligned} \begin{array}{ll} dV(u,v)&{}=(1-\frac{1}{u})du+\frac{1}{2u^2}(du)^2+(1-\frac{1}{v})dv+\frac{1}{2v^2}(dv)^2\\ &{}=LV(u,v)dt+(1-\frac{1}{u})\xi _1(t)u(t)dW_1(t)\\ &{}\quad +\,(1-\frac{1}{v})\xi _2(t)v(t)dW_2(t), \end{array} \end{aligned}$$
(A.2)

where

$$\begin{aligned} \begin{array}{lll} \displaystyle LV &{}=(1-\frac{1}{u})u\left[ R_{1e}+(R_1(0)-R_{1e})e^{-\alpha _1t}-r_1C_0(t)-a_1u(t)+\frac{b_1v(t)}{1+v(t)}\right] +\frac{\xi _1^2(t)}{2}\\ &{}\quad +\,(1-\frac{1}{v})v\left[ R_{2e}+(R_2(0)-R_{2e})e^{-\alpha _2t}-r_2C_0(t)-a_2v(t)+\frac{b_2u(t)}{1+u(t)}\right] +\frac{\xi _2^2(t)}{2}\\ &{}\le -a_1u^2-\left[ R_{1e}-\frac{\xi _1^2(t)}{2}+(R_1(0)-R_{1e})e^{-\alpha _1t}-r_1C_0(t)\right] \\ &{}\quad +\,[R_{1e}+(R_1(0)-R_{1e})e^{-\alpha _1t}-r_1C_0(t)+a_1+b_1]u\\ &{}\quad -\,a_2v^2-\left[ R_{2e}-\frac{\xi _2^2(t)}{2}+(R_2(0)-R_{2e})e^{-\alpha _2t}-r_2C_0(t)\right] \\ &{}\quad +\,\left[ R_{2e}+(R_2(0)-R_{2e})e^{-\alpha _2t}-r_2C_0(t)+a_2+b_2\right] v\\ &{}\le -\left[ R_{1e}-\frac{\xi _1^2(t)}{2}+(R_1(0)-R_{1e})e^{-\alpha _1t}-r_1C_0(t)\right] \\ &{}\quad -\,\left[ R_{2e}-\frac{\xi _2^2(t)}{2}+(R_2(0)-R_{2e})e^{-\alpha _2t}-r_2C_0(t)\right] +Q_1\\ &{}=-\left[ R_{1e}-\frac{\beta _1^2}{4\alpha _1}(1-e^{-2\alpha _1t})+(R_1(0)-R_{1e})e^{-\alpha _1t}-r_1C_0(t)\right] \\ &{}\quad -\,\left[ R_{2e}-\frac{\beta _2^2}{4\alpha _2}(1-e^{-2\alpha _2t})+(R_2(0)-R_{2e})e^{-\alpha _2t}-r_2C_0(t)\right] +Q_1, \end{array} \end{aligned}$$
(A.3)

and

$$\begin{aligned} \begin{array}{lll} Q_1 &{}=\sup _{(u,v)\in R_+^2}\{-a_1u^2+[R_{1e}+(R_1(0)-R_{1e})e^{-\alpha _1t}-r_1C_0(t)+a_1+b_1]u\\ &{}\quad -\,a_2v^2+[R_{2e}+(R_2(0)-R_{2e})e^{-\alpha _2t}-r_2C_0(t)+a_2+b_2]v\}. \end{array} \end{aligned}$$

It is not difficult to prove that LV is bounded. Substituting (A.3) into (A.2) yields that

$$\begin{aligned} \begin{array}{ll} dV(u,v) &{}\le \left\{ -\left[ R_{1e}-\frac{\beta _1^2}{4\alpha _1}(1-e^{-2\alpha _1t})+(R_1(0)-R_{1e})e^{-\alpha _1t}-r_1C_0(t)\right] \right. \\ &{}\quad {-}\,\left. \left[ R_{2e}{-}\frac{\beta _2^2}{4\alpha _2}(1-e^{-2\alpha _2t})+(R_2(0)-R_{2e})e^{-\alpha _2t}-r_2C_0(t)\right] {+}Q_1\right\} dt\\ &{}\quad {+}\,\left( 1-\frac{1}{u}\right) \xi _1(t)udW_1(t)+\left( 1-\frac{1}{v}\right) \xi _2(t)vdW_2(t). \end{array} \end{aligned}$$

The rest of the proof can be obtained by almost the same method in [45], and the details are omitted. This completes the proof.

