# American Institute of Mathematical Sciences

June  2017, 22(4): 1493-1508. doi: 10.3934/dcdsb.2017071

## Optimal harvesting of a stochastic delay competitive model

 1 School of Mathematical Science, Huaiyin Normal University, Huaian 223300, China 2 School of Mathematics and Statistics, Northeast Normal University, Jilin 130024, China

Received  November 2014 Revised  April 2016 Published  February 2017

Fund Project: This research is supported by National Natural Science Foundation of China (Nos. 11301207, 11571136, and 11171081), Natural Science Foundation of Jiangsu Province (Nos. BK2011407 and BK20130411), Qing Lan Project of Jiangsu Province (2014), Project Funded by China Post-doctoral Science Foundation (2015M571349, 2016T90236), Jiangsu Province “333 High-Level Personnel Training Project”, Science and Technology Support Plan Project of Huaian (HAR2015013).

In this paper an $n$-species stochastic delay competitive model with harvesting is proposed. Some dynamical properties of the model are considered. We first establish sufficient conditions for persistence in the mean of the species. Then asymptotic stability in distribution of the harvesting model is studied. Next the optimal harvesting effort and the maximum harvesting yield are given by using the ergodic approach. Finally the analytical results are illustrated through simulation figures using MATLAB followed by discussions and conclusions.