Appendix B: The proof of Theorem 3.1

From (2.1) and Lemma 2.2, for any \(\varepsilon >0\), there exists a constant \(T>0\) such that for all \(t\ge T\),

$$\begin{aligned} {\bar{q}}_i-\frac{\varepsilon }{2}\le \langle q_i(t)\rangle \le {\bar{q}}_i+\frac{\varepsilon }{2},\ {\tilde{C}}_0(t)-\frac{\varepsilon }{2r_i}\le C_0(t)\le {\tilde{C}}_0(t)+\frac{\varepsilon }{2r_i},\ i=1,2. \end{aligned}$$
(B.1)

Applying Itô’s formula to (1.6) gives

$$\begin{aligned} d\ln u(t)= & {} \left[ R_{1e}-\frac{1}{2}\xi _1^2(t)+(R_1(0)-R_{1e})e^{-\alpha _1t}-r_1C_0(t)-a_1u(t)\right. \\&+\left. \,\frac{b_1v(t)}{1+v(t)}\right] dt+\xi _1(t)dW_1 \end{aligned}$$

and

$$\begin{aligned} d\ln v(t)= & {} \left[ R_{2e}-\frac{1}{2}\xi _2^2(t)+(R_2(0)-R_{2e})e^{-\alpha _2t}-r_2C_0(t)-a_2v(t)\right. \\&+\left. \,\frac{b_2u(t)}{1+u(t)}\right] dt+\xi _2(t)dW_2. \end{aligned}$$

Integrating both sides from 0 to t, one has

$$\begin{aligned} \begin{array}{ll} \ln \frac{u(t)}{u(0)} &{}=\int _0^t[R_{1e}-\frac{1}{2}\xi _1^2(s)]ds+\int _0^t(R_1(0)-R_{1e})e^{-\alpha _1s}ds\\ &{}\quad -\,r_1\int _0^tC_0(s)ds-a_1\int _0^tu(s)ds+b_1\int _0^t\frac{v(s)}{1+v(s)}ds+\int _0^t\xi _1(s)dW_1(s) \end{array} \end{aligned}$$

and

$$\begin{aligned} \begin{array}{ll} \ln \frac{v(t)}{v(0)} &{}=\int _0^t[R_{2e}-\frac{1}{2}\xi _2^2(s)]ds+\int _0^t(R_2(0)-R_{2e})e^{-\alpha _2s}ds\\ &{}\quad -\,r_2\int _0^tC_0(s)ds-a_2\int _0^tv(s)ds+b_2\int _0^t\frac{u(s)}{1+u(s)}ds+\int _0^t\xi _2(s)dW_2(s). \end{array} \end{aligned}$$

Dividing t to both sides of above two equations yields that

$$\begin{aligned} \frac{1}{t}\ln \frac{u(t)}{u(0)}= & {} \big \langle q_1(t)\big \rangle +f_1(t)-r_1\big \langle C_0(t)\big \rangle -a_1\big \langle u(t)\big \rangle \nonumber \\&+\, b_1\big \langle \frac{v(t)}{1+v(t)}\big \rangle +\frac{1}{t}\int _0^t\xi _1(s)dW_1(s) \end{aligned}$$
(B.2)

and

$$\begin{aligned} \frac{1}{t}\ln \frac{v(t)}{v(0)}= & {} \big \langle q_2(t)\big \rangle +f_2(t)-r_2\big \langle C_0(t)\big \rangle -a_2\big \langle v(t)\big \rangle \nonumber \\&+\, b_2\big \langle \frac{u(t)}{1+u(t)}\big \rangle +\frac{1}{t}\int _0^t\xi _2(s)dW_2(s), \end{aligned}$$
(B.3)

where

$$\begin{aligned} f_i(t)=(R_i(0)-R_{ie})(1-e^{-\alpha _it})/\alpha _it,\ i=1,2. \end{aligned}$$

Obviously, \(\lim _{t\rightarrow +\infty }f_i(t)=0\).

  1. (i)

    Substituting (B.1) into (B.2), we can deduce that

    $$\begin{aligned} \begin{array}{lll} \frac{1}{t}\ln \frac{u(t)}{u(0)} &{}\le {\bar{q}}_1+\frac{\varepsilon }{2}+f_1(t)-r_1\langle {\tilde{C}}_0(t)-\frac{\varepsilon }{2r_1}\rangle +b_1+\frac{1}{t}\int _0^t\xi _1(s)dW_1(s)\\ &{}\le {\bar{q}}_1+\varepsilon +f_1(t)-r_1\langle {\tilde{C}}_0(t)\rangle +b_1+\frac{1}{t}\int _0^t\xi _1(s)dW_1(s). \end{array} \end{aligned}$$
    (B.4)

    Note that \(\lim _{t\rightarrow +\infty }f_i(t)=0\). Using the strong law of large number of local martingale (see [46]) yields \(\lim _{t\rightarrow +\infty }t^{-1}\int _0^t\xi _1(s)dW_1(s)=0\). It follows from (B.4) that \(\limsup _{t\rightarrow +\infty }t^{-1}\ln u(t) \le {\bar{q}}_1-r_1\delta +b_1\). If \({\bar{q}}_1-r_1\delta +b_1<0\), then u(t) will be extinct exponentially. Similarly, when \({\bar{q}}_2-r_2\delta +b_2<0\), specie v(t) is also extinct exponentially.

  2. (ii)

    If \({\bar{q}}_1-r_1\delta +b_1<0\), then it follows from (1) in Definition 2.1 that \(\lim _{t\rightarrow +\infty }u(t)=0\) a.s. Thus, for any \(\varepsilon >0\), there is a constant T such that for all \(t\ge T\),

    $$\begin{aligned} -\frac{\varepsilon }{2}\le b_2\langle \frac{u}{1+u}\rangle \le \frac{\varepsilon }{2},\ \ -\frac{\varepsilon }{2}\le t^{-1}\ln v(0)\le \frac{\varepsilon }{2}. \end{aligned}$$
    (B.5)

    Combining (B.1), (B.3) and (B.5) yields

    $$\begin{aligned} \begin{array}{lll} \ln v(t) &{}\le \frac{\varepsilon }{2}t+({\bar{q}}_2+\frac{\varepsilon }{2})t+tf_2(t)-r_2\langle {\tilde{C}}_0(t)-\frac{\varepsilon }{2r_2}\rangle t\\ &{}\quad -\,a_2\int _0^tv(s)ds+\frac{\varepsilon }{2}t+\int _0^t\xi _2(s)dW_2(s)\\ &{}\le tf_2(t)+({\bar{q}}_2-r_2\langle {\tilde{C}}_0(t)\rangle +2\varepsilon )t-a_2\int _0^tv(s)ds\\ &{}\quad +\,\int _0^t\xi _2(s)dW_2(s) \end{array} \end{aligned}$$
    (B.6)

    and

    $$\begin{aligned} \ln v(t)\ge & {} tf_2(t)+({\bar{q}}_2-r_2\langle {\tilde{C}}_0(t)\rangle -2\varepsilon )t -a_2\int _0^tv(s)ds\nonumber \\&+\,\int _0^t\xi _2(s)dW_2(s). \end{aligned}$$
    (B.7)

    Noticing that the function \(tf_i(t)=(R_i(0)-R_{ie})(1-e^{-\alpha _it})/\alpha _i\) is bounded as \(t\rightarrow +\infty \). Making use of (1) in Lemma 2.3 to (B.6), we can deduce that \(\langle v(t)\rangle ^*\le ({\bar{q}}_2-r_2\delta +2\varepsilon )/a_2\). Similarly, we have from (2) in Lemma 2.3 and (B.7) that \(\langle v(t)\rangle _*\ge ({\bar{q}}_2-r_2\delta -2\varepsilon )/a_2\). Since \(\varepsilon \) is arbitrary, we obtain a desired assertion that \(\lim _{t\rightarrow +\infty }\langle v(t)\rangle =({\bar{q}}_2-r_2\delta )/a_2\).

  3. (iii)

    The proof of (iii) coincides with that of (ii), and hence is omitted.

  4. (iv)

    Substituting (B.1) into (B.2), one has for \(t\ge T\),

    $$\begin{aligned} \ln u(t)\le & {} \ln u(0)+tf_1(t)+\big ({\bar{q}}_1-r_1\langle {\tilde{C}}_0(t)\rangle +b_1+\varepsilon \big )t\\&\quad -\,a_1\int _0^tu(s)ds+\int _0^t\xi _1(s)dW_1(s) \end{aligned}$$

    and

    $$\begin{aligned} \ln u(t)\ge & {} \ln u(0)+tf_1(t)+({\bar{q}}_1-r_1\langle {\tilde{C}}_0(t)\rangle -\varepsilon )t\\&\quad -\,a_1\int _0^tu(s)ds+\int _0^t\xi _1(s)dW_1(s). \end{aligned}$$

    Applying (1) and (2) in Lemma 2.3 to the above two inequalities, we can obtain that

    $$\begin{aligned} ({\bar{q}}_1-r_1\delta )/{a_1}\le \langle u(t)\rangle _*\le \langle u(t)\rangle ^*\le ({\bar{q}}_1-r_1\delta +b_1)/{a_1}. \end{aligned}$$

    Similarly, we can prove

    $$\begin{aligned} ({\bar{q}}_2-r_2\delta )/{a_2}\le \langle v(t)\rangle _*\le \langle v(t)\rangle ^*\le ({\bar{q}}_2-r_2\delta +b_2)/{a_2}, \end{aligned}$$

    which completes the claim.