Citation: Meng Liu, Chuanzhi Bai. Optimal harvesting of a stochastic delay competitive model. Discrete & Continuous Dynamical Systems - B, 2017, 22 (4) : 1493-1508. doi: 10.3934/dcdsb.2017071
##### References:
 [1] L. H. R. Alvarez and L. A. Shepp, Optimal harvesting of stochastically fluctuating populations, J. Math. Biol., 37 (1998), 155-177. doi: 10.1007/s002850050124. Google Scholar [2] A. Bahar and X. Mao, Stochastic delay population dynamics, Int. J. Pure Appl. Math., 11 (2004), 377-400. Google Scholar [3] J. Bao, Z. Hou and C. Yuan, Stability in distribution of neutral stochastic differential delay equations with Markovian switching, Statist. Probab. Lett., 79 (2009), 1663-1673. doi: 10.1016/j.spl.2009.04.006. Google Scholar [4] J. Bao, X. Mao, G. Yin and C. Y. uan, Competitive Lotka-Volterra population dynamics with jumps, Nonlinear Anal., 74 (2011), 6601-6616. doi: 10.1016/j.na.2011.06.043. Google Scholar [5] 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. Google Scholar [6] I. Barbalat, Systems dequations differentielles d'osci d'oscillations nonlineaires, Revue Roumaine de Mathematiques Pures et Appliquees, 4 (1959), 267-270. Google Scholar [7] J. R. Beddington and R. M. May, Harvesting natural populations in a randomly fluctuating environment, Science, 197 (1977), 463-465. Google Scholar [8] D. K. Bhattacharya and S. Begum, Bionomic equilibrium of two species system, Math. Biosci., 135 (1996), 111-127. Google Scholar [9] C. A. Braumann, Variable effort harvesting models in random environments: Generalization to density-dependent noise intensities, Math. Biosci., 177&178 (2002), 229-245. doi: 10.1016/S0025-5564(01)00110-9. Google Scholar [10] N. Bruti-Liberati and E. Platen, Monte Carlo simulation for stochastic differential equations, Encyclopedia of Quantitative Finance, 10 (2010), 23-37. doi: 10.1080/14697680902814233. Google Scholar [11] K. S. Chaudhuri and S. Saha Roy, On the combined harvesting of a prey-predator system, J. Biol. Syst., 4 (1996), 376-389. Google Scholar [12] C. W. Clark, Bioeconomic Modelling and Fisheries Management, Wiley, New York, 1985. Google Scholar [13] H. Crauel and M. Gundlach, Stochastic Dynamics, Springer-Verlag, New York, 1999. doi: 10.1007/b97846. Google Scholar [14] T. C. Gard, Stability for multispecies population models in random environments, Nonlinear Anal., 10 (1986), 1411-1419. doi: 10.1016/0362-546X(86)90111-2. Google Scholar [15] K. Gopalsamy, Stability and Oscillations in Delay Differential Equations of Population Dynamics, Kluwer Academic, Dordrecht, 1992. doi: 10.1007/978-94-015-7920-9. Google Scholar [16] D. J. Higham, An algorithmic introduction to numerical simulation of stochastic differential equations, SIAM Rev., 43 (2001), 525-546. doi: 10.1137/S0036144500378302. Google Scholar [17] D. Jiang, C. Ji, X. Li and D. O'Regan, Analysis of autonomous Lotka-Volterra competition systems with random perturbation, J. Math. Anal. Appl., 390 (2012), 582-595. doi: 10.1016/j.jmaa.2011.12.049. Google Scholar [18] D. Jiang and N. Shi, A note on nonautonomous logistic equation with random perturbation, J. Math. Anal. Appl., 303 (2005), 164-172. doi: 10.1016/j.jmaa.2004.08.027. Google Scholar [19] T. K. Kar, Influence of environmental noises on the Gompertz model of two species fishery, Ecological Modelling, 17 (2004), 251-272. Google Scholar [20] Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Academic Press, Boston, 1993. Google Scholar [21] R. Lande, S. Engen and B. E. Saeher, Optimal harvesting of fluctuating populations with a risk of extinction, Am. Nat., 145 (1995), 728-745. Google Scholar [22] W. Li and K. Wang, Optimal harvesting policy for general stochastic Logistic population model, J. Math. Anal. Appl., 368 (2010), 420-428. doi: 10.1016/j.jmaa.2010.04.002. Google