Appendix C: The proof of Theorem 3.2

Let’s first prove \(\liminf _{t\rightarrow +\infty }{\mathcal {P}}\{\sqrt{u^2(t)+v^2(t)}\ge \varphi \}\ge 1-\varepsilon \). Define \(V_1(u,v)=1/(u+v)\). Applying Itô’s formula leads to

$$\begin{aligned} \begin{array}{ll} dV_1 &{}=-V_1^2(du+dv)+V_1^3(du+dv)^2\\ &{}=-V_1^2\big \{u(t)\left[ R_{1e}+(R_1(0)-R_{1e})e^{-\alpha _1t}-r_1C_0(t)-a_1u(t)+\frac{b_1v(t)}{1+v(t)}\right] \\ &{}\quad +\,\xi _1(t)u(t)dW_1(t)+v(t)\left[ R_{2e}+(R_2(0)-R_{2e})e^{-\alpha _2t}-r_2C_0(t)-a_2v(t)\right. \\ &{}\quad +\left. \,\frac{b_2u(t)}{1+u(t)}\right] dt+\xi _2(t)v(t)dW_2(t)\big \}+V_1^3[\xi _1^2(t)u^2(t)+\xi _2^2(t)v^2(t)]dt. \end{array} \end{aligned}$$

By the assumption of Theorem 3.2, we can choose a constant \(\mu >0\) such that

$$\begin{aligned} \min _{i=1,2}\{R_{ie}-r_i{\widetilde{C}}_0^*\}-\frac{1}{2}\max _{i=1,2}\left\{ \frac{\beta _i^2}{2\alpha _i}(1-e^{-2\alpha _it})\right\} >\frac{\mu }{2}\max _{i=1,2}\left\{ \frac{\beta _i^2}{2\alpha _i}(1-e^{-2\alpha _it})\right\} . \end{aligned}$$

Define \(V_2=(1+V_1)^{\mu }\). By Itô’s formula we get that

$$\begin{aligned} \begin{array}{ll} dV_2 &{}=\mu (1+V_1)^{\mu -1}dV_1+\frac{1}{2}\mu (\mu -1)(1+V_1)^{\mu -2}(dV_1)^2\\ &{}=\mu (1+V_1)^{\mu -2}[(1+V_1)dV_1+\frac{1}{2}(\mu -1)(dV_1)^2]\\ &{}=\mu (1+V_1)^{\mu -2}\left\{ -(1+V_1)V_1^2\left[ u\left( R_{1e}+(R_1(0)-R_{1e})e^{-\alpha _1t}-r_1C_0(t)-a_1u(t)\right. \right. \right. \\ &{}\quad +\left. \left. \,\frac{b_1v(t)}{1+v(t)}\right) +v\left( R_{2e}+(R_2(0)-R_{2e})e^{-\alpha _2t}-r_2C_0(t)-a_2v(t)+\frac{b_2u(t)}{1+u(t)}\right) \right] \\ &{}\quad +\left. \,(1+V_1)V_1^3[\xi _1^2(t)u^2(t)+\xi _2^2(t)v^2(t)]+\frac{1}{2}(\mu -1)V_1^4[\xi _1^2(t)u^2(t)+\xi _2^2(t)v^2(t)]\right\} dt\\ &{}\quad -\,\mu (1+V_1)^{\mu -1}V_1^2\big (\xi _1(t)u(t)dW_1+\xi _2(t)v(t)dW_2\big )\\ &{}=\mu (1+V_1)^{\mu -2}J_1(u,v)dt-\mu (1+V_1)^{\mu -1}V_1^2\big (\xi _1(t)u(t)dW_1+\xi _2(t)v(t)dW_2\big ), \end{array} \end{aligned}$$

where

$$\begin{aligned} \begin{array}{ll} \displaystyle J_1(u,v) &{}\displaystyle =-(1+V_1)V_1^2\left[ u\left( R_{1e}+(R_1(0)-R_{1e})e^{-\alpha _1t}-r_1C_0(t)-a_1u(t)\right. \right. \\ &{}\displaystyle \ \ \ +\left. \frac{b_1v(t)}{1+v(t)}\right) +\left. v\left( R_{2e}+(R_2(0)-R_{2e})e^{-\alpha _2t}-r_2C_0(t)-a_2v(t)+\frac{b_2u(t)}{1+u(t)}\right) \right] \\ &{}\displaystyle \ \ \ +V_1^3[\xi _1^2(t)u^2(t)+\xi _2^2(t)v^2(t)]+\frac{\mu +1}{2}V_1^4[\xi _1^2(t)u^2(t)+\xi _2^2(t)v^2(t)]\\ &{}\displaystyle \le -(1+V_1)V_1^2[u\big (R_{1e}+(R_1(0)-R_{1e})e^{-\alpha _1t}-r_1{\widetilde{C}}_0^*(t)-\varepsilon \big )\\ &{}\displaystyle \ \ \ +v\big (R_{2e}+(R_2(0)-R_{2e})e^{-\alpha _2t}-r_2{\widetilde{C}}_0^*(t)-\varepsilon \big )]\\ &{}\displaystyle \ \ \ -(1+V_1)V_1^2\left( -a_1u^2(t)+u(t)\frac{b_1v(t)}{1+v(t)}-a_2v^2(t)+v(t)\frac{b_2u(t)}{1+u(t)}\right) \\ &{}\displaystyle \ \ \ +V_1\max \limits _{i=1,2}\left\{ \frac{\beta _i^2}{2\alpha _i}(1-e^{-2\alpha _it})\right\} +\frac{\mu +1}{2}V_1^2\max \limits _{i=1,2}\left\{ \frac{\beta _i^2}{2\alpha _i}(1-e^{-2\alpha _it})\right\} \\ &{}\displaystyle \le -(1+V_1)V_1^2\left[ \min _{i=1,2}\{R_{ie}+(R_i(0)-R_{ie})e^{-\alpha _it}-r_i{\widetilde{C}}_0^*-\varepsilon \}(u(t)+v(t))\right] \\ &{}\displaystyle \ \ \ +(1+V_1)V_1^2\left( a_1u^2(t)-u\frac{b_1v(t)}{1+v(t)}+a_2v^2(t)-v(t)\frac{b_2u(t)}{1+u(t)}\right) \\ &{}\displaystyle \ \ \ +V_1\max \limits _{i=1,2}\left\{ \frac{\beta _i^2}{2\alpha _i}(1-e^{-2\alpha _it})\right\} +\frac{\mu +1}{2}V_1^2\max \limits _{i=1,2}\left\{ \frac{\beta _i^2}{2\alpha _i}(1-e^{-2\alpha _it})\right\} \\ &{}\displaystyle \le \max \{a_1,a_2\}-V_1^2\left[ \min _{i=1,2}\left\{ R_{ie}-r_i{\widetilde{C}}_0^*-\varepsilon \right\} -\frac{\mu +1}{2}\max \limits _{i=1,2}\left\{ \frac{\beta _i^2}{2\alpha _i}(1-e^{-2\alpha _it})\right\} \right. \\ &{}\displaystyle \ \ \ +\left. \min _{i=1,2}\left\{ (R_i(0)-R_{ie})e^{-\alpha _it}\right\} \right] +V_1\left( \max \limits _{i=1,2}\left\{ \frac{\beta _i^2}{2\alpha _i}(1-e^{-2\alpha _it})\right\} +\max \{a_1,a_2\}\right) . \end{array} \end{aligned}$$

Assign \(\kappa >0\) sufficiently small such that

$$\begin{aligned} \min _{i=1,2}\{R_{ie}-r_i{\widetilde{C}}_0^*-\varepsilon \}-\frac{1+\mu }{2}\max _{i=1,2}\left\{ \frac{\beta _i^2}{2\alpha _i}(1-e^{-2\alpha _it})\right\}>\frac{\kappa }{\mu }>0. \end{aligned}$$
(C.1)