Scholar [23] M. Liu and C. Bai, Optimal harvesting of a stochastic logistic model with time delay, J. Nonlinear Sci., 25 (2015), 277-289. doi: 10.1007/s00332-014-9229-2. Google Scholar [24] M. Liu and C. Bai, Optimal harvesting of a stochastic mutualism model with Lévy jumps, Appl. Math. Comput., 276 (2016), 301-309. doi: 10.1016/j.amc.2015.11.089. Google Scholar [25] M. Liu and C. Bai, Analysis of a stochastic tri-trophic food-chain model with harvesting, J. Math. Biol., 73 (2016), 597-625. doi: 10.1007/s00285-016-0970-z. Google Scholar [26] M. Liu and M. Fan, Permanence of stochastic lotka-volterra systems, J. Nonlinear Sci. , (2016). doi: 10.1007/s00332-016-9337-2. Google Scholar [27] M. Liu and M. Fan, Stability in distribution of a three-species stochastic cascade predator-prey system with time delays, IMA J. Appl. Math. , (2016). doi: 10.1093/imamat/hxw057. Google Scholar [28] D. Ludwig and J. M. Varah, Optimal harvesting of a randomly fluctuating resource Ⅱ: Numerical methods and results, SIAM J. Appl. Math., 37 (1979), 185-205. doi: 10.1137/0137012. Google Scholar [29] E. M. Lungu and B. $\emptyset$ksendal, Optimal harvesting from a population in a stochastic crowded environment, Math. Biosci., 145 (1997), 47-75. doi: 10.1016/S0025-5564(97)00029-1. Google Scholar [30] Y. Lv, R. Yuan and Y. Pei, Dynamics in two nonsmooth predator-preymodels with threshold harvesting, Nonlinear Dyn., 74 (2013), 107-132. doi: 10.1007/s11071-013-0952-2. Google Scholar [31] X. Mao and C. Yuan, Stochastic Differential Equations with Markovian Switching, Imperial College Press, London, 2006. doi: 10.1142/p473. Google Scholar [32] R. M. May, Stability and Complexity in Model Ecosystems, Princeton University Press, New York, 2001.Google Scholar [33] M. Mesterton-Gibbons, On the optimal policy for combined harvesting of predator and prey, Nat. Resour. Model. , 3 (1988), 303. Google Scholar [34] D. Pal, G. S. Mahaptra and G. P. Samanta, Optimal harvesting of prey-predator system with interval biological parameters: A bioeconomic model, Math. Biosci., 241 (2013), 181-187. doi: 10.1016/j.mbs.2012.11.007. Google Scholar [35] D. Prato and J. Zabczyk, Ergodicity for Infinite Dimensional Systems, Cambridge University Press, Cambridge, 1996. Google Scholar [36] D. L. Ragogin and G. Brown, Harvest polices and non-market valuation in a predator prey system, J. Environ. Econ. Manag., 12 (1985), 155-168. Google Scholar [37] D. Ryan and F. Hanson, Optimal harvesting of a logistic population in an environment with stochastic jumps, J. Math. Biol., 24 (1986), 259-277. doi: 10.1007/BF00275637. Google Scholar [38] Q. S. Song, R. Stockbridge and C. Zhu, On optimal harvesting problems in random environments, SIAM J. Control Optim., 49 (2011), 859-889. doi: 10.1137/100797333. Google Scholar [39] Y. Zhang and Q. Zhang, Dynamic behavior in a delayed stage-structured population model with stochastic fluctuation and harvesting, Nonlinear Dynam., 66 (2011), 231-245. doi: 10.1007/s11071-010-9923-z. Google Scholar [40] C. Zhu and G. Yin, On hybrid competitive Lotka-Volterra ecosystems, Nonlinear Anal., 71 (2009), e1370-e1379. doi: 10.1016/j.na.2009.01.166. Google Scholar [41] X. Zou, W. Li and K. Wang, Ergodic method on optimal harvesting for a stochastic Gompertz-type diffusion process, Appl. Math. Lett., 26 (2013), 170-174. doi: 10.1016/j.aml.2012.08.006. Google Scholar [42] X. Zou and K. Wang, Optimal harvesting for a Logistic population dynamics driven by a Lévy process, J. Optim. Theory Appl., 161 (2014), 969-979. doi: 10.1007/s10957-013-0451-0. Google Scholar [43] X. Zou and K. Wang, Optimal harvesting for a stochastic regime-switching logistic diffusion system with jumps, Nonlinear Anal. Hybrid Syst., 13 (2014), 32-44. doi: 10.1016/j.nahs.2014.01.001. Google Scholar