We continue to define \(V_3=e^{\kappa t}V_2\). The Itô’s formula then gives that

$$\begin{aligned} \begin{array}{ll} \displaystyle dV_3 &{}\displaystyle =\kappa e^{\kappa t}V_2dt+e^{\kappa t}dV_2\\ &{}\displaystyle \le \kappa e^{\kappa t}(1+V_1)^{\mu }dt+e^{\kappa t}\mu (1+V_1)^{\mu -2}\left\{ \max \{a_1,a_2\} +V_1\left( \max \limits _{i=1,2}\left\{ \frac{\beta _i^2}{2\alpha _i}(1-e^{-2\alpha _it})\right\} \right. \right. \\ &{}\displaystyle \ \ \ +\left. \max \{a_1,a_2\}\right) -V_1^2\left[ \min _{i=1,2}\{R_{ie}-r_i{\widetilde{C}}_0^*-\varepsilon \} -\frac{\mu +1}{2}\max \limits _{i=1,2}\left\{ \frac{\beta _i^2}{2\alpha _i}(1-e^{-2\alpha _it})\right\} \right. \\ &{}\displaystyle \ \ \ +\left. \left. \min _{i=1,2}\{(R_i(0)-R_{ie})e^{-\alpha _it}\}\right] \right\} dt-e^{\kappa t}\mu (1+V_1)^{\mu -1}V_1^2\big (\xi _1(t)u(t)dW_1+\xi _2(t)v(t)dW_2\big )\\ &{}\displaystyle \le e^{\kappa t}(1+V_1)^{\mu -2}\left\{ \kappa (1+V_1)^2-\mu V_1^2\left[ \min _{i=1,2}\{R_{ie}-r_i{\widetilde{C}}_0^*-\varepsilon \}\right. \right. \\ &{}\displaystyle \ \ \ -\left. \frac{\mu +1}{2}\max \limits _{i=1,2}\left\{ \frac{\beta _i^2}{2\alpha _i}(1-e^{-2\alpha _it})\right\} +\min _{i=1,2}\{(R_i(0)-R_{ie})e^{-\alpha _it}\}\right] \\ &{}\displaystyle \ \ \ +\left. \mu V_1\left( \max \limits _{i=1,2}\{\frac{\beta _i^2}{2\alpha _i}(1-e^{-2\alpha _it})\}+\max \{a_1,a_2\}\right) +\mu \max \{a_1,a_2\}\right\} dt\\ &{}\displaystyle \ \ \ -\mu e^{\kappa t}(1+V_1)^{\mu -2}V_1^2\big (\xi _1(t)u(t)dW_1+\xi _2(t)v(t)dW_2\big )\\ &{}\displaystyle =e^{\kappa t}J_2(u,v)dt-\mu e^{\kappa t}(1+V_1)^{\mu -2}V_1^2\big (\xi _1(t)u(t)dW_1+\xi _2(t)v(t)dW_2\big ), \end{array} \end{aligned}$$

where

$$\begin{aligned} \begin{array}{ll} \displaystyle J_2(u,v) &{}\displaystyle =\mu (1+V_1)^{\mu -2}\left\{ \frac{\kappa }{\mu }(1+V_1)^2-V_1^2\left[ \min _{i=1,2}\{R_{ie}-r_i{\widetilde{C}}_0^*-\varepsilon \}\right. \right. \\ &{}\displaystyle \ \ \ -\left. \frac{\mu +1}{2}\max \limits _{i=1,2}\left\{ \frac{\beta _i^2}{2\alpha _i}(1-e^{-2\alpha _it})\right\} +\min _{i=1,2}\left\{ (R_i(0)-R_{ie})e^{-\alpha _it}\right\} \right] \\ &{}\displaystyle \ \ \ +\left. V_1\left( \max \limits _{i=1,2}\left\{ \frac{\beta _i^2}{2\alpha _i}(1-e^{-2\alpha _it})\right\} +\max \{a_1,a_2\}\right) +\max \{a_1,a_2\}\right\} . \end{array} \end{aligned}$$

It is easy to verify that \(J_2(u,v)\) is upper bounded as \(t\rightarrow +\infty \), denoted as \(J:=\sup _{(u,v)\in R_+^2}J_2(u,v)<+\infty \). Thus,

$$\begin{aligned} dV_3\le e^{\kappa t}Jdt-\mu e^{\kappa t}(1+V_1)^{\mu -2}V_1^2(\xi _1udW_1+\xi _2vdW_2). \end{aligned}$$
(C.2)

Integrating both sides of (C.2) and then taking expectation, one has

$$\begin{aligned} {\mathbb {E}}[V_3(u(t),v(t))]={\mathbb {E}}[e^{\kappa t}(1+V_1)^{\mu }]\le (1+V_1(u(0),v(0)))^{\mu }+\frac{J}{\kappa }e^{\kappa t}. \end{aligned}$$

Dividing both sides of \(e^{\kappa t}\) gives that

$$\begin{aligned} \limsup _{t\rightarrow +\infty }{\mathbb {E}}[V_1^{\mu }]\le \limsup _{t\rightarrow +\infty }{\mathbb {E}}[(1+V_1)^{\mu }]\le \frac{J}{\kappa }. \end{aligned}$$

Therefore

$$\begin{aligned} (u+v)^{\mu }\le 2^{\mu }(\max {\{u,v\}})^{\mu }=2^{\mu }(\max {\{u^2,v^2\}})^{\frac{\mu }{2}}\le 2^{\mu }\Big (\sqrt{u^2(t)+v^2(t)}\Big )^{\mu }, \end{aligned}$$

from which we conclude that

$$\begin{aligned} \limsup _{t\rightarrow +\infty }{\mathbb {E}}[\frac{1}{2^{\mu }\big (\sqrt{u^2(t)+v^2(t)}\big )^{\mu }}]\le \limsup _{t\rightarrow +\infty }{\mathbb {E}}[V_1^{\mu }]\le \frac{J}{\kappa }. \end{aligned}$$

Then

$$\begin{aligned} \limsup \limits _{t\rightarrow +\infty }{\mathbb {E}}[\frac{1}{\sqrt{u^2(t)+v^2(t)}^{\mu }}]\le 2^{\mu }\frac{J}{\kappa }:=K. \end{aligned}$$

For any given \(\varepsilon >0\), let \(\varphi =\varepsilon ^{\frac{1}{\mu }}/K^{\frac{1}{\mu }}\), by Chebyshev’s inequality (see [46]), we have

$$\begin{aligned} {\mathcal {P}}\big (\sqrt{u^2(t)+v^2(t)}<\varphi \big ){=}{\mathcal {P}}\{\sqrt{u^2(t){+}v^2(t)}^{{-}\mu }>\varphi ^{-\mu }\} \le {\mathbb {E}}[\sqrt{u^2(t)+v^2(t)}^{-\mu }]/\varphi ^{-\mu }, \end{aligned}$$

and then \(\limsup _{t\rightarrow +\infty }{\mathcal {P}}\{\sqrt{u^2(t)+v^2(t)}<\varphi \}\le \varphi ^{\mu }K=\varepsilon \). Thus, a desirable result,

\(\liminf _{t\rightarrow +\infty }{\mathcal {P}}\{\sqrt{u^2(t)+v^2(t)}\ge \varphi \}\ge 1-\varepsilon \), is obtained. Next, we will verify that \(\liminf _{t\rightarrow +\infty }{\mathcal {P}}\{\sqrt{u^2(t)+v^2(t)}\le \phi \}\ge 1-\varepsilon \). Define \(V_4(u,v)=u^{\theta }+v^{\theta }\), \(\theta >1\). Using Itô’s formula we have

$$\begin{aligned} \begin{array}{lll} dV_4 &{}=\theta u^{\theta -1}du+\frac{1}{2}\theta (\theta -1)u^{\theta -2}(du)^2 +\theta v^{\theta -1}dv+\frac{1}{2}\theta (\theta -1)v^{\theta -2}(dv)^2\\ &{}=\theta u^\theta \left[ R_{1e}+(R_1(0)-R_{1e})e^{-\alpha _1t} -r_1C_0(t)-a_1u(t)+\frac{b_1v(t)}{1+v(t)}+\frac{1}{2}(\theta -1)\xi _1^2(t)\right] dt\\ &{}\quad +\,\theta v^\theta \left[ R_{2e}+(R_2(0)-R_{2e})e^{-\alpha _2t} -r_2C_0(t)-a_2v(t)+\frac{b_2u(t)}{1+v(t)}+\frac{1}{2}(\theta -1)\xi _2^2(t)\right] dt\\ &{}\quad +\,\theta u^\theta \xi _1(t)dW_1(t)+\theta v^\theta \xi _2(t)dW_2(t)\\ &{}\le \theta u^\theta \big [R_{1e}+(R_1(0)-R_{1e})e^{-\alpha _1t} -a_1u+b_1+\frac{1}{2}(\theta -1)\xi _1^2(t)\big ]dt\\ &{}\quad +\,\theta v^\theta \big [R_{2e}+(R_2(0)-R_{2e})e^{-\alpha _2t} -a_2v+b_2+\frac{1}{2}(\theta -1)\xi _2^2(t)\big ]dt\\ &{}\quad +\,\theta u^\theta \xi _1(t)dW_1(t)+\theta v^\theta \xi _2(t)dW_2(t). \end{array} \end{aligned}$$