show all references

##### References:
 [1] L. H. R. Alvarez and L. A. Shepp, Optimal harvesting of stochastically fluctuating populations, J. Math. Biol., 37 (1998), 155-177. doi: 10.1007/s002850050124. Google Scholar [2] A. Bahar and X. Mao, Stochastic delay population dynamics, Int. J. Pure Appl. Math., 11 (2004), 377-400. Google Scholar [3] J. Bao, Z. Hou and C. Yuan, Stability in distribution of neutral stochastic differential delay equations with Markovian switching, Statist. Probab. Lett., 79 (2009), 1663-1673. doi: 10.1016/j.spl.2009.04.006. Google Scholar [4] J. Bao, X. Mao, G. Yin and C. Y. uan, Competitive Lotka-Volterra population dynamics with jumps, Nonlinear Anal., 74 (2011), 6601-6616. doi: 10.1016/j.na.2011.06.043. Google Scholar [5] 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. Google Scholar [6] I. Barbalat, Systems dequations differentielles d'osci d'oscillations nonlineaires, Revue Roumaine de Mathematiques Pures et Appliquees, 4 (1959), 267-270. Google Scholar [7] J. R. Beddington and R. M. May, Harvesting natural populations in a randomly fluctuating environment, Science, 197 (1977), 463-465. Google Scholar [8] D. K. Bhattacharya and S. Begum, Bionomic equilibrium of two species system, Math. Biosci., 135 (1996), 111-127. Google Scholar [9] C. A. Braumann, Variable effort harvesting models in random environments: Generalization to density-dependent noise intensities, Math. Biosci., 177&178 (2002), 229-245. doi: 10.1016/S0025-5564(01)00110-9. Google Scholar [10] N. Bruti-Liberati and E. Platen, Monte Carlo simulation for stochastic differential equations, Encyclopedia of Quantitative Finance, 10 (2010), 23-37. doi: 10.1080/14697680902814233. Google Scholar [11] K. S. Chaudhuri and S. Saha Roy, On the combined harvesting of a prey-predator system, J. Biol. Syst., 4 (1996), 376-389. Google Scholar [12] C. W. Clark, Bioeconomic Modelling and Fisheries Management, Wiley, New York, 1985. Google Scholar [13] H. Crauel and M. Gundlach, Stochastic Dynamics, Springer-Verlag, New York, 1999. doi: 10.1007/b97846. Google Scholar [14] T. C. Gard, Stability for multispecies population models in random environments, Nonlinear Anal., 10 (1986), 1411-1419. doi: 10.1016/0362-546X(86)90111-2. Google Scholar [15] K. Gopalsamy, Stability and Oscillations in Delay Differential Equations of Population Dynamics, Kluwer Academic, Dordrecht, 1992. doi: 10.1007/978-94-015-7920-9. Google Scholar [16] D. J. Higham, An algorithmic introduction to numerical simulation of stochastic differential equations, SIAM Rev., 43 (2001), 525-546. doi: 10.1137/S0036144500378302. Google Scholar [17] D. Jiang, C. Ji, X. Li and D. O'Regan, Analysis of autonomous Lotka-Volterra competition systems with random perturbation, J. Math. Anal. Appl., 390 (2012), 582-595. doi: 10.1016/j.jmaa.2011.12.049. Google Scholar [18] D. Jiang and N. Shi, A note on nonautonomous logistic equation with random perturbation, J. Math. Anal. Appl., 303 (2005), 164-172. doi: 10.1016/j.jmaa.2004.08.027. Google Scholar [19] T. K. Kar, Influence of environmental noises on the Gompertz model of two species fishery, Ecological Modelling, 17 (2004), 251-272. Google Scholar [20] Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Academic Press, Boston, 1993. Google Scholar [21] R. Lande, S. Engen and B. E. Saeher, Optimal harvesting of fluctuating populations with a risk of extinction, Am. Nat., 145 (1995), 728-745. Google Scholar [22] W. Li and K. Wang, Optimal harvesting policy for general stochastic Logistic population model, J. Math. Anal. Appl., 368 (2010), 420-428. doi: 10.1016/j.jmaa.2010.04.002. Google Scholar [23] M. Liu and C. Bai, Optimal harvesting of a stochastic logistic model with time delay, J. Nonlinear Sci., 25 (2015), 277-289. doi: 10.1007/s00332-014-9229-2. Google Scholar [24] M. Liu and C. Bai, Optimal harvesting of a stochastic mutualism model with Lévy jumps, Appl. Math. Comput., 276 (2016), 301-309. doi: 10.1016/j.amc.2015.11.089. Google Scholar [25] M. Liu and C. Bai, Analysis of a stochastic tri-trophic food-chain model with harvesting, J. Math. Biol., 73 (2016), 597-625. doi: 10.1007/s00285-016-0970-z. Google Scholar [26] M. Liu and M. Fan, Permanence of stochastic lotka-volterra systems, J. Nonlinear Sci. , (2016). doi: 10.1007/s00332-016-9337-2. Google Scholar [27] M. Liu and M. Fan, Stability in distribution of a three-species stochastic cascade predator-prey system with time delays, IMA J. Appl. Math. , (2016). doi: 