We finally define \(V_5(u,v)=e^tV_4\). Applying Itô’s formula, one can show that

$$\begin{aligned} \begin{array}{lll} dV_5 &{}=e^tV_4dt+e^tdV_4\\ &{}\le e^t(u^\theta +v^\theta )dt+e^t\big \{\theta u^\theta \left[ R_{1e}+(R_1(0)-R_{1e})e^{-\alpha _1t}-a_1u+b_1\right. \\ &{}\quad +\left. \,\frac{1}{2}(\theta -1)\xi _1^2(t)\right] dt+\theta v^\theta \left[ R_{2e}+(R_2(0)-R_{2e})e^{-\alpha _2t}-a_2v+b_2\right. \\ &{}\quad +\left. \,\frac{1}{2}(\theta -1)\xi _2^2(t)\right] dt+\theta u^\theta \xi _1(t)dW_1(t)+\theta v^\theta \xi _2(t)dW_2(t)\big \}\\ &{}\le e^t\left\{ \theta u^\theta \left[ R_{1e}+(R_1(0)-R_{1e})e^{-\alpha _1t} -a_1u+b_1+\frac{1}{2}(\theta -1)\xi _1^2(t)+\frac{1}{\theta }\right] \right. \\ &{}\quad +\,\theta v^\theta \left[ R_{2e}+(R_2(0)-R_{2e})e^{-\alpha _2t} \left. -a_2v+b_2+\frac{1}{2}(\theta -1)\xi _2^2(t)+\frac{1}{\theta }\right] dt\right\} \\ &{}\quad +\,\theta e^t(u^\theta \xi _1(t)dW_1(t)+v^\theta \xi _2(t)dW_2(t))\\ &{}=e^tJ_3(u,v)dt+\theta e^t(u^\theta \xi _1(t)dW_1(t)+v^\theta \xi _2(t)dW_2(t)), \end{array} \end{aligned}$$
(C.3)

where

$$\begin{aligned} \begin{array}{lll} J_3(u,v) &{}=-a_1\theta u^{\theta +1}+u^\theta \theta \left[ R_{1e}+(R_1(0)-R_{1e})e^{-\alpha _1t} +b_1+\frac{1}{2}(\theta -1)\xi _1^2(t)+\frac{1}{\theta }\right] \\ &{}\quad -\,a_2\theta v^{\theta +1}+v^\theta \theta \left[ R_{2e}+(R_2(0)-R_{2e})e^{-\alpha _2t} +b_2+\frac{1}{2}(\theta -1)\xi _2^2(t)+\frac{1}{\theta }\right] . \end{array} \end{aligned}$$

It is easy to know that \(J_3(u,v)\) is upper bounded, denoted as \(K_1:=\sup _{(u,v)\in R_+^2}J_3<+\infty \).

Integrating both sides of (C.3) shows that

$$\begin{aligned} e^tV_4(u(t),v(t))-V_4(u(0),v(0))\le & {} \int _0^te^sK_1ds+\theta \int _0^te^su^\theta (s)\xi _1(s)dW_1\\&\quad +\theta \int _0^te^sv^\theta (s)\xi _2(s)dW_2. \end{aligned}$$

Taking expectation, one has

$$\begin{aligned} {\mathbb {E}}[e^tV_4]-V_4(u(0),v(0))\le K_1{\mathbb {E}}[e^t]. \end{aligned}$$

Namely, \(\limsup _{t\rightarrow +\infty }{\mathbb {E}}[u^\theta +v^\theta ]\le K_1\). Assign \(\phi =(2^{\theta -1}K_1)^\frac{1}{\theta }/\varepsilon ^{\frac{1}{\theta }}\), we derive, by Chebyshev’s inequality (see [46]) again, that

$$\begin{aligned} {\mathcal {P}}\{\sqrt{u^2(t)+v^2(t)}^\theta >\phi ^\theta \}\le \phi ^{-\theta }{\mathbb {E}} [\sqrt{u^2(t)+v^2(t)}^\theta ]\le \phi ^{-\theta }{\mathbb {E}}[(u+v)^\theta ]\le 2^{\theta -1}\phi ^{-\theta }{\mathbb {E}}[u^\theta +v^\theta ]. \end{aligned}$$

Then we know that \(\limsup _{t\rightarrow +\infty }{\mathcal {P}}\{\sqrt{u^2(t)+v^2(t)}>\phi \}\le \varepsilon \). So the required assertion follows as \(\liminf _{t\rightarrow +\infty }{\mathcal {P}}\{\sqrt{u^2(t)+v^2(t)}\le \phi \}\ge 1-\varepsilon \).

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Ning, W., Liu, Z., Wang, L. et al. A stochastic mutualism model with saturation effect and impulsive toxicant input in a polluted environment. J. Appl. Math. Comput. 65, 177–197 (2021). https://doi.org/10.1007/s12190-020-01387-8

Download citation

Keywords

  • Impulsive stochastic mutualism model
  • Polluted environment
  • Ornstein–Uhlenbeck process
  • Permanence
  • Extinction

Mathematics Subject Classification

  • 92D25
  • 34A37
  • 60H10