10.1093/imamat/hxw057. Google Scholar [28] D. Ludwig and J. M. Varah, Optimal harvesting of a randomly fluctuating resource Ⅱ: Numerical methods and results, SIAM J. Appl. Math., 37 (1979), 185-205. doi: 10.1137/0137012. Google Scholar [29] E. M. Lungu and B. $\emptyset$ksendal, Optimal harvesting from a population in a stochastic crowded environment, Math. Biosci., 145 (1997), 47-75. doi: 10.1016/S0025-5564(97)00029-1. Google Scholar [30] Y. Lv, R. Yuan and Y. Pei, Dynamics in two nonsmooth predator-preymodels with threshold harvesting, Nonlinear Dyn., 74 (2013), 107-132. doi: 10.1007/s11071-013-0952-2. Google Scholar [31] X. Mao and C. Yuan, Stochastic Differential Equations with Markovian Switching, Imperial College Press, London, 2006. doi: 10.1142/p473. Google Scholar [32] R. M. May, Stability and Complexity in Model Ecosystems, Princeton University Press, New York, 2001.Google Scholar [33] M. Mesterton-Gibbons, On the optimal policy for combined harvesting of predator and prey, Nat. Resour. Model. , 3 (1988), 303. Google Scholar [34] D. Pal, G. S. Mahaptra and G. P. Samanta, Optimal harvesting of prey-predator system with interval biological parameters: A bioeconomic model, Math. Biosci., 241 (2013), 181-187. doi: 10.1016/j.mbs.2012.11.007. Google Scholar [35] D. Prato and J. Zabczyk, Ergodicity for Infinite Dimensional Systems, Cambridge University Press, Cambridge, 1996. Google Scholar [36] D. L. Ragogin and G. Brown, Harvest polices and non-market valuation in a predator prey system, J. Environ. Econ. Manag., 12 (1985), 155-168. Google Scholar [37] D. Ryan and F. Hanson, Optimal harvesting of a logistic population in an environment with stochastic jumps, J. Math. Biol., 24 (1986), 259-277. doi: 10.1007/BF00275637. Google Scholar [38] Q. S. Song, R. Stockbridge and C. Zhu, On optimal harvesting problems in random environments, SIAM J. Control Optim., 49 (2011), 859-889. doi: 10.1137/100797333. Google Scholar [39] Y. Zhang and Q. Zhang, Dynamic behavior in a delayed stage-structured population model with stochastic fluctuation and harvesting, Nonlinear Dynam., 66 (2011), 231-245. doi: 10.1007/s11071-010-9923-z. Google Scholar [40] C. Zhu and G. Yin, On hybrid competitive Lotka-Volterra ecosystems, Nonlinear Anal., 71 (2009), e1370-e1379. doi: 10.1016/j.na.2009.01.166. Google Scholar [41] X. Zou, W. Li and K. Wang, Ergodic method on optimal harvesting for a stochastic Gompertz-type diffusion process, Appl. Math. Lett., 26 (2013), 170-174. doi: 10.1016/j.aml.2012.08.006. Google Scholar [42] X. Zou and K. Wang, Optimal harvesting for a Logistic population dynamics driven by a Lévy process, J. Optim. Theory Appl., 161 (2014), 969-979. doi: 10.1007/s10957-013-0451-0. Google Scholar [43] X. Zou and K. Wang, Optimal harvesting for a stochastic regime-switching logistic diffusion system with jumps, Nonlinear Anal. Hybrid Syst., 13 (2014), 32-44. doi: 10.1016/j.nahs.2014.01.001. Google Scholar
Solution of model (26) with parameter values given in Table 1 and initial conditions $x_1(\theta)=0.6+0.1\sin \theta, x_2(\theta)=0.2-0.08\sin \theta$, $-10\leq\theta\leq0.$
Distribution of model (26) with the parameter values given in Table 1 and initial conditions $x_1(\theta)=0.5-0.1\sin \theta, x_2(\theta)=0.2+0.08\sin \theta$, $-10\leq\theta\leq0$. (a) is with $t=100$; (b) is with $t=400$; (c) is with $t=700$; (d) is with $t=1000$.
$E\big[h_1x_1(t)+h_2x_2(t)\big]$ of model (26) with the parameter values given in Table 2 and initial conditions $x_1(\theta)=0.5-0.1\sin \theta, x_2(\theta)=0.2+0.08\sin \theta$, $-10\leq\theta\leq0$. Green line is with $h_1=h_1^\ast=0.3366,~h_2=h_2^\ast=0.2310$, red line is with $h_1=0.45,~h_2=0.1$, and blue line is with $h_1=0.1,~h_2=0.25$.
Parameter values for Fig.1 and Fig.2
 Parameter Value $r_1$ 0.5 $r_2$ 0.4 $h_1$ 0.02 $h_2$ 0.055 $a_{11}$ 0.5 $a_{12}$ 0.2 $a_{21}$ 0.25 $a_{22}$ 0.4 $\tau_1$ 10 $\tau_2$ 8 $\sigma_1$ 0.4 $\sigma_2$ 0.3
 Parameter Value $r_1$ 0.5 $r_2$ 0.4 $h_1$ 0.02 $h_2$ 0.055 $a_{11}$ 0.5 $a_{12}$ 0.2 $a_{21}$ 0.25 $a_{22}$ 0.4 $\tau_1$ 10 $\tau_2$ 8 $\sigma_1$ 0.4 $\sigma_2$ 0.3
Parameter values for Fig.3
 Parameter Value $r_1$ 0.8 $r_2$ 0.5 $a_{11}$ 0.5 $a_{12}$ 0.2 $a_{21}$ 0.1 $a_{22}$ 0.4 $\tau_1$ 10 $\tau_2$ 10 $\sigma_1^2$ 0.2 $\sigma_2^2$ 0.2
 Parameter Value $r_1$ 0.8 $r_2$ 0.5 $a_{11}$ 0.5 $a_{12}$ 0.2 $a_{21}$ 0.1 $a_{22}$ 0.4 $\tau_1$ 10 $\tau_2$ 10 $\sigma_1^2$ 0.2 $\sigma_2^2$ 0.2
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